On a canonical construction of tessellated surfaces from finite groups

Accepted Manuscript On a canonical construction of tessellated surfaces from finite groups

Mark Herman, Jonathan Pakianathan

PII: DOI: Reference:

S0166-8641(17)30291-2 http://dx.doi.org/10.1016/j.topol.2017.05.014 TOPOL 6148

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Topology and its Applications

Received date: Revised date: Accepted date:

28 April 2014 31 May 2017 31 May 2017

Please cite this article in press as: M. Herman, J. Pakianathan, On a canonical construction of tessellated surfaces from finite groups, Topol. Appl. (2017), http://dx.doi.org/10.1016/j.topol.2017.05.014

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On a canonical construction of tessellated surfaces from finite groups. Mark Herman and Jonathan Pakianathan June 5, 2017 Abstract In this paper we study an elementary functorial construction from the category of finite non-abelian groups to the category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a finite collection of compact, connected, oriented smooth surfaces equipped with a closed-cell structure which is face and edge transitive. These tessellated surfaces are best viewed as regular or dual quasiregular maps, i.e., cellular graph embeddings into the surface with a high degree of symmetry. This construction in fact exhibits the noncommutative part of the group’s multiplication table as equivalent to a collection of such maps. It generally results in a large collection of maps per group, for example when the construction is applied to Σ6 it yields 4477 maps of 27 distinct genus. We study the distribution of these maps in various groups. We also show that extensions of groups result in branched coverings between the component surfaces in their decompositions. Finally we exploit functoriality to obtain interesting faithful, orientation preserving actions of subquotients of these groups and their automorphism groups on these surfaces and maps. Keywords: Riemann surface tessellations, regular graph maps, strong symmetric genus. 2010 Mathematics Subject Classification. Primary: ; Secondary: .

The work of the second author was partially supported by NSA Grant H98230-15-1-0319.

Contents 1 Introduction 1.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 The complexes X(G), Y (G) and M (G) 2.1 Closed stars of vertices . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Functoriality and Inn(x, y) . . . . . . . . . . . . . . . . . . . . .

4 6 10

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2.3

The map M (x, y) and its edge and face transitivity . . . . . . . .

3 Examples 3.1 Dihedral groups . . . . . . . . . . . . . . . . . . 3.2 Analysis of group action of Aut(D8 ) on M (D8 ) 3.3 Aut(Inn(x, y)) . . . . . . . . . . . . . . . . . . 3.4 Quaternions . . . . . . . . . . . . . . . . . . . . 3.5 Extraspecial p-groups . . . . . . . . . . . . . . 3.6 Products of dihedral groups . . . . . . . . . . . 4 Extensions of Groups and Branched Covers. 4.1 General Extensions . . . . . . . . . . . . . . . 4.2 Central Extensions . . . . . . . . . . . . . . . 4.3 Example: Extension of Σ3 to Σ4 . . . . . . . 4.4 Example: Central Extension of P SL(2, F3 ) to 5 Application: Group Actions on Surfaces 5.1 Branched covers over the sphere and Belyi’s 5.2 Genus zero actions . . . . . . . . . . . . . . 5.3 Extraspecial groups . . . . . . . . . . . . . . 5.4 Miscellaneous examples . . . . . . . . . . .

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Appendices

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A Basic Structure A.1 Pseudosurfaces . . . . . . . . . . . . . . . . A.2 Structure Theorem for X(G) . . . . . . . . . A.3 Group Theoretic Analysis of Closed Stars of A.4 Miscellaneous facts about the maps M (G) . A.5 Valence Two and Doubling . . . . . . . . . A.6 A Finiteness Theorem . . . . . . . . . . . . B Data for Cell Complexes of Several Groups

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Introduction

There is a long tradition in the area of algebraic combinatorics of associating geometries or simplicial complexes to algebraic objects such as groups as a means of studying them, see for example [B, Bu, Ca, Ch, N, Stan]. In this paper we explore such a construction which takes a finite nonabelian group G and constructs a 2-dimensional simplicial complex X(G) in a canonical manner from it. This complex turns out to have a nice geometric structure as a union of finitely many pseudosurfaces which pairwise intersect in finitely many points. These pseudosurfaces are compact, connected and oriented and hence surfaces with at most singularities due to self-point-intersections, for example a

2

sphere with 3 points identified or a torus with three meridian circles squashed to points. The singularities of X(G) can be resolved in a functorial way to yield an associated complex Y (G) which is a disjoint union of finitely many compact, connected, oriented, triangulated 2-manifolds (we’ll call these Riemann surfaces in this paper for brevity at the expense of slight abuse of notation as we will not be using complex structure at all. There is however a unique complex structure on these surfaces compatible with the construction which is explained towards the end of the paper.) The triangulated Riemann surfaces in Y (G) hence form a natural set of invariants for the group and indeed we shall see that they are equivalent to the noncommutative part of the group’s multiplication table if labels are maintained. Additionally, the Riemann surfaces arising via this construction are equipped with interesting closed-cell structures naturally associated to their triangulations. These are best viewed as graph embeddings into the surface and in this context are called closed-cell maps in the literature. We furthermore show that the maps that arise in our construction are all regular (that is to say the map automorphism group acts transtively on the flags of the map, i.e. the triplets consisting of a vertex in an edge in a face) or dual quasi-regular (the map automorphism group acts transitively on edges and faces but might have two orbits of vertices). These maps generalize the classical Platonic solids which are regular maps into the sphere and there is an extensive literature on them, see [GT], [Bi], [CD], [CJT], [CST], [GW], [G], [JS], [K], [LS], [O], [OPPT], [RSJTW], [S], [Sah], [ST], [STW], [SS], [Tu]. The collection of maps obtained from a given group G in our construction will be denoted M (G). Many famous theorems of mathematics concern maps. Kuratowski’s theorem on planar graphs can be viewed as one determining which graphs admit maps into the sphere. The 4-color theorem establishes an upper bound on the chromatic number of such graphs. Given a finite graph, its cellular embeddings into orientable surfaces correspond to rotation systems on the graph and determining the distribution of genii of surfaces into which a given graph embeds is a problem with a vast literature, see [GT] for basic references. Constructions that yield such regular maps are not new, see [Cox, Cox2, MS] in addition to the references above. There are classical constructions using triangle or Fuchsian groups, hyperbolic geometric tessellations of the Poincare disk and Cayley graphs. There are also branched cover and fundamental group arguments to construct such objects. The construction of this paper provides a direct and elementary connection between regular and dual-quasi regular maps and the basic structure of any non-abelian group. In fact it displays these objects as geometrically equivalent to the non-commutative part of the group’s multiplication table and so provides yet another setting where such maps arise naturally. As the construction is functorial, the action of Aut(G) on G induces a (simplicial) action of Aut(G) on this collection of triangulated surfaces Y (G) and the resulting collection of maps M (G). Thus, one can also use this construction to find faithful actions of subquotients of Aut(G) on various surfaces and hence 3

obtain embeddings of these subquotients into the group of orientation preserving diffeomorphisms of Xg , the surface of genus g. We explore some examples of orientation preserving group actions afforded by this construction in this paper. We discuss some examples pertaining to the strong symmetric genus of the group, i.e., the smallest genus orientable closed surface for which the group acts on faithfully via orientation preserving diffeomorphisms. See Section 5 for details. We attempt to quantify the number and type of surface components that occur in the map decomposition M (G) for a given group G. For example when G is the alternating group on 7 letters, the construction M (G) discussed in this paper results in 16813 surface components of 58 distinct genus, with even more distinct cell-structures. Tables summarizing the various maps that occur in M (G) for various groups G can be found at the end of the paper. Finally we show that extensions of groups yield branched coverings between their constituent surfaces in their decompositions. We also show that all of our surfaces are branched covers over the sphere, ramified over at most 3 points and thus aposteriori have a unique complex structure compatible with the Aut(G) ¯ action and can be defined as projective algebraic varieties over Q.

1.1

Acknowledgement

The authors would like to thank Dinesh Thakur, Tom Tucker and Erg¨ un Yal¸cın for useful discussions concerning the topics in this paper. We would especially like to thank the referee of the first draft of this paper for a very detailed and useful referee report. The work of the second author was partially supported by NSA Grant H98230-15-1-0319.

2

The complexes X(G), Y (G) and M (G)

Let G be a finite nonabelian group and let Z(G) be its center. We will now describe the construction of X(G) in detail - the reader is encouraged to draw pictures to help with the definitions in this section as at first they seem elaborate though with some study, the nice structure of the complex will emerge. Before we construct the simplicial complex X(G), we construct an associated subgraph G1 = (V1 , E1 ). The vertices of the graph G1 are given by the noncentral elements of G, i.e., V1 = G − Z(G) and are labelled as type 1 vertices, thus a typical vertex is labelled (x, 1) where x ∈ G − Z(G). We declare [(x, 1), (y, 1)] to be an edge in the simplicial graph G1 if and only if x and y do not commute in G, i.e., xy = yx. This graph is the 1-skeleton of the simply connected complex BN C(G) considered in [PY] and hence is path connected but we provide a simple proof of this here: Lemma 2.1. G1 = (V1 , E1 ) is a path connected graph with diameter 2 and each vertex (x, 1) ∈ G1 has valence |G| − |C(x)| ≥ |G| 2 .

4

Proof. Let (x, 1), (y, 1) be two vertices in G1 . Then x, y are not central in G so their centralizer subgroups C(x) and C(y) are proper subgroups of G and hence have at most half the elements of G in them. Thus |C(x) ∪ C(y)| < |G| as C(x), C(y) each have at most half the elements of G in them and they have the identity element in common. Thus there must exist z ∈ / C(x) ∪ C(y) which means there is a path of length two from (x, 1) to (y, 1) in G1 through (z, 1) as z does not commute with either x or y. Since (x, 1), (y, 1) were arbitrary vertices of the graph G1 , we conclude that G1 is path connected with diameter 1 or 2. To rule out the diameter 1 case, note that since G is non-abelian there exists a non-central α ∈ G such that α = α−1 , hence (α, 1), (α−1 , 1) do not share an edge in G1 . We now extend the graph G1 to get the simplicial complex X(G). First we include a second set of vertices V2 which also consists of the noncentral elements of G but with label 2, written as (x, 2), for x ∈ G − Z(G). Thus the vertex set of X(G) consists of the disjoint union of V1 and V2 , i.e., two copies of the non central elements of G. To describe a simplicial complex, it is enough to describe its maximal faces. The maximal faces of X(G) are of the form [(x, 1), (y, 1), (xy, 2)] where x and y do not commute; we orient these 2-simplices in the order indicated also. Note that the three vertices x, y, xy of this face pairwise do not commute in G. Notice every edge in the graph G1 lies in two distinct faces of X(G), i.e., [(x, 1), (y, 1), (xy, 2)] and [(y, 1), (x, 1), (yx, 2)]. Also note that the orientations on these two faces cancel along this edge. Furthermore if α is a noncentral element, there must be an element x which does not commute with it by definition. Then y = x−1 α does not commute with x and [(x, 1), (y, 1), (α, 2)] is a face of X(G). Thus every type 2 vertex does lie in a face also. Notice from this argument that for any α, x which do not commute, the edge [(x, 1), (α, 2)] is part of the complex. Furthermore such an edge lies again in exactly two faces [(x, 1), (x−1 α, 1), (α, 2)] and [(αx−1 , 1), (x, 1), (α, 2)] whose orientations cancel along that edge. To summarize, we have defined a 2-dimensional simplicial complex X(G) whose vertex set V is the disjoint union of V1 and V2 where V1 = {(x, 1)|x ∈ G − Z(G)} and V2 = {(x, 2)|x ∈ G − Z(G)} and whose edges are of the form [(x, 1), (y, 1)] and [(x, 1), (y, 2)] where x and y do not commute. Finally the (oriented) faces in the complex are of the form [(x, 1), (y, 1), (xy, 2)] and every edge lies in exactly two faces whose orientations cancel along that edge. Note in particular every face of X(G) shares an edge with the graph G1 considered earlier. Finally note that X(G) is path-connected as any point in X(G) lies in a face and any point in a face can be connected by a straight line path to a point on the graph G1 . As we have seen that the graph G1 is path-connected, it follows that any two points in X(G) are connected by a path. We summarize the properties of X(G) here that will be relevant to the rest of the analysis of this complex in this section. We postpone the proof of the 5

Figure 1: The face [(x, 1), (y, 1), (xy, 2)] and two adjacent faces xy, 2

y1 xy, 1 x, 1

y, 1

yx, 2

functoriality of this construction until later. Proposition 2.2. Let G be a finite nonabelian group and Z(G) be its center. The complex X(G) is a compact, 2-dimensional simplicial complex such that: (1) Every edge lies in exactly two faces. (2) The faces can be oriented so that along any edge, the two orientations of the adjacent faces cancel. (3) It is path-connected, thus in particular every vertex lies in an edge. As X(G) is the union of triangles [(x, 1), (y, 1), (xy, 2)] encoding the noncommuting products of the group G, we will call X(G) the geometric realization of the non-commuting part of the group G’s multiplication table. As long as vertex labels and orientations are maintained, X(G) clearly encodes this information and indeed is equivalent to it. Appendix A.1 shows that any simplicial complex that satisfies the criteria of Proposition 2.2 like X(G) does, is a union of finitely many compact, connected, oriented, pseudosurfaces which pairwise intersect in a finite collection of vertices. As explained in this appendix, these pseudosurfaces are manifolds away from their vertices, i.e., every non-vertex point has an open neighborhood homeomorphic to an open disk. However, at a vertex they may be singular and we analyze the structure of these singularities in the next section.

2.1

Closed stars of vertices

Throughout this section, we will be concerned with the structure of the closed star of vertices in a simplicial complex X which satisfies the conditions of Proposition 2.2. Throughout this section X is such a complex. Definition 2.3 (m-stars). Let m ≥ 3. Let V = {0} ∪ {e2πik/m |0 ≤ k < m, k an integer } be the vertex set containing zero and the mth roots of unity viewed within the complex plane. Consider the collection of edges E obtained by joining 0 to the m, mth roots of unity. This simplicial graph Gm = (V, E) will 6

be called the m-star graph. Note it is star-convex so as a space, it is contractible to its middle point 0. Definition 2.4 (m-disks). Let m ≥ 3. Let V be the vertex set containing 0 and the mth roots of unity viewed within the complex plane. Let Dm be the convex hull of V (i.e. an m-gon) triangulated using its m boundary edges and the m edges joining 0 to the mth roots of unity as edge set. Dm has m faces (2-simplices) consisting of the m triangles cut out by the edges mentioned above. Dm is homeomorphic to the standard closed unit disk in the complex plane but triangulated by this specific triangulation. We will call such a triangulated disk, an m-disk or a disk of type m in this paper. 0 is called the center of this m-disk. The boundary of a m-disk is a triangulated circle which we will call an m-circle. Definition 2.5 ((m1 , m2 , . . . , mk )-disk bouquet). Let m1 , . . . , mk be integers ≥ 3. A simplicial complex T is called a (m1 , m2 , . . . , mk )-disk bouquet if it is simplicially isomorphic to the simplicial complex obtained by taking k disjoint disks of types m1 , . . . , mk respectively and identifying their centers to a common center vertex. The individual disks in a disk bouquet are called the sheets of the bouquet. We have seen neighborhoods of nonvertex points of X look like open disks in R2 , we are now ready to describe neighborhoods of vertices in X, including their simplicial structure. Proposition 2.6 (Closed stars of vertices). Let X be a simplicial complex satisfying the conditions of Proposition 2.2 and let v be a vertex of X. Then there ¯ exist integers k ≥ 1, m1 , . . . , mk ≥ 3 such that the closed star of v, St(v) is a (m1 , m2 , . . . , mk )-disk bouquet. The link of v, Lk(v) is a disjoint union of k circles, of simplicial types mi , 1 ≤ i ≤ k. Proof. The vertex v is contained in a face σ1 . Let E be the edge opposite v in this face. Note that by the structure of simplicial complexes, no other face can contain both v and E or it would have to be equal to σ1 . Also note that if w ¯ is a vertex of E, only at most two faces in St(v) can contain w as they must contain the edge [v, w]. Now take an edge of σ1 that contains v. As every edge lies in exactly two faces, there is a unique face σ2 that shares this edge with σ1 and is not σ1 . Note v is contained in this face also. Repeating the argument we can find a sequence of faces, each containing v of the form σ1 , . . . , σk where σi and σi+1 share an edge for 1 ≤ i ≤ k − 1. Eventually by finiteness and as every edge lies in exactly two faces, we must have σk share an edge with the first face σ1 for some k = m1 . The union of faces σ1 , . . . , σm1 hence forms a m1 -disk ¯ centered at v. If this exhausts all faces in St(v) we are done. If not pick another unused face containing v and proceed to find a m2 disk centered at v using a similar process. Proceed in this way till one has found k disks all centered at v and there are no unused faces containing v. (This must occur eventually as there are finite number of faces in X). Note that as no two faces which have v as a vertex, can intersect in an edge opposite v, and all the edges adjacent to v can only be in two faces which are 7

part of the same disk, it is clear that the disks obtained are disjoint except for the common vertex v. Thus the union of all faces containing v, i.e. the closed star of v, forms a (m1 , m2 , . . . , mk )-disk bouquet. mi ≥ 3 for all i by general structure conditions of simplicial complexes. The statement on links follows immediately from this so we are done. In the complex X(G), one does in general have neighborhoods of vertices which are disk bouquets with more than one sheet. For type 2 vertices this can occur as a given element can be written as a non-commuting product in more than one way i.e. α = xy = uv = ab as in Figure 2 below: Figure 2: A representative (m1 , m2 , m3 )-disk bouquet. y

Α  ab  xy  uv

x

Α , 2 b

a

u v

Corollary 2.7. Let X be a simplicial complex satisfying the conditions of Proposition 2.2 and let Y = X − {Vertex set of X}. Then the components of Y are punctured Riemann surfaces. We may fill in each distinct puncture of Y with a distinct point to obtain a disjoint union of finitely many Riemann surfaces which are in bijective correspondence with the pseudomanifold components of X. Proof. From proposition A.2 we know that Y is an oriented 2-dimensional manifold with finitely many connected components in bijective correspondence with the pseudosurface components of X (see appendix A.1 for definition). Note from Proposition 2.6, we see that there is an open neighborhood of each puncture homeomorphic to a punctured disk. From this it is easy to see that if the punctures are filled in with distinct points, we will obtain components which are connected, oriented 2-manifolds with finite triangulations (and hence compact). Adding these distinct points does not change connected components so we result in a finite disjoint union of Riemann surfaces, which are in bijective correspondence with the pseudomanifold components of X. Definition 2.8. Let G be a finite nonabelian group. Then X(G) satisfies the conditions of Proposition 2.2 and so we may remove its vertices and fill in the 8

Figure 3: A representative simplicial complex X(G).





Figure 4: The resulting desingularization Y (G) of the complex X(G) in figure 3. 















resulting punctures with distinct points. The resulting space will be denoted by Y (G) and is a disjoint union of Riemann surfaces which are in bijective correspondence with the pseudosurface components of X(G). As a Riemann surface is determined up to homeomorphism by its genus g which is a nonnegative integer, the quantity mg (G), the number or Riemann surfaces of genus g that occur in the complex Y (G) is a natural invariant of the group G. Y (G) will be called the desingularization of X(G). During the desingularization, the new added points are given labels identical to the label of the original point that yielded the associated puncture. Thus, the reader is warned that in the desingularization Y (G) it is possible to have multiple vertices with the same label even on the same component. Indeed identifying points with the same label in Y (G) recovers the original singular manifold X(G). However edge and face sets are not changed in going from X(G) to Y (G) and so there is a unique component in Y (G) containing a given edge [(x, 1), (y, 1)] corresponding to a given noncommuting pair of elements x, y of the group. As X(G) was the geometric realization of the noncommutative part of the group G’s multiplication table, its desingularization Y (G) also encodes this information.

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2.2

Functoriality and Inn(x, y)

A group homomorphism f : G → H is said to be injective on commutators if the restriction f : G → H is injective where G is the commutator subgroup of G. It is easy to check that the composition of two homomorphisms which are injective on commutators is also injective on commutators. If f : G → H is such a homomorphism then g1 commutes with g2 if and only if f (g1 ) commutes with f (g2 ) for all g1 , g2 ∈ G. Let C denote the category of finite groups and homomorphisms which are injective on commutators. Let D be category of finite oriented 2-dimensional simplicial complexes and orientation preserving simplicial maps. Here by an oriented 2-dimensional simplicial complex we mean one whose faces (2-simplices) have all been given an orientation and when we say a simplicial map preserves orientation we mean it takes 2-simplices to 2-simplices in a manner that preserves orientation of the individual faces. Proposition 2.9. The construction X(G) is part of a covariant functor from the category C of finite groups and homomorphisms injective on commutators to the category D of finite oriented 2-dimensional simplicial complexes and orientation preserving simplicial maps. In particular, if G1 is isomorphic to G2 , then X(G1 ) is simplicially isomorphic to X(G2 ) and if H ≤ G, then X(H) is a subcomplex of X(G). Proof. We have already described X on the level of objects so let f : G → H be a homomorphism injective on commutators. For vertices (either of type 1 or type 2) define X(f )((v, i)) = (f (v), i) for i = 1, 2. As f takes noncommuting elements to noncommuting elements, it takes edges of X(G) to those of X(H). If [(x, 1), (y, 1), (xy, 2)] is an oriented face of X(G), then [(f (x), 1), (f (y), 1), (f (xy), 2)] is an oriented face of X(H) as f (xy) = f (x)f (y). Thus X(f ) defines an orientation preserving simplicial map between X(G) and X(H). It is now easy to check that X respects compositions and identity maps and defines a covariant functor from C to D as desired. The rest follows readily. Corollary 2.10. For G a finite nonabelian group, Aut(G) acts on X(G) through orientation preserving simplicial automorphisms. In particular, G acts on X(G) simplicially by conjugation. Furthermore, an anti-automorphism of G like θ(g) = g −1 induces an orientation reversing simplicial automorphism of X(G). The covariant functor X induces another functor Y , the desingularization of X. We have already described the desingularization Y (G) on the level of objects. For a homomorphism f : G → H which is injective on commutators, we have already defined an orientation preserving simplicial map X(f ) : X(G) → X(H). As such a map takes faces to faces, it is easy to see that it takes pseudomanifold components to pseudomanifold components and induces a continuous map X(G)−{vertices} → X(H)−{vertices} which takes punctures to punctures. As each puncture arises from a unique sheet of the closed star of a unique vertex, or equivalently from a unique circle of the link of a unique vertex, to see if there is 10

a well-defined continuous extension of X(f ) to a simplicial map Y (G) → Y (H), we need only note that each circle in a link of the vertex (v, i) must map to a unique circle in the link of the vertex (f (v), i) under X(f ). We then define Y (f ) in such a way as to map the puncture associated to a particular circle in the link of (v, i) in X(G) to the puncture associated to the circle in the link (f (v), i) in X(H) which X(f ) takes the first circle to. After doing this, it is easy to check Y (f ) is a well-defined orientation preserving simplicial map from Y (G) to Y (H) and that the construction preserves compositions and identity maps. Thus we have proven: Proposition 2.11. The construction Y (G) is part of a covariant functor Y from C to D. From this it follows that if H is a subgroup of G, then the manifold components of Y (H) are a subset of those of Y (G). Thus mg (H) ≤ mg (G) for every genus g. Thus we have proven: Corollary 2.12 (Monotonicity). Let H ≤ G be finite non-abelian groups. Then mg (H) ≤ mg (G) for every genus g ≥ 0. The next corollary follows from monotonicity as every finite group is a subgroup of a symmetric group Σn . It shows that a particular genus g surface can occur as a component of Y (G) for finite groups if and only if it can occur for symmetric groups. Corollary 2.13. If mg (G) > 0 for some genus g ≥ 0 and finite non-abelian group G, then mg (Σn ) > 0 for some n ≥ 3. We also record the important fact that every component of Y (G) originates from a component of Y (H) where H ≤ G is a non-abelian subgroup generated by 2 elements. Corollary 2.14. Let G be a finite non-abelian group. Then X(G) = ∪H∈A X(H) where A denotes the collection of subgroups of G which are non-abelian and generated by 2-elements. Similar statements hold for Y (G). Proof. Starting from a triangle in X(G), say [(x, 1), (y, 1), (xy, 2)] it is easy to verify by induction that all triangles in the pseudosurface component which contains that triangle have vertices which lie in the subgroup H =< x, y > which is generated by x and y. As x and y don’t commute, H is non-abelian and generated by 2-elements. As X(G) is the union of its pseudosurface components, the corollary is proved. Let N C2 = {(x, y) ∈ G × G|xy = yx} be the set of non-commuting pairs in G. Note as the edges and faces of X(G) are exactly the same as those in Y (G) (only vertex sets changed), for any pair (x, y) ∈ N C2 , there is a unique edge [(x, 1), (y, 1)] in X(G) and hence in Y (G). We will call the unique Riemann surface component of Y (G) containing this edge, Y (x, y). It is clear that different pairs in N C2 might determine the same component, so we introduce an 11

equivalence relation on N C2 declaring (x, y) ∼ (x , y  ) when Y (x, y) = Y (x , y  ). Then we may state:  Y (x, y). Y (G) = (x,y)∈N C2 /∼

In general the action of Aut(G) on Y (G) permutes the surfaces in this decomposition amongst their genus type. Given non-commuting elements x, y ∈ G we define H(x, y) to be the subgroup generated by x, y. Define Inn(x, y) to be the subgroup of Inn(H(x, y)) ⊆ Aut(H(x, y)) generated by cx and cy . Here ca denotes conjugation by a. By functoriality we know that cx and cy induce orientation preserving, simplicial automorphisms of Y (G). Note that the sheet about (xy, 2) with link circle: (x, 1), (y, 1), (y −1 xy, 1), . . . lies in Y (x, y). A simple computation shows that the elements on this link circle consist of the α = xy conjugation orbits of x and y and that the element right before the link circle comes back to (x, 1) is (xyx−1 , 1). As c−1 takes y the edge [(x, 1), (y, 1)] to the edge [(y −1 xy, 1), (y, 1)] that also lies in Y (x, y), it follows that cy must preserve this component. Similarly as cx takes the edge [(x, 1), (y, 1)] to the edge [(x, 1), (xyx−1 , 1)] which also lies in Y (x, y) we have cx preserves this component. It follows that Inn(x, y) acts on the triangulated Riemann surface Y (x, y) via orientation preserving simplicial automorphisms. This action is faithful as any element of Aut(H(x, y)) which fixes the reference edge [(x, 1), (y, 1)] of Y (x, y) pointwise, must fix group elements x and y and hence H(x, y). This will be a crucial observation for the next section.

2.3

The map M (x, y) and its edge and face transitivity

Let G be a finite nonabelian group and (x, y) ∈ N C2 be a non-commuting pair of elements. As in the last section, we denote by Y (x, y) the triangulated Riemann surface determined by this non-commuting pair of elements. We have seen that Inn(x, y) ⊆ Aut(H(x, y)) acts faithfully on Y (x, y) via orientation preserving simplicial automorphisms. We now define a map M (x, y) into the surface Y (x, y) by taking the triangulation of Y (x, y) and suppressing (removing from the graph) all type 2 vertices and edges between type 2 and type 1 vertices. Thus in the map M (x, y) we embed the graph consisting only of the part of the 1-skeleton of Y (x, y) consisting of type 1 vertices and the edges between them into the underlying surface of Y (x, y). The (closed) faces in this cell structure are the sheets about type 2 vertices in Y (x, y). As we have seen that these are triangulated disks, this map M (x, y) is a closed-cell structure map which is stronger than the cellularity required to be a map in general.

12

 Thus as a topological space, Y (G) = (x,y)∈N C2 /∼ M (x, y). Note furthermore if vertex labels and orientations are recorded in M (x, y) then the triangulation of Y (x, y) can be recovered as follows: To recover the label of the type 2 vertex in the middle of a face, take any two adjacent vertices on the boundary of the face and multiply them in the order determined by the orientation. Then a barycentric subdivision of the face with respect to this type 2 vertex recovers the triangulation of that sheet. In this way Y (x, y) and all the type 2 vertex labels can be recovered from the map M (x, y). Thus no information is lost in this process. Thus as we have seen X(G) and Y (G) are both determined by and determine the noncommutative part of the group G’s multiplication table, the same is true for the collection of maps M (x, y) provided we maintain vertex labels and orientation information. We will denote this collection of maps by  M (x, y). M (G) = (x,y)∈N C2 /∼

It is clear that the action of Inn(x, y) on Y (x, y) induces one on the cellcomplex M (x, y). We now establish that this action is always edge and face transitive. Note this means that there are at most two orbits of vertices as any vertex is in the same orbit as one of the two vertices on a representative edge. We will see later that there are examples of this construction which are regular, and there are those who are not, so in general we only have edge and face transitivity of the orientation preserving automorphism group of the map. It is clear the action of Aut(G) cannot provide an automorphism of the map M (x, y) taking x to y when these elements have, say, different order in G. It is possible for the map to still be regular in this situation (we will see examples and discuss this more later) but in general it need not be regular. Theorem 2.15. For any finite nonabelian group G and noncommuting pair (x, y) ∈ N C2 , the action of Inn(x, y) on M (x, y) is face and edge transitive. Proof. We use the triangulation of Y (x, y) to help us describe the action of Inn(x, y) on M (x, y). See figure 5 below for accompanying picture. Let F be the face of M (x, y) containing the triangle [(x, 1), (y, 1), (xy, 2)] and F  the face of M (x, y) containing the triangle [(y, 1), (x, 1), (yx, 2)]. F and F  are adjacent faces in M (x, y). It is easy to check that cx−1 ∈ Inn(x, y) fixes (x, 1) and takes (xy, 2) to (yx, 2) and so provides an automorphism of M (x, y) taking F to F  . A simple induction argument now shows that Inn(x, y) acts face-transitively on M (x, y). Thus in particular, every edge of M (x, y) is in the Inn(x, y)-orbit of an edge of the face F . As cxy ∈ Inn(x, y) acts by fixing (xy, 2) and rotating the sheet about it by two triangle notches, we see that every edge of M (x, y) is in the Inn(x, y) orbit of at least one of the edges e = [(x, 1), (y, 1)] or e = [(y, 1), (y −1 xy, 1)]. As cy−1 rotates edge e to edge e fixing (y, 1) we see that Inn(x, y) indeed acts edge-transitively as claimed.

13

Figure 5: Images of the “reference face” containing the shaded triangle [(x, 1), (y, 1), (α, 2)]. Labels for type 1 vertices are suppressed, for example, (x, 1) is simply denoted by x. Conjugation by α−1 = (xy)−1 rotates the face counterclockwise about (α, 2), i.e., the shaded triangle has image [(xα , 1), (y α , 1), (α, 2)] (our notation is xα = α−1 xα). Conjugation by x−1 rotates the reference face clockwise about (x, 1) into the adjacent face, i.e., the shaded triangle has image [(x, 1), (y x , 1), (αx = yx, 2)]. Similarly, conjugation by y rotates the reference face counterclockwise about (y, 1) into the adjacent face, i.e., the shaded triangle −1 −1 has image [(xy , 1), (y, 1), (αy = yx, 2)].

Α  xy, 2 yΑ yx

x

xΑ  Α 1 xΑ

y

1

yx, 2 1 yx  Αx  Α y

xy

Refer to [STW] for a complete analysis of edge transitive maps. It follows immediately from edge-transitivity that Inn(x, y) is an index at most two subgroup of Aut0 M (x, y), the group of orientation preserving automorphisms of the map M (x, y). As the map M (x, y) is edge and face transitive, all its faces consist of n-gons for the same n and there are either one or two vertex valencies. We will denote its Schl¨ afli symbol as {n, λ} if all faces are n gons and all vertex valencies are λ and we will denote it as {n, λ1 − λ2 } in the case where all faces are n gons and all vertex valencies are either λ1 or λ2 .

3 3.1

Examples Dihedral groups

Let D2n , n ≥ 3 denote the dihedral group of order 2n. It consists of n rotations and n reflections which are elements of order 2. It is generated by two reflections σ1 , σ2 whose product is a rotation of order n. Since a product of an even number of reflections is a rotation and the product of an odd number of reflections is a reflection, each triangle in the triangulation of X(D2n ) has vertices which consist of two reflections and a rotation. Let τ denote a generator of the cyclic subgroup of rotations of order n and σ a fixed reflection, then στ σ = τ −1 . The set of reflections is then {στ k |0 ≤ k < n}. The center of D2n is trivial if n is

14

n

odd and has order two generated by τ 2 if n is even. Let τ k be a noncentral rotation. A simple computation shows that the sheets centered at the vertex (τ k , 2) have rim vertices of the form (στ s , 1), (στ s+k , 1), (στ s+2k , 1), . . . . Thus these sheets form d-gons where d is the order of τ k and there are nd of them corresponding to the cosets of the cyclic group of order d generated by τ k in the cyclic group of n rotations. Each of these sheets fits together with a corresponding sheet of its inverse (τ −k , 2) to make a sphere which is the suspension of a d-gon. In the cell structure of M (D2n ), the corresponding Schl¨afli symbol is {d, 2}, i.e. the map is the embedded d-cycle in the sphere. Let φ(d) be Euler’s Phi function, denoting the number of primitive dth roots of unity or equivalently the number of generators of a cyclic group of order d. Then when n is odd, for every divisor 1 < d|n we have φ(d) 2 pairs of elements of afli symbol {d, 2} as mentioned order d, each pair leading to nd spheres of Schl¨ above. When n is even we only need to exclude the case d = 2 which corresponds to the nontrivial central rotation of order 2. The analysis so far accounts for all 2-sheets centered at rotations. Now a completely similar analysis shows that a sheet about (τ k , 1) is of the form (στ s , 1), (στ s+k , 2), (στ s+2k , 1), (στ s+3k , 2), . . . and hence consists of d type 1 vertices and d type 2 vertices where d is the order of τ 2k .The number of such distinct sheets as s varies is dn and each of these fits together with a paired sheet about (τ −k , 1) to form a sphere which is the suspension of a 2d gon. In the corresponding cell structure in M (D2n ), the 2-cells are 4-gons, the d equatorial vertices have valency 2 and the north and south poles (τ ±k , 1) afli symbol is {4, 2-d } where d is the order of have valency d . Thus the Schl¨ 2k k τ . Now when τ has order d then τ 2k has order d = d when d is odd and d = d2 when d is even. The cell structure {4, 2-d } is the double of that of {2, d } which is denoted D{2, d } - see section A.5 of the appendix for details. These observations can then be put together to get the following theorem: Theorem 3.1. Let D2n , n ≥ 3 denote the dihedral group of order 2n. Then: (1) All components in M (D2n ) are spheres (genus g = 0). spherical components of (2) If n is odd, then for any 1 < d|n there are φ(d)n 2d φ(d)n afli symbol {d, 2} and another 2d spherical components with M (D2n ) with Schl¨ Schl¨ afli symbol {4, 2-d} = D{2, d}. spherical components of (3) If n is even, then for any 3 ≤ d|n there are φ(d)n 2d afli symbol {d, 2}. For every odd such d there is another φ(d)n M (D2n ) with Schl¨ 2d spherical components with symbol {4, 2-d} = D{2, d} and for every even such d spherical components with symbol {4, 2- d2 }. there is another φ(d)n d Corollary 3.2. M (D6 ) = M (Σ3 ) consists of two spheres with cell structure of type {3, 2} and {4, 2-3} = D{2, 3}. M (D8 ) consists of three spheres with cell structure of type {4, 2}.

15

3.2

Analysis of group action of Aut(D8 ) on M (D8 )

Let D8 =< u, v|v 4 = u2 = 1, uvu = v 3 >. Note that v 2 is the unique nontrivial central element and that D8 has 6 noncentral elements. We have seen that Y (D8 ) consists of three octahedra as depicted in Figure 6 below. M (D8 ) consists of 3 maps with Schl¨ afli symbol {4, 2} (embedded 4-cycles in the sphere) as depicted in Figures 7 and 8 below. The 3 spheres in Figure 6 are Y (u, v), Y (uv, v) and Y (u, uv) respectively. Note that Inn(u, v) = Inn(uv, v) = Inn(u, uv) = Inn(D8 ) = C2 × C2 acts on each of these spheres individually, taking it back to itself. For example cu acts on the first octahedra Y (u, v) by rotating it 180◦ about the axis u − uv 2 while cv acts by rotating it 180◦ about the axis v − v 3 . Figure 6: Three octahedra of Y (D8 ). Labels for type 1 vertices are suppressed. For example, in the octahedron on the left u, v, uv 2 , v 3 denote type 1 vertices (u, 1), (v, 1), (uv 2 , 1), (v 3 , 1). uv, 2

uv2 , 2

v3

uv3

uv v

v

uv3

v3 uv2

u

v, 2

uv

u, 2

vu, 2

uv2

u

v3 , 2

Notice that the cell structure {4, 2} has 4 ∗ 2 ∗ 2 = 16 flags and is regular. Thus the orientation preserving automorphism group has order 8 and Inn(u, v) provides a subgroup of index 2. Similar comments hold for Inn(uv, v) and Inn(u, uv). Now as v and v 3 are the unique elements of order 4 in D8 , Aut(D8 ) = D8 must preserve the octahedron Y (u, uv) which has (v, 2) and (v 3 , 2) as its type 2 vertices. Thus we see that Aut(D8 ) = Aut◦ M (u, uv) and that indeed the automorphisms of D8 provide all the orientation preserving automorphisms of this map. On the other hand, Aut(D8 ) does not preserve the other two spheres Y (u, v) and Y (uv, v) - indeed any outer automorphism switches them. Thus to summarize, while Inn(D8 ) provides an index two subgroup of the orientation preserving automorphisms of all 3 maps in M (D8 ), Aut(D8 ) is this orientation preserving automorphism group for only one of them M (u, uv) while instead providing a swap map between the other two. Thus M (D8 ) provides examples of regular maps where Aut(D8 ) provides all of the automorphisms and also ones where it doesn’t. 16

Figure 7: A sphere of M (D8 ). The hollow points are given for reference but are not vertices of the cell structure map since type 2 vertices are removed when constructing M (x, y). See section 2.3. Labels are again suppressed for type 1 vertices. v, 2

uv3 uv2

u uv

v3 , 2

Figure 8: More spheres of M (D8 ). uv, 2

uv2 , 2

v3

v3 uv2

u

uv3

uv

v

v

u, 2

vu, 2

17

This example shows that in general while Inn(x, y) is an index at most two subgroup of Aut◦ (M (x, y)), Aut(G) might be able to provide more automorphisms or not with both cases evenoccurring in the same group. k Note that in general if M (G) = i=1 Miji where the M1 , . . . , Mk are distinct ji maps and Mi indicates that map Mi occurs ji times, then Aut◦ (M (G)) = k ◦ i=1 Aut (Mi )Σji is given as a product of wreath products. Thus even though Aut(G) fails in general to provide all orientation preserving automorphisms of the global collection of maps M (G), Inn(G) does provide an index at most two subgroup of the individual map automorphism groups Aut◦ (Mi ). This is sufficient for most analysis as the global automorphisms of the collection of maps are determined from the individual map automorphism groups via wreath product constructions. In this case, Aut◦ (M (D8 )) = D8  Σ3 = (D8 × D8 × D8 )  Σ3 where D8 is the orientation preserving automorphism group of each individual map with symbol {4, 2}. Thus Aut(D8 ) = D8 while reasonably giving us a large chunk of the orientation preserving automorphisms of the individual three maps, fails to give a significant chunk of the automorphisms of the full collection. In practice this failing is not that important as maps in the collection are often analyzed individually anyway.

3.3

Aut(Inn(x, y))

As we saw in the last example, in general Inn(x, y) is an index at most two subgroup of Aut◦ (M (x, y)), the orientation preserving automorphisms of the corresponding map, and Aut(G) might or might not be able to provide the missing automorphisms. It follows by the work in [STW] and ([ST], Proposition 4.2) that all elements of Aut◦ (M (x, y)) are provided by Aut(Inn(x, y)) and we refer the interested reader to these works for details.

3.4

Quaternions

Let Q8 = {±1, ±i, ±j, ±k} denote the quaternionic group of order 8. One component of Y (Q8 ) is an octahedron with (k, 2), (−k, 2) as north and south pole and with the 4 vertices (i, 1), (j, 1), (−i, 1), (−j, 1) along the equator. There are two more similar components obtained by cyclically permuting the roles of i, j and k. In the corresponding cell structure these three spheres have Schl¨afli symbol {4, 2}. Thus M (D8 ) and M (Q8 ) are isomorphic as cell complexes. It is not hard to check that Y (D8 ) and Y (Q8 ) are isomorphic as simplicial complexes and so are X(D8 ) and X(Q8 ) which consist of three octahedra which pairwise meet in a pair of antipodal vertices. These facts reflect the fact that D8 and Q8 are N C-isomorphic, i.e., there is a bijection between the noncentral elements of both groups which preserve noncommuting products. Due to this, it is not a surprise that the geometric realizations of the noncommutative parts of their group tables are isomorphic.

18

Despite the fact that M (D8 ) = M (Q8 ), the group action of Aut(Q8 ) on M (Q8 ) is quite different than that of Aut(D8 ) on M (D8 ) as we will see now. Inn(Q8 ) = C2 ×C2 acts very similarly on the 3 spheres as Inn(D8 ) = C2 ×C2 did. It provides an index 2 subgroup of each of their orientation preserving automorphism groups as before. However Aut(Q8 ) = Σ4 = (C2 × C2 )  Σ3 acts globally on M (Q8 ) as follows (as direct computations can check): Aut(Q8 ) permutes the three spheres according to its projection onto Σ3 . For each sphere, the transposition of Σ3 that swaps the other two spheres, generates together with C2 × C2 a subgroup isomophic to D8 that preserves the given sphere and so provides the full orientation preserving automorphism group of that individual sphere map. Thus Aut(Q8 ) provides a subgroup isomorphic to D8 as the full automorphism group of each individual map in the collection M (Q8 ) and also provides permutations amongst the maps in the collection M (Q8 ). It still does not provide all orientation preserving automorphisms of the collection (there are |D8  Σ3 | = 83 ∗ 6 of those). Figure 9: Three octahedra of M (Q8 ). Labels are again suppressed for type 1 vertices. k, 2

i, 2

 j, 2

j

k

i

i

i

j

j k

j

3.5

i  i, 2

 k, 2

k

k

 j, 2

Extraspecial p-groups

Let p be an odd prime and Fp be the field of p elements. Consider U3 (p) the group of 3 × 3 upper triangular matrices with entries in Fp and 1’s on the diagonal. Thus ⎡ ⎤ 1 a b U3 (p) = {⎣0 1 c ⎦ | a, b, c ∈ Fp } 0 0 1 It is easy to see that U3 (p) has order p3 and exponent p (any nonidentity element has order p). To see this just note that any matrix in U3 (p) can be written as I + A where A is a strictly upper triangular 3 × 3 matrix and so A is nilpotent with A3 = 0. Then (I + A)p = I follows from the binomial theorem and the 19

  fact that p1 and p2 are congruent to zero modulo p. G = U3 (p) is sometimes 3 called the extra special ⎡ p-group⎤of exponent p⎡ and order ⎤ p . 1 1 0 1 0 0 If one sets x = ⎣0 1 0⎦ and y = ⎣0 1 1⎦ then one can readily 0 0 1 0 0 1 (p) and have commutator [x, y] = xyx−1 y −1 = check that x and y generate U 3 ⎡ ⎤ 1 0 1 ⎣0 1 0⎦ which is central and in fact generates the center of U3 (p) which is a 0 0 1 cyclic group of order p. From this one can easily verify that U3 (p) has presentation < x, y|xp = y p = [x, y]p = [[x, y], x] = [[x, y], y] = 1 >. The abelianization of U3 (p) is an elementary abelian p-group of rank 2 (i.e. vector space of dimension 2 over Fp ) generated by the images x ¯, y¯ of x and y. Thus U3 (p) fits into a central short exact sequence: 1 → C → U3 (p) → E = Fp × Fp → 1 where C is the center of U3 (p) and is cyclic of order p. It is clear from this short exact sequence that C = F rat(G), the Frattini subgroup of G. Corresponding to this extension is a commutator map [·, ·] : E × E → C which takes two elements of E, lifts them to U3 (p) and looks at their commutator which lies in C. This map is readily checked to be well-defined and bilinear, alternating, and nondegenerate (see [BrP]). From this map it follows that any two elements u, v that do not commute in U3 (p) must map to a basis in E under the abelianization and hence generate U3 (p) by properties of Frattini quotients. They clearly satisfy the same presentation that x and y did and hence there must be an automorphism of U3 (p) taking any noncommuting pair of elements to any other noncommuting pair of elements. Finally let us note that if α and β are conjugate in U3 (p) they must map to the same element in the abelianization E. Thus β = αc where c is some element in the center C. In particular conjugate elements commute in U3 (p) and every noncentral element has exactly p elements in its conjugacy class. In fact if x is a non central element, xC is its conjugacy class. We summarize these observations in the next lemma as we will use them in determining the 2-cell structure of the components of M (U3 (p)). Note that the definition of α-conjugacy class is found in definition A.5. Lemma 3.3. Let p be an odd prime and G = U3 (p) be the extraspecial group of order p3 and exponent p. (1) If (u, v), (u , v  ) are pairs of noncommuting elements in G, then there exists an automorphism φ of G such that φ(u) = u and φ(v) = v  . (2) If α, β are conjugate in G, then α and β commute. (3) Every noncentral element α has exactly p elements in its conjugacy class. (4) If α does not commute with x then the α-conjugacy class of x consists of exactly p elements. Proof. Parts (1), (2) and (3) were proved in the paragraph before the lemma. Part (4) follows as the size of an α-conjugacy class must divide the order of α. 20

If α does not commute with x then α is not the identity element and hence has order p and the α-conjugacy class of x must have size > 1 and dividing the order of α. As |α| = p is prime, this size must be p. The next theorem determines the structure of M (U3 (p)) completely. Theorem 3.4. Let p be an odd prime and G = U3 (p), the extra special p-group of order p3 and exponent p. Then in M (G) we have: 2 2 −p) (1) All components are isomorphic as cell-complexes and there are (p −1)(p 2 of them. (2) The cell structure of each component of M (G) is a regular map which Schl¨ afli symbol {2p, p}. (3) These cell structures hence tessellate the Riemann surface of genus g = p(p−3) + 1 with 2p-gons. The face, edge and vertex count of this regular tessela2 tion is given by F = p, E = p2 , V = 2p and vertex valency p. Proof. If T is the component of Y (G) determined by the triangle [(x, 1), (y, 1), (xy, 2)] and T  is the component of Y (G) determined by the triangle [(u, 1), (v, 1), (uv, 2)], then Lemma 3.3 guarantees the existence of a group automorphism and hence simplicial automorphism of Y (G) (and also X(G)) which takes one triangle to the other and hence induces a simplicial isomorphism of the component T with the component T  . This simplicial isomorphism induces a cell-isomorphism between the cell structures of these two components in M (G) also. In the component determined by the triangle [(x, 1), (y, 1), (xy, 2)], all type 2vertices are conjugate to xy. Since the x-conjugacy class of xy must all occur as type 2-vertices by Proposition A.6, there are at least p of these in the component. However xy only has p-conjugates so all conjugates of xy occur. As x and y do not commute, they are not conjugate by Lemma 3.3. Thus the sheets in the closed star of (xy, 2) in that component consist of two distinct xy-conjugacy classes of triangles each of order p and hence forms a 2p-gon in the corresponding cell-structure. Now there is a 2p-gon face in the component for each type2 vertex in that component and these consist of the p conjugates of (xy, 2) with possible multiplicities due to the desingularization process (there can be more than one type-2 vertex labeled with the same group element in a given component because of the desingularization). We wish to show there is no multiplicity of type 2-vertices in a fixed component and hence that there are exactly p of these faces. To do this, by symmetry (we have already shown there are automorphisms which will take any triangle to any other in X(G)), it is enough to show that there is exactly one sheet about a type-2 vertex labelled (xy, 2) in the pseudo-manifold component of X(G) that contains the triangle [(x, 1), (y, 1), (xy, 2)] (before desingularization). In other words there is only one disk in the closed star of (xy, 2) which is a bouquet of disks that lies in that pseudomanifold component. Note that all the type-1 vertices in the given pseudomanifold-component are conjugate to either x or y and hence of the form xz a , yz b where z is a generator of the center and 0 ≤ a, b < p. Thus any triangle in a sheet about a type-2 vertex labeled (xy, 2) in the given 21

pseudo-manifold component of X(G) (before desingularization) has to be of the form [(xz a , 1), (yz b , 1), (xy, 2)] or of the form [(yz b , 1), (y −1 xyz a , 1), (xy, 2)] which forces b ≡ −a mod p. This means there there are at most 2p such triangles and hence exactly one such sheet in that pseudo manifold component. Thus there are no multiply labelled type 2-vertices in a given component in Y (G) and we can conclude that each component of M (G) has cell-structure given by exactly p many 2p-gon faces. By part (5) of Theorem A.10 and Lemma 3.3 we see that the cell structure of each component forms a regular map. By Theorem A.8 the vertex valency is the size of the x-conjugacy class of xy which is p. Thus this regular abstract 3-polytope has Schl¨afli symbol {2p, p}. As F = p, the formulas in part (3) of the same theorem can then be used to yield the stated values of E, V and g. As V = 2p in the cell structure, we see that the p conjugates of (x, 1) and (y, 1) occur without multiplicity in the component (alternatively one can mimic the proof used for type 2-vertices earlier). Thus each component consists exactly of two noncommuting conjugacy classes in U3 (p) and is determined by the unordered pair of these. These pairs in turn are determined uniquely by the two linearly independent vectors in the abelianization E of U3 (p) that they project to. Thus the number of components of Y (U3 (p)) is the same as the number of unordered pairs of noncommuting conjugacy classes which is the same as the number of unordered basis of E, a Fp -vector space of dimension 2. 2 2 −p) This number is easily computed as (p −1)(p . 2 Note when p = 3 in the last theorem, we see that U3 (3) provides a regular tesselation of the torus by 3 hexagons with vertex valancy 3. When p = 5, U3 (5) provides a regular tesselation of the surface of genus 6 by five 10-gons with vertex valancy 5. Corollary 3.5. As one varies the construction X(G) over all finite nonabelian groups, the set of genuses of components that occur is infinite. Proof. Theorem 3.4 shows that the set of genuses obtained when looking at extraspecial groups over all odd primes p is infinite and so the corollary follows.

For G a group, let Aut(G) denote the extended automorphism group which consists of automorphisms and anti-automorphims of the group G under com

position. Aut(G) is a subgroup of Aut(G) of index 2 (hence normal) as any anti-automorphism of G is the composition of the inversion map I(x) = x−1 with an automorphism. Furthermore it is easy to verify that if φ is an automorphism then φ(x−1 )−1 = φ(x) and so φ commutes with I. Thus I gener

ates a central subgroup of order 2 complementary to Aut(G) in Aut(G) and so ∼

Aut(G) = Aut(G) × Z/2Z. The anti-automorphism group of U3 (p) acts transitively on the set of components of M (G). If H is the stabilizer of a component in this action, then H

22

acts transitively on the flags in the cell-structure of this component by Theorem A.10 and its proof. As the group elements represented by the vertices of this component generate the group U3 (p), this action is also faithful.

3.6

Products of dihedral groups

We thank the referee for mentioning this example which shows that all surfaces of genus a multiple of 4 can arise from the construction discussed in this paper. Let m, n ≥ 3 be odd integers. Let σi , τi , i = 1, 2, be the reflection and generating rotation of D2m and D2n respectively. Let x = (τ1 , σ2 ) and y = (σ1 , τ2 ) in D2m × D2n . Then since n, m are odd we have x2 = (τ12 , 1), y 2 = (1, τ22 ), xm = (1, σ2 ), y n = (σ1 , 1) and so x and y generate all of D2m × D2n . As this group has trivial center when m, n ≥ 3 are odd, we conclude that Inn(x, y) = D2m × D2n has order 4mn. Inn(x, y) acts transitively on the edges and faces of M (x, y) by Theorem 2.15. Furthermore as x and y are easily seen to be not conjugate in D2m × D2n , there are two orbits of vertices V1 (orbit of x) and V2 (orbit of y). The centralizer C(x) of x in D2m × D2n is C(τ1 ) × C(σ2 ) which is easily seen to be cyclic generated by x and hence of order 2m, hence |V1 | = 4mn 2m = 2n. Similarly |V2 | = 4mn 2n = 2m. As C(x) ∩ C(y) is the center of the group which is trivial, the number of edges, |E|, is equal to |Inn(x, y)| = 4mn by the edge transitivity of the action of Inn(x, y) on M (x, y). An explicit computation shows that a 2-cell sheet in M (x, y) containing the edge [x, y] (1-labels surpressed) and with central point label (xy, 2) has four vertices on its boundary given by cyclic ordering of x = (τ1 , σ2 ), y = (σ1 , τ2 ), (τ1−1 , σ2 τ22 ), (τ12 σ1 , τ2−1 ). Thus all faces are 4-gons. It follows that 4|F | = 2|E| and so |F | = 2mn. Computing the Euler characteristic we have: 2 − 2g = |V1 | + |V2 | − |E| + |F | = 2n + 2m − 4mn + 2mn and so the genus of M (x, y) is given by g = (m − 1)(n − 1). Thus when m = 3 and n varies over all odd integers ≥ 3 we generate surfaces of genus g for every g which is a positive integer multiple of 4. This gives another infinite family of surfaces occuring from this construction and is another example establishing Corollary 3.5.

4 4.1

Extensions of Groups and Branched Covers. General Extensions

Now we will consider what happens to the surface decompositions of groups under extension of groups: Let π

1→N →Γ→G→1

23

be an extension. Note that if x ˆ, yˆ in Γ do not commute but their images x, y in G commute, then the surface in Y (Γ) corresponding to the edge [(ˆ x, 1), (ˆ y , 1)] has no corresponding surface in Y (G). However what we will see is that if the images in G do not commute, then there is a nice correspondence between the component corresponding to the edge [(ˆ x, 1), (ˆ y , 1)] in Y (Γ) and the component corresponding to the edge [(x, 1), (y, 1)] in Y (G) given by a branched covering. Thus let x, y be a non-commuting pair of elements in G, and let S be the component of M (G) that is determined by the edge [(x, 1), (y, 1)]. Fix one choice of lift x ˆ of x and yˆ of y in Γ. Then the set of all lifts of x to Γ is the set {ˆ xn|n ∈ N } of cardinality |N | with a similar statement for y. Thus there are a total of |N |2 lifts of the edge [(x, 1), (y, 1)] to Y (Γ) consisting of edges of the form [(ˆ xn1 , 1), (ˆ y n2 , 1)] for n1 , n2 ∈ N . Let Tn1 ,n2 be the component determined y n2 , 1)] in Y (G). As n1 , n2 ranges over N , we get |N |2 by the edge [(ˆ xn1 , 1), (ˆ of these components but they need not be distinct components, as more than one lift edge can live in the same component. Let T1 , . . . , Tk be a list of distinct components arising this way where the component Ti contains mi of the lift k edges. Thus i=1 mi = |N |2 . We will now focus on a single lift component T with m lift edges. Picking a lift edge in T , we get a reference triangle [(ˆ xn1 , 1)(ˆ y n2 , 1), (ˆ xn1 yˆn2 , 2)] in T which maps onto the triangle [(x, 1), (y, 1), (xy, 2)] in S under a simplicial map induced by π. A simple induction using the fact that an edge in any component determines two triangles adjacent to the edge via algebraic relations preserved by the quotient homomorphism π, shows that π induces a non-degenerate (faces map to faces) simplicial map from T onto S. Now suppose S consists of n-gon faces in the 2-cell structure. Consider the ngon sheet containing the edge [(x, 1), (y, 1)] in S and a polygonal face lying above it in T containing one of the lifts of the reference edge. Starting at the triangle y n2 , 1), (α = x ˆn1 yˆn2 , 2)] and proceeding around the vertex (α, 2) we [(ˆ xn1 , 1), (ˆ find after moving by n-triangles along the sheet, we arrive at a triangle lying above the triangle [(x, 1), (y, 1), (xy, 2)] which need not be the starting triangle. Like a spiral staircase, we have arrived at a location directly above where we started. However by finiteness, there must be a positive integer such that after moving n -triangles along the sheet we arrive back at the starting triangle. Thus the face type of the component T is n , a multiple of the face type of the component S. Note that this means that on this polygonal face of T , aside from at the midpoint, the map π is an to 1 map. Thus restricted to this face, π induces a to 1 branched cover over the corresponding polygonal face of S, branched only over the midpoint which has only one preimage (ramification ). By face transitivity of the cell structures, this is true for all faces of T for a fixed value of independent of the face. Also note that this face will contain exactly lifts of the reference edge in S as our cell-structures are closed (no self-identifications on rims). However different faces in T covering a given face in S can have intersecting rims - though their interiors are disjoint. Thus if there are t faces covering a given face in S, this number is independent of the face chosen by face transitivity and the map π is a t branched covering map over the interior of a face of S, branched over the midpoint which has only t 24

lifts (ramification index ). To complete the picture we have to deal with what is happening on the rim of these faces. First note that two distinct faces in T lying above a given face in S cannot have rims which intersect along an edge. This is because if we had two adjacent faces in T intersecting say in the edge [(w, 1), (z, 1)] mapping to the same face in S, this means that their midpoints (wz, 2) and (zw, 2) map to the same element in S which means that the images of w and z commute in G. This does not happen as mentioned earlier as the map π : T → S is well-defined and is a non-degenerate simplicial map. Thus any intersections between two of these faces along the rim happens only at vertices. Thus the lifts of the reference edge within each of the t lift faces are all distinct and must be the full set m of lifts of the reference edge in that component, i.e., t = m. We hence can consider the sheet centered about (x, 1) in S under the triangulation structure of Y (G) which consists of 2λ1 triangles, where λ1 is the valency of (x, 1) in the corresponding cell structure (so λ1 2-cells touch (x, 1)). Lifting this sheet to T one again gets a spiral staircase picture which shows that ˆ 1 = x λ1 and π ˆ 1 is the valency of (ˆ x, 1) in T under the cell-structure, then λ if λ restricted to the sheet about (ˆ x, 1) in T is a x to 1 branched cover over the corresponding sheet about (x, 1) in S, branched only over (x, 1) with ramification index x . Similar arguments work over (y, 1) though with potentially a different ramification index y . As our cell-structures have at most two orbits of vertices represented by (x, 1) and (y, 1), we conclude that the map π : T → S is indeed a branched covering map, branched at most over type 1 and type 2 vertices in the Y (G) structure of the component S with at most 3 types of ramification index, , x and y . Though sheets about distinct type 1-vertices do not have to have disjoint interiors in general, the sheets about the lifts of (x, 1) in T must all have disjoint interiors as if they didn’t, there would be an edge of the form [(xn1 , 1), (xn2 , 1)] in T which maps to a commuting pair in S, contradicting the earlier argument that π induces a well-defined map to S under our assumptions. Thus if there are tx sheets in T lying above the sheet about (x, 1) in S, we conclude tx x is the generic π-preimage size. Similar comments hold for y in place of x. Comparing with earlier expressions for the total covering number we have tx x = ty y = t = m is the generic preimage size of the branched cover π : T → S. As there are m lifts of the reference edge of S to T , considering the points above the interior of the edge, we see that the total covering number is equal to m and hence that

x , y and divide m. We summarize what we have shown in the following theorem: π

Theorem 4.1. Let 1 → N → Γ → G → 1 be an extension of finite groups and suppose x, y do not commute in G. Let S be the component determined by [(x, 1), (y, 1)] in M (G). Then the |N |2 -lifts of the edge [(x, 1), (y, 1)] to M (Γ) determine k distinct components T1 , . . . Tk where Tj contains mj of these edge lifts.Then: k (1) j=1 mj = |N |2 . 25

(2) If T is a component containing m lifts of the reference edge then π induces a m-fold branched cover T → S which is branched only over type 1 and 2 vertices in S with at most three ramification indices denoted (over the type 2 vertices),

x and y (over the at most two orbits of type 1 vertices) which are divisors of afli symbol m. Furthermore if the Schl¨ afli symbol of S is {n, λ1 -λ2 } then the Schl¨ of T is {n , λ1 x -λ2 y }. Note that all the ramification that occurs can be deduced from the change in Schl¨ afli symbols as we lift the component S to the component T . The value   of t = m  can be computed from face counts as F = tF where F is the number of faces in T and F is the number of faces in S. Further note that edge and vertex counts are related by E  = mE, V1 = mx V1 , V2 = my V2 .

4.2

Central Extensions

A little more can be said in the case of central extensions π

1 → C → Γ → G → 1. In the case of a component S in G with two orbits of vertices under conjugation, we find the component lifts Tci ,cj , ci , cj ∈ C have the same property. Fix c, c ∈ C. We can define a map Ψc,c on vertices which takes any type 1 vertex in the first orbit and multiplies it by c and any type 1 vertex in the second orbit and multiplies it by c and finally takes any type 2 vertex and multiplies it by cc . Notice this map takes the vertices [(α, 1), (β, 1), (αβ, 2)] to [(αc, 1), (βc , 1), (αβcc , 2)] and hence induces a well-defined simplicial map of the union of the Tci ,cj to itself which we will call the “central twist” Ψc,c . In fact C ×C acts on the union of Tci ,cj via these maps transitively permuting these components amongst themselves. Thus the genus and cell type of each of these lift components is the same and hence each of them contain the same number m of lifts of the reference edge. If there are k distinct lifts then mk = |C|2 . Furthermore when considering the ramification picture over a n-gon 2-cell of S, one finds upon traversing along a lift sheet in T for n triangle steps starting at xyˆ, 2)] [(ˆ x, 1), (ˆ y , 1), (ˆ xyˆ, 2)] one returns to a triangle of the form [(ˆ xc, 1), (ˆ y c−1 , 1), (ˆ where c ∈ C is a “monodromy element”. A simple computation using that movement along the sheet rim is accomplished by conjugations, then finds that the face type in T , which is a n -gon, will have , the ramification index equal to the order of this monodromy element. Thus in particular divides the exponent of the group C. Similar considerations work for x and y . On the other hand if S has a single conjugation orbit of vertices, so can its lifts Tci ,cj and so we can no longer distinguish into two kinds of type 1 vertices. Thus only the central twists of the form Ψc,c are still well-defined. These yield an action of C on the lifts Tci ,cj which is no longer transitive but breaks up into |C| many groupings of components each containing |C| lifts of the reference edge. Within each grouping, the components must have the same genus, cell-structure and branching structure over S. If ki denotes the number of components in one of the groupings and mi the common covering number then ki mi = |C|. The 26

arguments about monodromy still work so and x still divide the exponent of C. We record these observations in the next theorem: Theorem 4.2. Let 1 → C → Γ → G → 1 be a central extension of finite groups. Let S be a component of M (G), then in addition to the results of Theorem 4.1 one has: (1) If S has two orbits of vertices under conjugation then all the components lying above S in M (Γ) have the same genus and cell type and branching structure over S. If there are k distinct such components and each yields a m-fold branched cover over S then mk = |C|2 . The ramification indices , x and y divide the exponent of C. (2) If S has a single orbit of vertices under conjugation then the components lying above S break up into |C| groupings. Within each grouping all components have the same genus, cell-structure and branching structure. If mi and ki denote the covering number and number of components in a given grouping then we have mi ki = |C|. The ramification indices , x still divide the exponent of C.

4.3

Example: Extension of Σ3 to Σ4

Here we present the data in theorem 4.1 for π

1 → N → Σ4 → Σ3 → 1, where N = {(), (12)(34), (13)(24), (14)(23)} ∼ = Z2 × Z2 . From tables 6 and 8 (see appendix), we see there are two (genus 0) components of M (Σ3 ), which we refer to by their face count and Schl¨afli symbol as {3, 2}2 and {4, 2-3}3 (the superscript refers to the face count). We similarly refer to the components of M (Σ4 ). The {3, 2}2 component of M (Σ3 ) is covered by six components of M (Σ4 ): {6, 2-4}8 (,x ,y )=(2,1,2)

{3, 4}8

(,x ,y )=(1,2,2)

{3, 2}2

(,x ,y )=(1,1,1)

{3, 2}2 (four of this type) On the left are the components in M (Σ4 ) which map under π to the component of M (Σ3 ) on the right. The ramification indices ( , x , y ) are those referenced in theorem 4.1 and along with face counts of the components, determine all the relevant data. For example, there are four (1, 1, 1) coverings which are simply one-to-one (no branching), while the (1, 2, 2) covering is branched over all type 27

1 vertices (but not type 2). The face counts tell us, for instance, that for the (2, 1, 2) covering, there are t = 4 faces in the M (Σ4 ) component over each face in the M (Σ3 ) component (since an 8 face component maps to a 2 face component). As expected, there are |N |2 = 16 total edge lifts of each edge in the M (Σ3 ) component. Indeed, the (2, 1, 2) covering accounts for 8 of them, the (1, 2, 2) covering accounts for 4, and the (1, 1, 1) coverings each account for 1. The {4, 2-3}3 component of M (Σ3 ) is covered by seven components of M (Σ4 ): {8, 2-3}6 (,x ,y )=(2,1,1)

{8, 3-4}6 (,x ,y )=(2,1,2)

{4, 2-3}3 (,x ,y )=(1,1,2)

{4, 3-4}12

(,x ,y )=(1,1,1)

{4, 2-3}3 (four of this type) Note that for all other components of M (Σ4 ) there is no corresponding x, 1), (ˆ y , 1)] surface in M (Σ3 ). For these components of M (Σ4 ), each edge [(ˆ “maps to a commuting pair” under π.

4.4

Example: Central Extension of P SL(2, F3 ) to SL(2, F3 )

Here we present the data in theorem 4.2 for the central extension π

1 → C → SL(2, F3 ) → P SL(2, F3 ) → 1. From tables 2 and 7, we see there are five (genus 0) components of M (P SL(2, F3 )), which we refer to by their face count and Schl¨afli symbol as in the previous example. Here, the only branching occurs over the two {3, 3}4 components of M (P SL(2, F3 )) given below:

28

{6, 3}4

{6, 3}4 (,x ,y )=(2,1,1)

{3, 3}4

(,x ,y )=(1,1,1)

(,x ,y )=(2,1,1)

{3, 3}4

{3, 3}4

(,x ,y )=(1,1,1)

(,x ,y )=(1,1,1)

{3, 3}4

(,x ,y )=(1,1,1)

{3, 3}4

{3, 3}4

The remaining three components of M (P SL(2, F3 )) are covered in a one-toone fashion, i.e. ( , x , y ) = (1, 1, 1), by isomorphic components in M (SL(2, F3 )). Note that for all other components of M (SL(2, F3 )) there is no corresponding surface in M (P SL(2, F3 )). For these components of M (SL(2, F3 )), each edge [(ˆ x, 1), (ˆ y , 1)] “maps to a commuting pair” under π.

5

Application: Group Actions on Surfaces

By functoriality, Aut(G) acts on Y (G) (recall this is the desingularized construction under the original triangulation) via orientation preserving simplicial automorphisms. Using standard equivalences between the PL and smooth categories in dimension 2, we can view this action as one through orientation preserving diffeomorphisms on the corresponding smoothened surfaces constituting Y (G). This action also preserves the modified cell structures of these surfaces in M (G). If S is one of the surface components, then Aut(G) permutes this component amongst others. All components in the same Aut(G)-orbit as the component S must have the same genus and cell-structure type. Let us denote the component stabilizer as Aut(G)S , thus Aut(G)S consists of the elements of Aut(G) which map the component S back to itself. The index of Aut(G)S in Aut(G) gives the number of components in the component orbit of S and Aut(G)S acts on the surface S itself via orientation preserving simplicial automorphisms. Suppose S = Y (x, y) is the component of Y (G) containing the reference edge [(x, 1), (y, 1)]. If Autx and Auty denote the Aut(G)-stabilizers of the vertices (x, 1) and (y, 1) respectively then Autx ∩ Auty is the kernel of the action of Aut(G)S on S and hence is a normal subgroup of Aut(G)S . Let QS be defined by the short exact sequence 1 → Autx ∩ Auty → Aut(G)S → QS → 1 then QS is a sub-quotient of Aut(G) and QS acts faithfully and orientation preservingly on the surface S. 29

We will call the group QS the sub-quotient of Aut(G) associated to the component S. The next theorem discusses some important structural properties of this subquotient. Theorem 5.1. Let G be a finite nonabelian group and let S = Y (x, y) be the component of Y (G) containing the reference edge [(x, 1), (y, 1)]. Let Aut(G)S be the Aut(G)-stablizer of the component S and QS the associated sub-quotient. Then the following facts hold regarding the faithful QS -action on S: (1) As the component S is edge transitive, the set-stablizers of edges form a conjugacy class. The set-stabilizer of our reference edge in QS will be denoted by Qe . Thus Qe is the subgroup of elements of QS which take the edge [(x, 1), (y, 1)] back to itself as a set. Then |Qe | ≤ 2. (2) All vertex stabilizers (of type 1 vertices) are conjugate to Qx and/or Qy where Qx and Qy denote the stabilizers of the vertex (x, 1) and (y, 1) respectively and furthermore Qx ∩ Qy = {1}. Let λ1 and λ2 denote the valencies of the vertices (x, 1) and (y, 1) then Qx and Qy are cyclic groups of order λ1 and λ2 generated by cx and cy respectively where ct denotes conjugation by t. (3) As the component S is face transitive in the closed cell-structure, the setstabilizers of faces in the closed cell structure of the component form a conjugacy class. The set-stabilizer of a reference face containing the triangle [(x, 1), (y, 1), (xy, 2)] will be denoted QF . QF is also the vertex stabilizer of the type 2 vertex (xy, 2) in the center of that face. If the faces of S in the cellstructure are n-gons, then QF is a cyclic group of order either n or n2 . (4) The group QS acts on the surface S with cyclic stabilizer groups. The only points which can have nontrivial stabilizer groups are vertices of type 1 or vertices of type 2 or midpoints of edges joining two type 1 vertices. Proof. Proof of (1): A simplicial map which takes the reference edge back to itself as a set either fixes the end vertices, in which case it becomes the trivial element in QS or it flips them. If τ is such an automorphism which takes x to y and vice-versa, then any other element which does the same differs from τ by an element which fixes the component pointwise and hence projects to the same thing as τ does in QS . Thus Qe is cyclic generated by τ¯, the image of τ , and τ¯ clearly has order two. Proof of (2): All type 1 vertices in S are conjugate to a vertex in the reference edge. Qx is exactly the image of Autx in QS and similarly Qy is the image of Auty . As Autx ∩Auty was the kernel of the action, Qx ∩Qy = {1}. Recall we saw that cx , conjugation by x permuted the 2-cell faces indecent to (x, 1) transitively and that there were λ1 2-cell faces in the orbit where λ1 was the valency of x. If γ is an automorphism of the component S which takes (x, 1) to itself, then γ must induce a permutation of the faces incident to (x, 1). Thus the composition of γ with a suitable power of cx will fix the reference edge [(x, 1), (y, 1)] pointwise and hence induce the trivial automorphism of the component. Thus in QS , γ¯ , the image of γ, lies in the cyclic group generated by c¯x . Thus Qx is cyclic and is generated by the image of cx in QS and has order λ1 . Similar arguments work for y in place of x. This proves (2). 30

Proof of (3): Let τ denote an automorphism that fixes the reference face setwise. Then the composition of τ and a suitable power of the conjugation cxy takes this face to itself and the reference edge either to itself or an adjacent edge in the same face. (as there are at most 2 xy-conjugacy orbits of triangles in each cell face). The latter case need only be considered if the reference edge and the adjacent edge are not xy-conjugate. If this latter case exists then there exists an automorphism τ  which fixes the face and takes the reference edge to an adjacent edge. The cyclic group generated by τ  then acts transitively on the edges of the face. Any other automorphism which fixes the face is readily seen to be equal to the composite of a power of τ  and an automorphism which fixes the reference edge pointwise. Thus the image of τ  generates QF and QF is a cyclic group of order n. On the other hand, if there is no automorphism of the component which takes the reference edge to an adjacent edge, then any automorphism that fixes the face setwise is equal to the composite of a power of cxy and an automorphism which acts trivially on the component. In this case, QF is cyclic generated by cxy and has order n2 . Proof of (4): Recall that any element of Aut(G)S fixing an edge of S pointwise, must act trivially on all of S and hence project to the trivial element in QS . Thus in the faithful QS action on S with its 2-cell structure, any nontrivial element that fixes an edge (between two type 1 vertices) setwise must flip the end-vertices. Thus the only points with nontrivial stabilizers on such an edge are the midpoint (stabilizer is at most a cyclic group of order 2) and the two end vertices (cyclic stabilizers of order λ1 and λ2 ). Considering the triangulation structure underlying the 2-cell structure of the component, each triangle is of the form [(a, 1), (b, 1), (ab, 2)] and can only be taken to itself by an automorphism that fixes (ab, 2). Such an automorphism either fixes the edge [(a, 1), (b, 1)] pointwise in which case it induces the trivial automorphism of S or it flips it. However flipping an edge and taking the triangle back to the same triangle violates orientation preservingness. Thus no nontrivial element of QS fixes any triangle as a set and this shows that QF , the face stabilizer of the n-gon face has no nontrivial element fix anything besides the midpoint of the face (ab, 2). Putting this all together, we see that the only points with nontrivial stabilizers in the QS action on S are the type 1 vertices, the midpoints of the 2-cell faces (type 2 vertices) and possibly the midpoints on the edges (between type 1 vertices). Each of these stabilizer groups has been shown to be cyclic. Theorem 5.1 goes through similarly when the group Inn(G) is used in place of Aut(G). Another remark is that the size of a finite group of orientation preserving homeomorphisms of Xg , the closed surface of genus g is bounded in general by 84(g − 1) when g ≥ 2. This upper bound can be achieved for infinitely many g but also cannot be achieved for infinitely many g. A Riemann surface with a finite conformal automorphism group achieving this bound is called a Hurwitz surface. This result was proven first by Hurwitz for conformal maps and then generalized to the case of homeomorphisms in [TT]. For more on these issues and for a list of open questions regarding these surfaces see [F]. Thus for any 31

sub quotient Q associated to a component of genus g ≥ 2 in the construction M (G), we can conclude that |Q| ≤ 84(g − 1). We record this as a corollary: Corollary 5.2 (Hurwitz bound). Let Q be a sub quotient associated to a component of M (G) of genus g ≥ 2. Then |Q| ≤ 84(g − 1). This upper bound is only achieved for genus g where a Hurwitz surface exists.

5.1

Branched covers over the sphere and Belyi’s Theorem

We will next prove that if Q is a sub quotient acting on a component S arising in the construction M (G), then the orbit quotient map S → S/Q is a branched covering over the sphere, branched (or equivalently ramified) over exactly 3 branch points. Theorem 5.3 (Components are branched over the Riemann Sphere). Let G be a finite nonabelian group and let S = M (x, y) be a component occurring in M (G) determined by the edge [(x, 1), (y, 1)]. Let QS be the associated sub-quotient and |E| be the number of edges of S. (Note this counts only edges joining type 1 vertices) Then: (1) The quotient map π : S → S/Q is a branched |Q|-fold covering map, branched over exactly three points. S/Q is homeomorphic to the 2-sphere. (2) In the case that the component consists of two Q-orbits of vertices, then the three branch points in S/Q consist of the image of (x, 1), the image of (y, 1) and the image of the center of faces with ramification indices λ1 , λ2 and n2 respectively (note λ1 = λ2 is possible in this case also). In this case, Q is a subgroup of index ≤ 2 of the orientation preserving automorphisms of the cell structure of S and |Q| = |E|. In the case λ1 = λ2 , Q is the whole group of orientation preserving automorphisms of the cell-structure of S. (3) In the case that the component consists of a single Q-orbit of vertices, then the three branch points in S/Q consist of the image of (x, 1), the image of the midpoints of edges, and the image of the center of faces with ramification indices λ, 2 and n respectively. In this case, Q is the group of orientation preserving automorphisms of the cell structure of S and |Q| = 2|E|. Proof. By Theorem 5.1, outside the finite number of points with nontrivial stabilizer groups, the Q-action is free and the map π : S → S/Q is a covering map. Thus it is clear π is a |Q|-fold branched cover, branched only over the images of orbits with nontrivial stabilizer and S/Q is hence a orientable, compact, connected surface of genus g¯ to be determined. Furthermore, note that there are two Q-orbits of vertices in S (in the cellstructure so we are only discussing “type 1” vertices) if and only if there is no element of Q flipping an edge. This is because if an element of Q flips an edge, as we know the action is edge transitive, it follows it is also vertex transitive. On the other hand, if Q acts vertex transitively, by composing the element of Q that maps x to y with a suitable power of y-conjugations (which permute all the edges incident to (y, 1)), we will arrive at an element of Q which flips the edge [(x, 1), (y, 1)]. 32

The rest of the proof, breaks up into two cases, one where the Q-action on vertices is transitive and one where it has two orbits. We will do the two orbit case and leave the easier one orbit case to the reader as the proof is similar. In the case that the Q-action on vertices has two orbits, Qe = {1} as no element of Q can flip an edge. Thus there are exactly 3 orbits with nontrivial stabilizers, that of (x, 1), that of (y, 1) and that of the midpoint of 2-cell faces, for example (xy, 2). The corresponding ramification indices (size of stabilizer groups) are |Qx | = λ1 , |Qy | = λ2 and |QF | = n2 by Theorem 5.1. As Q acts faithfully and edge transitively, without edge flips, |Q| = |E| and Q can have at most index ≤ 2 in the group of orientation preserving automorphisms of the cell-structure of S. (The only possible nontrivial coset of Q being that by an automorphism flipping an edge. Thus this cannot occur when λ1 = λ2 and in this case Q is the full group of orientation preserving automorphisms of the cell structure of S.) The Riemann-Hurwitz formula applied to the branched cover π : S → S/Q yields:

2 − 2g

= =





|Q| |Q| 2|Q| |Q|(2 − 2¯ g ) − |Q| − − |Q| − − |Q| − λ1 λ2 n

1 1 2 |Q| g + + − 1 − 2¯ λ1 λ2 n

On the other hand using |Q| = |E| in Theorem A.8 yields

1 1 2 2 − 2g = |Q| + + −1 λ1 λ2 n Comparing this with the equation above it we conclude g¯ = 0 i.e. S/Q is a sphere. This completes the proof. Note as a consequence of Theorem 5.3, each component S is a branched cover S → S/Q over the sphere, branched over exactly 3 points. As the sphere has a unique complex structure (that of the Riemann sphere), and its conformal automorphism group is sharply 3-transitive, we can assume those 3 points are 0, 1 and ∞. Then, away from the finite number of points with nontrivial Q-stablizers in S, the complex structure of the sphere will lift to a unique complex structure on S such that the deck transformations in Q act via conformal maps. One of course has to discuss the lifting of the complex structure about the ramified points also but as this is not the focus of this paper, we will not do so here. However this shows that the various Q-actions on the individual components indeed connect with the Fuchsian group constructions obtained via complex analysis classically for the group Q and that one can indeed picture these components as genuine Riemann surfaces being acted on by Q via conformal automorphisms. Furthermore a theorem of Belyi (see [Bel, JSi]), characterizes Riemann surfaces with 3-fold branched covers over the Riemann sphere as exactly the Riemann surfaces that can be defined as projective varieties over 33

¯ Thus we have established that the surfaces in our construction are of these Q. types and hence connected to the corresponding theory of Dessins d’enfants but we will not pursue these directions here. As another consequence, the sub quotient Q associated to a given component X either for the G or Aut(G) action must be a group generated by two elements. This follows from considering the Q-covering over the sphere minus the three branch points which yields a short exact sequence: 1 → π1 (X − ramified points) → π1 (S 2 − 3 points) → Q → 1 and the fact that π1 (S 2 − 3 points) = π1 (R2 − 2 points) is a free group on two generators. Let us now look at some examples of the actions constructed via the functor M (G).

5.2

Genus zero actions

In [TT] the concept of strong symmetric genus of a finite group was introduced. The strong symmetric genus of a finite group G is the minimal genus g ≥ 0 such that G acts faithfully via orientation preserving homeomorphisms on the closed surface of genus g. A complete classification of groups with a given strong symmetric genus has been obtained for genus g ≤ 25 in the papers [MZ1], [MZ3], [FLTX]. There exist a finite, nonzero number of finite groups (up to isomorphism) of any given strong symmetric genus g ≥ 2 (see [MZ2]). Note if Q is one of the sub quotients corresponding to a component S of genus g obtained via Theorem 5.1, then the strong symmetric genus of Q is ≤ g. We will now use our construction to recover all finite groups of strong symmetric genus 0. Though this list is classical and well known, we point out that this construction hands these actions to us without any need for inspiration. The group builds the surface canonically via the functor Y (G), the surface thus constructed comes with labels of all vertices by group elements and we merely have to calculate how the group acts on itself via conjugation to visualize the resulting action on the surface. Thus basically the construction removes any need for special inspiration to find these actions. Example 5.4 (Dihedral Groups). Let D2n be the dihedral group of order 2n where n ≥ 3 and let τ be a generator of the cyclic subgroup of “rotations”. Then as discussed in Section 3.1, there is a sphere component with Schl¨ afli symbol {n, 2} in M (D2n ) where the north and south pole of the sphere are (τ, 2) and (τ −1 , 2) and where all of the n reflections of D2n are laid out along the equator. The upper and lower hemispheres form the two n-gon cells in the corresponding cell structure. The stabilizer of this component under the conjugation action is the whole group D2n and the corresponding sub quotient Q is the quotient of D2n by its center which is D2m where m = n if n is odd and m = n2 if n is even. Picturing this sphere as the canonical sphere of radius 1 centered about the origin in R3 , it is easy to compute that conjugation by τ yields a rotation by 2π m radians about the axis going thru the poles. Conjugation by a reflection σ 34

yields a rotation by π radians about an axis going through the equator of the sphere. The resulting action is a faithful, orientation preserving action of D2m on the sphere which exhibits D2m , m ≥ 2 as a subgroup of SO(3) and shows that all dihedral and cyclic groups (which occur as the rotation subgroups) have strong symmetric genus zero. In the next example, the explicit construction involved in the functor M (G) would construct the necessary example and action explicitly but we will bypass explicit analysis and argue indirectly to just show existence. Example 5.5 (Tetrahedron). Looking at table 7 which lists the components in M (A4 ) we find two components of genus 0 and Schl¨afli symbol {3, 3} which are hence isomorphic as cell complexes to the tetrahedron. A4 acts on these two components by conjugation and the stabilizer of one of them is a subgroup of index ≤ 2 i.e., A4 or a subgroup of order 6. As A4 has no subgroups of order 6, A4 must be the component stabilizer. As the tetrahedron has 4 faces, the quotient Q corresponding to that component, which acts transitively on these faces must have 4||Q| which forces Q = A4 as A4 has no quotients of order exactly 4. Thus Q = A4 is contained in the orientation preserving automorphism group of the tetrahedron. On the other hand, the automorphism group of the tetrahedron embeds into Σ4 by considering the induced action on vertices of the tetrahedron, the orientation preserving automorphism group is hence a subgroup of A4 . Putting these two facts together, we see that A4 is the group of orientation preserving automorphisms of the tetrahedron and this action occurs in the construction M (A4 ). Using a regular tetrahedron centered at the origin, one hence finds A4 displayed as a subgroup of the rotation group SO(3). Before the next examples, we record a simple lemma about the construction M (G) for simple groups G. Lemma 5.6. Let G be a nonabelian simple group and let S be a component of M (G) whose type (genus and cell structure) occurs k times in M (G) where k! < |G|. Then the sub quotient Q associated to the component S is G. If the component is non-equivar then G = Q is the group of orientation preserving automorphisms of the cell-structure of S. Proof. GS is an index ≤ k subgroup of G. Considering the action of G on the left cosets of GS , we either have GS = G or the kernel of the action is a proper normal subgroup of G and hence is trivial. Thus in the latter case, G embeds in Σk which is a contradiction as k! < |G|. Thus G = GS . Again by simplicity and the fact that Q = 1, we now see that Q = G. The rest follows from Theorem 5.3. Example 5.7 (Octahedron and Cube). Looking at table 8, we find a unique component of M (Σ4 ) of genus 0 and Schl¨afli symbol {3, 4} which is isomorphic as a cell complex to the octahedron. As the component is unique and has 12 edges the corresponding sub quotient Q must equal Σ4 as 12||Q| and Σ4 has no normal 35

subgroups of order 2. Thus Σ4 embeds in the group of orientation preserving automorphisms of the octahedron. However the octahedron has 12 edges and |Σ4 | = 2|E|, thus Σ4 is the group of orientation preserving automorphisms of the octahedron and this action occurs in the construction. Using a regular octahedron centered at the origin, one hence finds Σ4 displayed as a subgroup of SO(3). Similarly from the table, there is a unique component in M (Σ4 ) of genus 0 and Schl¨ afli symbol {4, 3} which is isomorphic to the cube. Analogous arguments show that this component exhibits Σ4 as the group of orientation preserving symmetries of the cube (which of course also follows as the cube and octahedron are dual complexes). Example 5.8 (Dodecahedron and Icosahedron). Looking at table 9, we find two components of M (A5 ) of genus 0 and Schl¨afli symbol {5, 3} which are isomorphic to dodecahedra. As A5 is simple, the corresponding sub quotient is Q = A5 by lemma 5.6. As |A5 | = 60 = 2|E| we conclude that A5 is the group of orientation preserving symmetries of the dodecahedron and this action occurs in the construction. As usual, using a regular dodecahedron centered at the origin displays A5 as a subgroup of SO(3). The table also has two components isomorphic to icosahedra and analogous arguments show A5 is the group of orientation preserving symmetries of the icosahedron and that this action arises in M (A5 ). The preceding examples display cyclic groups, dihedral groups D2m (m ≥ 2), A4 , Σ4 and A5 as finite groups with strong symmetric genus 0. It is known that this is the complete list, in particular all finite subgroups of SO(3) are isomorphic to one of these groups.

5.3

Extraspecial groups

Let p be an odd prime and G be the extra special group of order p3 and exponent p. We saw in Section 3.5 that all the components in M (G) are isomorphic and + 1, Schl¨ afli symbol {2p, p} and p2 edges. Let are regular, of genus g = p(p−3) 2 S be one of these components, it follows from the previous analysis that the component stabilizer under the G conjugation action is all of G itself and the kernel of the resulting action is Z, the center of G. Thus the resulting sub ¯ = G/Z ∼ quotient Q = Cp × Cp acts faithfully on S where Cp denotes the cyclic group of order p. This is not too interesting as the strong symmetric genus of ¯ is actually 1, thus aside from the case p = 3 the elementary abelian group Q this does not display an example of an action achieving the strong symmetric genus. If we use the action of Aut(G) on S instead, it is not hard to show that the resulting sub quotient Q acting faithfully via orientation preserving homeomorphisms on S is Q = (Cp × Cp )  C2 where the action of C2 on Cp × Cp in the semi direct product is by coordinate exchange. As |Q| = 2p2 = 2|E|,

36

Q is the group of orientation preserving automorphisms of the corresponding regular cell-structure on S. It follows that the strong symmetric genus p(p−3) of (Cp × Cp )  C2 ∼ = Cp × D2p is no greater than 2 + 1 for all odd primes p. By comparison with the lists in [FLTX] we find that this is the strong symmetric genus for p = 3, 5, 7. We do not know if one also has equality for p ≥ 11.

5.4

Miscellaneous examples

By table 5, the simple group P SL(2, F7 ) which is isomorphic to P SL(3, F2 ) has two components in M which are surfaces of genus 3, Schl¨afli symbol {14, 2-3} = D{7, 3} and 168 edges. By simplicity, and lemma 5.6, P SL(2, F7 ) must equal the corresponding sub quotient Q. Also as the cell-structure is not equivar we see that P SL(2, F7 ) is the group of orientation preserving automorphisms of the cell structure. It also follows that it has strong symmetric genus no more than 3. A quick consultation of the lists in [FLTX] shows that the strong symmetric genus of P SL(2, F7 ) is indeed 3. Furthermore, as 84(g − 1) = 168 = |P SL(2, F7 )| this is an example which achieves the Hurwitz bound. Indeed the Riemann surface given by the Klein quartic is a Hurwitz surface of genus 3 with automorphism group P SL(2, F7 ) and is the smallest genus Hurwitz surface. Similar arguments also work for the genus 3 components with cell-structure {6, 2-7} = D{3, 7} and {4, 3-7} in the same table exhibiting P SL(2, F7 ) as the orientation preserving automorphism group of at least three non-isomorphic cell structures on the surface of genus 3. In table 13, the simple group A7 has 4 components of genus 136, Schl¨afli symbol {14, 2-4} = D{7, 4} and 2520 edges. Under the conjugation action, by ¯ for one of simplicity and lemma 5.6, A7 is equal to the associated sub quotient Q these components. Furthermore as we are in the non-equivar case, we see that A7 is the group of orientation preserving automorphisms of the corresponding cell structure on the surface of genus 136. In particular the strong symmetric genus of A7 is no more than 136. Consulting the work of Conder ([C]) who determined the strong symmetric genus of symmetric and alternating groups, we see that 136 is indeed the strong symmetric genus of A7 . Similar arguments hold also for the cell structures with Schl¨afli symbol {8, 2-7} = D{4, 7} and {4, 4-7} in the same table.

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Dept. of Mathematics University of Rochester, Rochester, NY 14627 U.S.A. E-mail address: [email protected]

Dept. of Mathematics University of Rochester, Rochester, NY 14627 U.S.A. E-mail address: [email protected]

40

Appendices A A.1

Basic Structure Pseudosurfaces

Recall that an n-dimensional pseudomanifold is a n-dimensional simplicial complex X which satisfies: (1) X is the union of its n-simplices. (2) Every n − 1 simplex in X lies in exactly two n-simplices. (3) For any two n-simplices σ0 , σk in X there is a sequence of n-simplices σ1 , . . . , σk−1 such that σi and σi+1 share a common (n−1)-face for 0 ≤ i ≤ k−1. Furthermore a pseudomanifold is orientable if one can orient all the n-simplices in such a way that for any n − 1 simplex τ in X, the two orientations from the two adjacent n-simplices cancel along τ . Note that conditions (1) and (3) imply that all pseudomanifolds are pathconnected. It is well known that every path connected, triangulated topological nmanifold is a n-dimensional pseudomanifold. However pseudomanifolds also allow certain types of singularities which we will talk about below. A 2-dimensional pseudomanifold will be called a pseudosurface. Proposition A.1. Let X be a simplicial complex satisfying the conditions of Proposition 2.2 such as X(G). Define an equivalence relation on the 2-faces of X by saying σ0 and σk are equivalent if there is a sequence of 2-faces σ1 , . . . , σk−1 such that σi and σi+1 share a common edge for 0 ≤ i ≤ k − 1. The union of faces in an equivalence class then gives an oriented compact pseudosurface. Furthermore X has finitely many equivalence classes and so is a union of finitely many pseudosurfaces any two of which can meet only in a finite set of vertices. Proof. As every vertex of X lies in an edge and every edge lies in a face (2simplex), it is clear that X is the union of its 2-simplices. The relation described on these faces is easily checked to be an equivalence relation and the equivalence classes partition the 2-simplices of X. As every edge of X lies in exactly two faces and faces can be oriented so that orientations cancel along the edges in common, it is clear that the union of the 2-simplices in one of the equivalence classes forms an oriented pseudosurface. It consists of finitely many simplices and is compact as X is. Thus X is a union of finitely many compact, oriented, pseudosurfaces as claimed. If Y and Z are two of these, the intersection Y ∩ Z is a subcomplex. If this complex contained any edge, it would bound two faces, one of which has to be in Y and one which has to be in Z, and these faces would be equivalent contradicting the construction of the pseudosurfaces Y and Z as unions from distinct equivalence classes of faces. Thus the intersection contains no edges and hence no faces and so consists of at most a finite collection of vertices as claimed. 41

Note from the proof above, it is clear that the pseudomanifolds whose union is X(G) can be algorithmically determined and hence are unique. We will call them the pseudosurface components of X(G). The following three examples are examples of simplicial complexes satisfying the hypothesis of Proposition 2.2 and give a good indication of the type of spaces that can be X(G): (1) A compact, connected, oriented, triangulated 2-manifold (which we will call a Riemann surface for brevity). Here there is a single pseudomanifold component which happens to be a manifold. (2) Any compact, oriented pseudosurface. For example a sphere with 3 points identified to a single point. This can be triangulated so that it is a compact, oriented pseudosurface and is a typical example of such. Every compact, oriented 2-dimensional pseudomanifold can be obtained from a triangulated Riemann surface by a finite number of vertex identifications. We will see this soon as a byproduct of other analysis. Thus basically in dimension 2 an oriented compact pseudomanifold is nearly an oriented compact connected manifold aside from some point self-intersections. (3) Take a torus and squash three meridian circles to three distinct points. This space can be triangulated so that it satisfies the hypothesis of Proposition 2.2. It has three pseudomanifold components which are spheres and any two of these pseudomanifold components intersect in a single vertex but no point lies in all three pseudomanifold components. Let X be any simplicial complex satisfying the conditions of Proposition 2.2. Notice that if x is a point in the interior of a face (2-simplex) then it is clear it has an open neighborhood homeomorphic to the unit open ball of R2 . If x is a point on an edge other than its two end points, then as every edge lies in exactly two faces, it is again clear that x has an open neighborhood homeomorphic to the unit open ball of R2 . Thus the only points of X that might not have open neighborhoods homeomorphic to open subsets of R2 are the vertices. In particular X−{Vertex set of X} is a 2-dimensional oriented topological manifold whose connected components, we will see later, are punctured Riemann surfaces. The next proposition shows that the connected components of X −{ vertex set of X} are in natural bijective correspondence with the pseudomanifold components of X. Proposition A.2. Let X be a simplicial complex satisfying the conditions of Proposition 2.2 and let Y = X − {Vertex set of X} be given the subspace topology. Then Y is a 2-dimensional oriented topological manifold and the connected components of Y are in natural bijective correspondence with the pseudosurface components of X. More precisely y1 , y2 ∈ Y lie in the same component of Y if and only if they lie in the same pseudosurface component of X. Proof. It is clear from the preceding paragraph that Y is an oriented 2-dimensional topological manifold. Fix y1 , y2 in Y . As X was the union of its pseudosurface components X1 , . . . Xm which pairwise intersected in at most finitely many vertex points, we see that Y is the disjoint union of X1 − V, . . . , Xm − V where 42

V is the vertex set of X. As each Xi is closed in X, each Xi − V is closed in Y = X − V and hence also open as there are a finite number of pseudosurface components. Thus if y1 , y2 lie in distinct pseudosurface components of X they must lie in distinct connected components of Y . Conversely, suppose that y1 , y2 lie in the same pseudosurface component Xj of X. Then there must be a face σ0 and a face σk both in Xj joined by a sequence of faces σ1 , . . . , σk−1 in Xj such that y1 ∈ σ0 and y2 ∈ σk and such that σi , σi+1 share a common edge for 0 ≤ i ≤ k − 1. As y1 , y2 are not vertices, it is clear we can construct a path along this sequence of faces which avoids vertices and joins y1 , y2 . Thus y1 , y2 lie in the same path component and hence component of Y . This concludes the proof.

A.2

Structure Theorem for X(G)

In this appendix section, we will prove a basic structure theorem for the complex X(G). Recall this is the initial complex of the construction, prior to desingularization. Theorem A.3. Let G be a finite nonabelian group. Let mg (G) denote the number of surfaces of genus g that occur in Y (G), the desingularization of X(G). Then X(G) is homotopy equivalent to a wedge product (bouquet) of Riemann surfaces and a finite number of circles where the surface of genus g occurs mg (G) times in the wedge product. In other words,   X(G)  ( Xg[mg ] ) (S 1 ∨ · · · ∨ S 1 ) g≥0 [m ]

where Xg g is the mg = mg (G)-fold bouquet of the Riemann surface Xg of genus g with itself. Proof. Let Z(G) be the simplicial complex obtained as follows: First take out all the vertices from X(G) resulting in a punctured oriented topological 2-manifold. Label each puncture point by the vertex in X(G) it resulted from (note if the closed star of v in X(G) is a bouquet of k disks, there will be k punctures with label v). Now for each vertex v in X(G) which lies in k sheets, i.e. whose closed star is a bouquet of k disks, take a disjoint k-star graph and identify its ends with the k points in the desingularization Y (G) filling in the punctures labelled by v. The result is a path connected space consisting of the Riemann surfaces in Y (G) connected by a finite number of star graphs (any two of which are disjoint from each other), one for each vertex in X(G). This is the space Z(G). Now note that if one collapses all the star graphs in Z(G) to their central points, one obtains a homotopy equivalence Z(G)  X(G). Now Z(G) consists of the finite set of Riemann surfaces X1 , . . . , Xm of Y (G) connected together with various star graphs. Note that any edge in the star graph can be thought of as obtained by adjunction of the unit interval via a gluing map on its two boundary points. By basic facts on adjunction spaces, we can move the point of attachment of the interval along a continuous path without changing the 43

homotopy type of the whole space. Thus if one has a k-star graph in Z(G) with k ≥ 3, note that two of the edges form an interval whose end points lie on the Riemann surfaces. Thus using adjunction deformations, we can move the central point attachment of the other k − 2 edges so that they attach to Riemann surfaces in Y (G) on both ends. Thus up to homotopy equivalence, Z(G) and hence X(G) is homotopy equivalent to Y (G) with a finite number of edges attached where both end points lie in Y (G) and whose interiors are disjoint. Now fix a basepoint xi in each Riemann surface Xi of Y (G). Using adjunction space deformations, we can up to homotopy equivalence assume all the edges have endpoints in the set {x1 , . . . , xm } as the Riemann surfaces Xi are path connected. At this stage as X(G) is connected, the union of these edges forms a connected graph with vertices {x1 , . . . , xm }. Collapsing a spanning tree of this graph (which is contractible) one obtains a final homotopy equivalence. Notice that under this collapse, the points x1 , . . . , xm become a common bouquet point to which the Riemann surfaces X1 , . . . Xm from the desingularization Y (G) are attached. Any edges of the graph not in the spanning tree of the graph become circles in this bouquet. The theorem is hence proven. Corollary A.4. The integral homology groups of X(G) are free abelian groups of finite rank. If βi denotes the ith Betti number, i.e., the rank of Hi (X(G); Z) then  β0 = 1, β2 = g≥0 mg (G) and β1 =



2gmg (G) + L

g≥0

where mg (G) denotes the number of times the surface of genus g occurs in the desingularization Y (G) of X(G) and L denotes the number of circles occurring in the homotopy decomposition of Theorem A.3. Proof. Follows immediately from Theorem A.3 and the known homology of Riemann surfaces.

A.3

Group Theoretic Analysis of Closed Stars of X(G)

Definition A.5. Let G be a finite group and α ∈ G. The cyclic subgroup Cα =< α > generated by α acts on G by conjugation. The orbits are called α-conjugacy classes and have size dividing the order of α. Two elements of the group are said to be α-conjugate if they lie in the same α-conjugacy class. An element {x} forms an α-conjugacy class of size 1 if and only if x commutes with α if and only if x ∈ C(α), the centralizer group of α. We will denote the conjugate of x by α−1 i.e., α−1 xα by xα . Let us study the closed star of a type 2 vertex v = (α, 2) in X(G). A face in ¯ St(v) is of the form σ1 = [(x, 1), (y, 1), (α, 2)] where x, y do not commute and 44

α = xy. Note α does not commute with either x or y. Now let us consider the adjacent face σ2 = [(y, 1), (z, 1), (α, 2)]. Then yz = α and so yz = xy i.e. z = y −1 xy = y −1 x−1 xxy = α−1 xα = xα . Thus the adjacent face is σ2 = [(y, 1), (xα , 1), (α, 2)]. Now taking the equation α = xy corresponding to the first face σ1 and conjugating it by α−1 we see that α = xα y α . Thus the face σ3 other than σ1 which is adjacent to σ2 which contains the vertex (α, 2) is σ3 = [(xα , 1), (y α , 1), (α, 2)]. Note that σ3 was obtained from σ1 by conjugating its 3 vertices by α−1 . We now have made an important observation. In a given sheet of the closed star of a type 2 vertex, given one triangle in the sheet, the triangle two away in the same sheet is obtained by conjugating all vertices by α−1 . Thus the entire sheet is made by α-conjugacy applied to the two initial triangles [(x, 1), (y, 1), (α, 2)] and [(y, 1), (xα , 1), (α, 2)]. ¯ Figure 10: A representative sheet in St(v) for v = (α, 2).

Α  xy, 2 yΑ , 1

xΑ , 1

x, 1 y, 1

There are two possibilities: Either the α-conjugacy orbits of these two triangles are distinct, the same size, and the sheet is made from two α-conjugacy orbits of triangles or these two triangles are α-conjugate and the sheet consists of a single α-conjugacy orbit of triangles. So sheets consist of either L or 2L triangles where L ≥ 2 divides the order of α. Furthermore, the number of triangles in the sheet is odd when there is one α-conjugacy orbit and even when there are two. This follows because the action of α-conjugacy always moves one 2 steps along the rim of a n-triangle sheet, thus there are two or one α-conjugacy classes in a sheet depending on whether the subgroup generated by 2 has index two or one in the cyclic group of order n. This is exactly determined by when n is even or odd respectively. Now we consider the closed star of vertices of type 1. Let w = (α, 1) be a type 1 vertex of X(G). Let [(x, 1), (α, 1), (xα, 2)] be a face containing w. [(α, 1), (xα , 1), (xα, 2)] and [(α, 1), (x, 1), (αx, 2)] are the adjacent faces containing w also. As (αx)α = xα, it is easy to check that the 2-labelled vertices in the part of Lk(w) in this sheet consists of the α-conjugacy orbit of αx. It also follows that the 1-labelled vertices in the part of Lk(w) in this sheet consists of the α-conjugacy orbit of x. The α-conjugacy orbit of the 1-vertices in the 45

link for this sheet has the same size as the α-conjugacy orbit of the 2-vertices in the link as the edges along the link of w alternate between type 1 and type 2 vertices. Furthermore the α-conjugacy class of the 2-vertices is uniquely determined from the one for the 1-vertices by the fact that there exist u in the type 2 vertex orbit and v in the type 1 vertex orbit whose “difference” v −1 u is α. We have thus proven: Proposition A.6 (Closed star of type 1 vertices). Let G be a nonabelian group ¯ and let w = (α, 1) be a type 1 vertex in X(G). Then each sheet of St(w) has 2 triangles where ≥ 2 divides the order of α. The vertices along Lk(w) in any sheet alternate between type 1 and type 2 vertices and there is a single α-conjugacy orbit of type 1 vertices and a single α-conjugacy orbit of type 2 vertices in the link of any given sheet. These orbits have the same size. One orbit determines the other by the fact the type 2 orbit contains u and the type 1 orbit contains v such that v −1 u = α. As each sheet contains one α-conjugacy class of size > 1 each of type 1 and type 2 vertices, the total number of sheets ¯ in St(w) is the total number of α-conjugacy classes of size > 1.

A.4

Miscellaneous facts about the maps M (G)

Corollary A.7. Let G be a finite nonabelian group then the number of components of Y (G) is greater or equal to the number of conjugacy classes of noncentral elements in G. Proof. There is at most one conjugacy class represented by the type 2 vertices in a given component. As there has to be at least one type 2 vertex for each noncentral element of G, the corollary follows. (Note the desingularization process can cause there to be more than one type 2-vertex corresponding to a given noncentral element of G. Also examples show a conjugacy class can be spread out over more than one component. Both these factors cause the inequality just proved to often not be equality.) Recall the modified cell structure of components denoted M (G) where the vertices are only the type 1 vertices, the edges only those joining two type 1 vertices and 2-faces being n-gons which were sheets about type 2 vertices in Y (G). These n-gons will carry an implicit label given by the original middle element of the corresponding sheet. Theorem 2.15 shows that a given component with this cell structure has face and edge-transitive cell automorphism group (bijections of vertices which carry edges to edges and boundaries of 2-faces to boundaries of 2-faces) and either one or two vertex orbits. The next theorem collects important formulas for the quantities in this component. Theorem A.8. Let G be a finite nonabelian group and x, y be non-commuting elements in G. Recall M (x, y) is the unique component of M (G) corresponding to the triangle [(x, 1), (y, 1), (xy, 2)]. Then M (x, y) is a compact, connected, oriented 2-manifold of genus g with cell structure consisting of 2-faces which

46

are all n-gons for some fixed n ≥ 3. The cell automorphism group always acts face and (unoriented) edge transitively on M (x, y). (1) If x and y lie in different α-conjugacy classes (n even case), then n = 2(size of α − conjugacy class of x) and the automorphism group acts with at most 2-orbits of vertices represented by (x, 1) and (y, 1) respectively. The valency of (x, 1) in M (x, y) with the 2-cell structure is given by λ1 ≥ 2, the size of the x-conjugacy class of α = xy. The valency of (y, 1) in M (x, y) in this modified 2cell structure is λ2 ≥ 2, the size of the y-conjugacy class of α = xy. In the case 2 V2 ≥ 2 to be the average valency of the component λ1 = λ2 we define λ = λ1 VV11 +λ +V2 where Vi denotes the number of vertices of valency λi in the component. In this case each edge in the component joins a vertex of valency λ1 with a vertex of valency λ2 and we also have E = λi Vi for i = 1, 2. Finally the average valency can be computed as either λ = 2E V or as the harmonic average of the valencies λ1 , λ2 , i.e., 2 1 1 = + . λ λ1 λ2 (2) If x and y lie in the same α-conjugacy class (n odd case), then n = size of α − conjugacy class of x and the automorphism group acts transitively on edges and vertices also. The common valency λ ≥ 2 of all vertices is given by the size of the x-conjugacy class of α. (3) If V, E, F denote the number of vertices, edges and 2-faces of the 2-cell structure of the component M (x, y) and the face type of the component is ngons, with average vertex valency λ then the following equations hold: nF = 2E nF = λV 2 − 2g = V − E + F = 2(

1 1 1 + − )E λ n 2

Proof. The formulas for n follow from results in previous sections concerning the number of triangles in a sheet about a type 2-vertex. The formulas for valency follow once one notes that the sheet centered about (xy, 2) and the one about (yx, 2) containing (x, 1) are adjacent (share an edge) and are conjugate to each other by conjugation by x−1 . Repeating this observation, one finds that the 2-faces containing (x, 1) in M (x, y) consist of those centered at the x-conjugacy orbit of (α = xy, 2). Thus the number of faces containing (x, 1) which is the same as the number of edges incident to (x, 1) in the 2-cell structure is given by the size of this x-conjugacy class. Similar arguments work for the vertex (y, 1). As the cell automorphism group of the component is edge transitive, every edge joins a vertex conjugate to (x, 1) to a vertex conjugate to (y, 1) where the edge [(x, 1), (y, 1)] is the original one determining the component. Thus each edge joins a vertex of valency λ1 to one of valency λ2 . When λ1 = λ2 note that each edge contributes a total of one to the valency count of vertices 47

of valence λ1 . As λ1 of these edges are incident on a given such vertex we find E = λ1 V1 . Similarly E = λ2 V2 . Note also that V = V1 + V2 . Now it follows that 2E = λ1 V1 + λ2 V2 = λV by definition so λ = 2E V . Now note: 1 V2 V 2 1 V1 + = = + = λ1 λ2 E E E λ and so λ is the harmonic average of λ1 and λ2 . Finally the first formula in (3) follows from the observation that each polygonal face contributes n edges to the component and each edge lies in exactly 2 such faces. We will prove the second formula in the harder case of two types of vertex valencies - the other case follows similarly. First recall that each edge will contribute two vertices, one of which is in the orbit of (x, 1) and the other in the orbit of (y, 1). Now each face contributes n vertices with an equal number of valency λ1 as with valency λ2 . Again taking account that a vertex of valency λi lies in λi such faces we get nF 2 = λi Vi . Adding these equations for i = 1, 2 yields nF = λ1 V1 + λ2 V2 = λV and so the second formula is proven. Finally the Euler characteristic of a Riemann surface of genus g is 2 − 2g and equals the alternating sum V − E + F in any cell decomposition. Thus the final formula follows upon plugging in the previous formulas. It follows from face and edge transitivity that many of the quantities in the last theorem have stringent divisibility conditions. The next proposition records these: Proposition A.9. Let G be a finite nonabelian group and let T be a component of M (G) with the 2-cell structure discussed in Theorem A.8 consisting of n-gon faces and with vertex, edge and face counts V, E, F respectively. Then: (1) E ≥ 3 and F ≥ 3 are divisors of |G|. (2) In the case of two distinct vertex orbits V1 , V2 , λ1 , λ2 ≥ 2 are divisors of E and hence of |G|. Furthermore n ≥ 4 is even and divides 2E and hence 2|G|. (3) In the case of one vertex orbit λ ≥ 2, V, n ≥ 3 are divisors of 2E and hence of 2|G|. (4) If either E or F is equal to |G| then the component T must be invariant under conjugation by G and G must have trivial center. In fact if the edge [(x, 1), (y, 1)] lies in the component then C(x) ∩ C(y) = {1}. (5) There are finitely many possibilities for all the data of the component (g, Vi , E, F, λi , n) given |G| determined by these simple divisibility conditions. Proof. G acts on M (G) cellullarly by conjugation. Thus it shuffles components around taking components to isomorphic components. The G-conjugacy orbit of the component T is thus a union of ≥ 1 components all isomorphic as cell complexes with T . Taking an edge in e in the component T , Theorem 2.15 shows that the G-conjugation orbit of e includes all the edges in T . It is then easy to see that the G-conjugation orbit of e is exactly the set of edges in the

components conjugate to T . Since each of these components has the same edge count E we have E = |G| |S| where S is the stabilizer subgroup of the edge 48

e. Thus |G| = E|S| and so E divides |G|. Furthermore E = |G| if and only if |S| = 1 and = 1 i.e. T is invariant under G-conjugation and G acts freely and transitively on the set of edges of T . As any element of C(x) ∩ C(y) fixes the edge [(x, 1), (y, 1)] under conjugation, we must have C(x) ∩ C(y) = {1} in this case and in particular G must have trivial center. An analogous argument works for faces and so (1) and (4) are proved. In case (2), we have E = λi Vi and 2E = nF and so the result follows. Also n is even in this case as the proof of Theorem A.8 shows. In case (3), we have 2E = λV = nF and that case follows immediately also. Finally (5) follows as there are only finitely many divisors of a positive integer |G| and 2 − 2g = V1 + V2 − E + F = V − E + F is an “alternating” sum of three or four of these divisors. The reader is warned that in general V = V1 + V2 does not divide |G| as examples in later sections show. Theorem A.10. Let G be a nonabelian group and let x, y be non-commuting elements in G. Recall all 2-cells of M (x, y) are n-gons. Then: (1) The map M (x, y) has face and edge transitive automorphism group. (2) All vertices have the same valency λ when the size of the x and y-conjugacy classes of α = xy are the same. The Schl¨ afli index is then {n, λ}. (3) The map has vertex transitive automorphism group if x and y are α-conjugate. (4) The map has flag-transitive automorphism group (i.e. is regular) if x and y are α-conjugate and there is an automorphism of the group G taking x to y −1 and y to x−1 (5) The results of (3) and (4) still hold when x and y are not α-conjugate as long as there exists an automorphism of the group G taking x to y and y to y −1 xy and another taking x to y −1 and y to x−1 . Proof. (1) and (2) follow from the work in Theorem 2.15 and the previous paragraphs. To see (3), note that any edge or vertex of the component can be mapped by an automorphism to lie in the 2-cell centered at (α, 2). Then α-conjugacy will move this image edge or vertex to any chosen representative edge or vertex on that 2-cell as long as all vertices on the boundary of this 2-cell are α-conjugate. This happens if and only if x and y are α-conjugate. For (4), given a reference flag v ∈ e ⊂ f (we will surpress the empty face and greatest face in this proof as they do not come into play) and another flag v  ∈ e ⊂ f  we can first apply an automorphism to take f  to f by face transitivity. Thus we many assume f = f  . Then as all vertices along the rim of the corresponding 2-cell are conjugate, we may conjugate fixing f so that e moves to e. Thus it remains to show that there is a automorphism of the polytope taking the flag (x, 1) ∈ [(x, 1), (y, 1)] ⊂ [(x, 1), (y, 1), (xy, 2)] to the flag (y, 1) ∈ [(x, 1), (y, 1)] ⊂ [(x, 1), (y, 1), (xy, 2)]. As such a map has to be orientation reversing, to achieve it using functoriality we have to use an antiautomorphism of the group. If I is the inversion map I(x) = x−1 , then any anti-automorphism is the composition of I with an automorphism. A quick calculation shows that if φ is an automorphism of G taking x to y −1 and y to x−1 , the simplicial automorphism arising from I ◦ φ does the job! Finally for 49

(5), the stated automorphism maps the edge [(x, 1), (y, 1)] to its adjacent edge on the sheet about (xy, 2). This together with previous comments gives vertex transitivity. Then regularity follows as in the proof of (4). The classical duality operation of maps has the effect of interchanging the roles of vertices and faces while leaving the edges alone. It takes a map of Schl¨ afli symbol {n, λ} to one of symbol {λ, n}. It corresponds to the classical dual cell structure construction behind Poincare duality of the component. The vertices in this new structure correspond to the centers of the 2-faces in the original. Edges are drawn between these vertices when the 2-faces they came from were adjacent. The new 2-faces hence come from arrangements of faces around vertices in the original and are λi -gons if the vertex had valency λi . Thus the dual of one of our complexes would be vertex and edge transitive and have either one or two orbits of faces. In the case of two orbits, the faces would consist of either λ1 -gons or λ2 -gons and around every vertex these types would alternate with total even vertex valency n. Thus all in all, the role of face type and valency interchange and the role of vertices and faces change under duality. Thus if the reader prefers, they can consider the dual to our construction which would be like a “soccerball”, consisting of at most two types of polygonal faces with vertex and edge transitive automorphism group. In other words the complexes that arise as components in the construction M (G) that have two valencies are in general dual to abstract quasiregular (i.e., vertex, edge transitive with two types of faces arranged alternatingly about a vertex) maps.

A.5

Valence Two and Doubling

In this section we will discuss the case when one or both valencies in Theorem A.8 are equal to two. Lemma A.11. Let X be a tesselated Riemann surface as those arising is Theorem A.8 i.e., edge and face transitive and with at most two orbits of vertices. If the valencies λ1 = λ2 = 2 then F = 2, E = V = n, g = 0 and X is a sphere obtained by gluing two n-gon faces along their common rim. Proof. λ1 = λ2 = 2 implies λ = 2. Using this in the equations in part (3) of Theorem A.8, we find 2 − 2g = 2E n . As this quantity is positive, this forces g = 0 and then E = n. Then nF = 2E forces F = 2 and λV = 2E forces E = V . The lemma follows. We now consider the case of a tesselated Riemann surface which is edge/face transitive and has two orbits of vertices of valency λ1 = 2 and λ2 = k > 2. Recall when we have two valencies we have an even face type 2s. We will denote the Schl¨ afli symbol of such a complex as {2s, 2-k} where 2s is the face type. If we have such a complex, the two orbits of vertices are distinguishable due to their distinct valencies. Notice each vertex of valency 2 lies in two faces and is adjacent to two vertices of valency k > 2 which also lie in both of these faces.

50

Thus we can remove all valency 2 vertices and combine the two incident edges to any of them into a single edge. This construction doesn’t change the underlying surface so the genus is unchanged. The face type changes from 2s-gon to sgon and the new edge count is half the original one. The new vertex count is equal to the count of valency k vertices in the original complex. The resulting complex is equivar with a single vertex valency k. Thus we have changed a {2s, 2-k} complex into an equivar {s, k} complex. Conversely given an equivar complex of the form {s, k} one can add a midpoint vertex to each edge to obtain a {2s, 2-k} one and it is easy to see these processes are inverse processes. We will refer to the {2s, 2-k} complex obtained from the {s, k} one as the “double” of the {s, k}-complex. We will sometimes write {2s, 2-k} = D{s, k}. Although the complexes are so similar, it is important to note that if both occur in the decomposition Y (G) for a given group G, they are distinct functorially, i.e., no group automorphism can interchange the two types. Also note that {4, 2-k} = D{2, k}; in this case only, after removing valency two vertices, one obtains an equivar complex whose faces are 2-gons. While the {2, k} complexes do not arise in M (G) since n < 3, the {2, k} complexes are duals of the {k, 2} complexes described in Lemma A.11. Since the doubling and duality operations are genus preserving, the {4, 2-k} complexes have genus zero (g = 0).

A.6

A Finiteness Theorem

In this section we show that except the genus one (g = 1), and the {n, 2} and {4, 2-k} families described above, for a given genus there are only finitely many distinct tesselations on the closed surface of genus g of the sort arising in Theorem A.8. Recall that these tesselations are closed cell structures, i.e., the closed cells are homeomorphic to 2-disks, that is to say, no self-identifications occur along the boundary of the faces. Theorem A.12. Let g, V, E, F, n, λ1 , λ2 be as in Theorem A.8. The distinct closed-cell tesselations on the closed surface of genus g which are edge and face transitive having n ≥ 3, can be categorized as follows: (i) For each fixed genus g ≥ 2, there are only finitely many possibilities for all the data (Vi , E, F, n, λi ). (ii) For g = 1, there are only finitely many possibilities for the Schl¨ afli symbol {n, λ} or {n, λ1 -λ2 }. There are infinitely many possibilities for Vi , E, and F . (iii) For g = 0, there are infinite families when λ1 = λ2 = 2 and {n, λ1 -λ2 } = {4, 2-k}, k ≥ 3. Otherwise, there are only finitely many possibilities for all the data (Vi , E, F, n, λi ). Proof. The {n, 2} (i.e. λ1 = λ2 = 2) and {4, 2-k} cases are described by Lemma A.11 and the discussion that followed. Furthermore, the doubling operation shows that the remaining valence two cases where λ1 = 2 and λ2 = k > 2, are in one-to-one correspondence with the equivar complexes {s, k}, where s ≥ 3. Hence, we need only prove the finiteness assertions for λ1 , λ2 ≥ 3.

51

We utilize the equations presented in Theorem A.8: nF nF

= =

2(1 − g) = V − E + F

=

2E λV 1 1 1 2( + − )E, λ n 2

(1) (2) (3)

where λ = λ1 = λ2 is the common valency in the equivar case, and λ1 = 2λ1 1 + 2λ1 2 in the case of two valencies. In general n ≥ 3, and in the two valency case n must be even (hence n ≥ 4). Let g ≥ 1. Using n ≥ 3 in Equation 3 we have 1 1 E( − ) ≤ g − 1 . 6 λ

(4)

3λ Since V ≥ 3 we have E = λV 2 ≥ 2 . Applying this to Equation 4, we obtain λ ≤ 2(2g + 1). Note that F ≥ 3 since we assume λ ≥ 3. Then n ≤ 2(2g + 1) by the exact same argument with the roles of λ and n interchanged. In particular, this proves that there are only finitely many possibilities for λ and n in the equivar case, for each g ≥ 1. It remains to show there are only finitely many possibilities for 3 ≤ λ1 < λ2 in the two valency case. Using λ1 < λ and n ≥ 4, Equation 3 becomes 1 1 E( − ) < g − 1 4 λ1

Since V1 ≥ 2 in this case, E = λ1 V1 ≥ 2λ1 . Applying this we obtain 3 ≤ λ1 < 2g + 2. Proceeding a similar manner using n ≥ 4, we have 1 1 1 − )≤g−1 E( − 4 2λ1 2λ2 and with λ1 ≥ 3 and E ≥ 2λ2 we obtain 3 < λ2 ≤ 6g. So for g ≥ 1 there are finitely many possibilities for λ1 , λ2 , n. For g ≥ 2, all of the data (Vi , E, F, n, λi ) is determined by g, n, λ1 , λ2 through equations 1-3. This proves (i). For g = 1, E is not determined by g, n, λ1 , λ2 , and there are infinitely many possibilities for E, V, and F . This proves (ii). For g = 0, equation 3 implies λ1 + n1 > 12 . Using n ≥ 3 we have λ < 6, and from λ ≥ 3 we obtain n < 6. So there are only finitely many possibilities for n and λ in the equivar case. In the case of two valencies 3 ≤ λ1 < λ2 , applying n ≥ 4 and λ1 < λ, λ1 + n1 > 12 implies λ11 + 14 > 12 or 3 ≤ λ1 < 4. So λ1 = 3. Using this with n ≥ 4, λ1 + n1 > 12 implies 16 + 2λ1 2 + 14 > 12 or λ2 < 6. So for g = 0 there are finitely many possibilities for λ1 , λ2 , n. This proves (iii) since all of the data (Vi , E, F, n, λi ) is determined by g, n, λ1 , λ2 through Equations 1-3. Corollary A.13. A closed cell tesselation on the closed surface of genus 0 which is edge and face transitive having n ≥ 3, must have data (Vi , E, F, n, λi ) given by one of the rows of the following: 52

Table 1: Possible Tesselations on the Riemann Surface of Genus 0 # Faces 2 k 4 4 6 6 8 8 12 12 12 20 20 30

Schl¨ afli Symbol {n, 2} {4, 2-k} {3, 3} {6, 2-3} = D{3, 3} {4, 3} {8, 2-3} = D{4, 3} {3, 4} {6, 2-4} = D{3, 4} {5, 3} {10, 2-3} = D{5, 3} {4, 3-4} {3, 5} {6, 2-5} = D{3, 5} {4, 3-5}

# Vertices n k+2 4 10 8 20 6 18 20 50 14 12 42 32

# Edges n 2k 6 12 12 24 12 24 30 60 24 30 60 60

Solid Type dual Hosahedron double Hosahedron Tetrahedron double Tetrahedron Cube double Cube Octahedron double Octahedron Dodecahedron double Dodecahedron Rhombic Dodecahedron Icosahedron double Icosahedron Rhombictriacontahedron

Note: All cases in table 1 are realized in our construction, in particular these cases exist. For example, these are generated by Σ5 , see table 10. Proof. The proof of Theorem A.12 provides an algorithm for finding all of the cases. We have the infinite families {n, 2} and {4, 2-k}. Considering λ1 , λ2 ≥ 3, in the equivar cases we found the bounds 3 ≤ n, λ ≤ 5, or n, λ = 3, 4, or 5. Then on each pair n, λ we use equation 3 to solve for E, or rule out the case if no integer solution is found. If an integer E is found, we then use equations 1 and 2 to solve for V and F . This leads to the well-known five Platonic Solids as shown in the chart. We also have the doubles of these five. In the two valency case 3 ≤ λ1 < λ2 we found n = 4, λ1 = 3, λ2 = 4 or 5. Again we use equation 3 to solve for E, then Equations 1 and 2 to solve for V and F . This provides all cases. Corollary A.14. If G is an odd order group then no surface of genus 0 (sphere) occurs in Y (G) and hence X(G) is a K(π, 1)-space. On the other hand, using the odd order theorem, it follows that for any nonabelian simple group G, there exists a surface of genus 0 (sphere) in the decomposition Y (G) and π2 (X(G)) = 0. Proof. The possible cell-structures of the surface of genus 0 that can arise in M (G) are captured in Corollary A.13. All of these have an even number of faces or an even number of edges which implies |G| is even if one of these occurs in M (G) by Proposition A.9. Thus if G is an odd order group, no spheres occur in Y (G) and hence X(G) is homotopy equivalent to a bouquet of closed surfaces of genus g ≥ 1 and circles and hence is a K(π, 1)-space, i.e., all higher homotopy groups vanish. On the other hand, if G is a nonabelian simple group, then by the odd order theorem, |G| is even and G possesses an element of order 2. If all elements of order two commuted with each other in G, they would form a 53

normal elementary abelian subgroup which is impossible as G is simple and so there exist two noncommuting elements of order two which generate a dihedral subgroup. As spheres occur in Y (H) when H is dihedral (see the example section under dihedral groups for a proof of this), spheres occur in Y (G) also by monotonicity. Thus X(G) is homotopy equivalent to a bouquet of a positive number of spheres with a K(π, 1)-space and so has π2 (X(G)) = 0. Note the odd order theorem was used in the 2nd part of the argument of the last corollary. In fact, if an independent argument could be made to show that π2 (X(G)) = 0 or equivalently that Y (G) contained a sphere when G is a nonabelian simple group then it would provide a proof of the odd order theorem. One can similarly solve for all possible Schl¨afli Symbols in higher genus cases. We present the genus g = 1 case in Corollaries A.15. Full lists of the possibilities in the regular case (which are limited to single valency) are available for genus 2 through 15 and are contained in work of Conder and Dobcs´anyi (see [CD]). The reader is warned that in [CD], cell structures do not have to be closed, that is to say that the interior of faces are open disks but their closure need not be disks in the space: self-identifications along the boundary are allowed. As noted in the proof of Theorem A.12, Equation 3 does not constrain E when g = 1, and there are infinitely many possibilities for V, E, and F . Corollary A.15. A closed-cell tesselation on the closed surface of genus 1 which is edge and face transitive having n ≥ 3, must have one of the following Schl¨ afli Symbols: {3, 6},

D{3, 6} = {6, 2-6},

{6, 3},

D{6, 3} = {12, 2-3},

{4, 4},

D{4, 4} = {8, 2-4},

{4, 3-6}

Note: All cases above exist as tesselations of the torus, however we are not sure if they all occur for M (G), for some group G.

B

Data for Cell Complexes of Several Groups

Given here is data for the collection of maps M (G) of various groups G. All data that follows was generated using SAGE. Table 2: SL(2, F3 ) - Cell Complexes of 21 Total Components Genus 0 (19 components)

1 (2 components)

# Faces 2 4 4 6 4

Schl¨ afli Symbol {4, 2} {3, 3} {6, 2-3} {4, 3} {6, 3}

# Vertices 4 4 10 8 8

54

# Edges 4 6 12 12 12

# Components of This Type 3 4 8 4 2

Table 3: SL(2, F5 ) - Cell Complexes of 341 Total Components Genus 0 (281 components)

1 (10 components) 4 (16 components) 5 (10 components) 9 (22 components) 13 (2 components)

# Faces 2 2 2 2 2 3 4 4 5 6 12 12 20 20 30 4

Schl¨ afli Symbol {3, 2} {4, 2} {5, 2} {6, 2} {10, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 2-5} {4, 3} {5, 3} {10, 2-3} {3, 5} {6, 2-5} {4, 3-5} {6, 3}

# Vertices 3 4 5 6 10 5 4 10 7 8 20 50 12 42 32 8

# Edges 3 4 5 6 10 6 6 12 10 12 30 60 30 60 60 12

# Components of This Type 20 15 24 10 12 40 20 40 48 20 4 8 4 8 8 10

12 12 30 12 20 12 20 12

{5, 5} {10, 2-5} {4, 5} {10, 3} {6, 3-5} {10, 3-5} {6, 5} {10, 5}

12 42 24 40 32 32 24 24

30 60 60 60 60 60 60 60

4 8 4 2 8 16 6 2

55

Table 4: SL(2, F7 ) - Cell Complexes of 1376 Total Components Genus 0 (784 components)

1 (92 components) 3 (294 components)

8 (24 components) 10 (32 components)

15 (12 components) 17 (10 components) 19 (14 components) 22 (8 components)

# Faces 2 2 2 2 3 4 4 4 6 6 8 8 12 4 7 3 6 7 8 24 24 56 56 84 42 56 24 24 42 42 84 42 56 24 56 24 24 84 42

Schl¨ afli Symbol {3, 2} {4, 2} {6, 2} {8, 2} {4, 2-3} {3, 3} {4, 2-4} {6, 2-3} {4, 3} {8, 2-3} {3, 4} {6, 2-4} {4, 3-4} {6, 3} {6, 3} {14, 3} {8, 3-4} {6, 3-7} {6, 4} {7, 3} {14, 2-3} {3, 7} {6, 2-7} {4, 3-7} {8, 3} {6, 3-4} {7, 4} {14, 2-4} {4, 7} {8, 2-7} {4, 4-7} {8, 3-4} {6, 4} {14, 3} {6, 3-7} {7, 7} {14, 2-7} {4, 7} {8, 4}

# Vertices 3 4 6 8 5 4 6 10 8 20 6 18 14 8 14 14 14 10 12 56 140 24 108 80 112 98 42 126 24 108 66 98 84 112 80 24 108 48 84

# Edges 3 4 6 8 6 6 8 12 12 24 12 24 24 12 21 21 24 21 24 84 168 84 168 168 168 168 84 168 84 168 168 168 168 168 168 84 168 168 168

# Components of This Type 56 42 28 42 112 56 84 112 56 56 28 56 56 28 64 64 56 128 14 4 8 4 8 8 8 16 4 8 4 8 8 8 4 2 8 4 8 2 8 continued on next page

56

Table 4: SL(2, F7 ) - continued Genus 24 (48 components) 31 (10 components) 33 (22 components) 40 (24 components) 49 (2 components)

# Faces 24 42 56 24 42 24 56 24 42 24

Schl¨ afli Symbol {14, 3-4} {8, 3-7} {6, 4-7} {14, 4} {8, 4-7} {14, 3-7} {6, 7} {14, 4-7} {8, 7} {14, 7}

# Vertices 98 80 66 84 66 80 48 66 48 48

57

# Edges 168 168 168 168 168 168 168 168 168 168

# Components of This Type 16 16 16 2 8 16 6 16 8 2

Table 5: P SL(2, F7 ) - Cell Complexes of 385 Total Components Genus 0 (245 components)

1 (16 components) 3 (72 components)

8 (6 components) 10 (10 components)

15 (3 components) 17 (2 components) 19 (4 components) 22 (2 components) 24 (12 components)

# Faces 2 2 3 4 4 6 6 8 8 12 7

Schl¨ afli Symbol {3, 2} {4, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 3} {8, 2-3} {3, 4} {6, 2-4} {4, 3-4} {6, 3}

# Vertices 3 4 5 4 10 8 20 6 18 14 14

# Edges 3 4 6 6 12 12 24 12 24 24 21

# Components of This Type 28 63 28 28 28 14 14 14 14 14 16

3 6 7 24 24 56 56 84 42 56 24 24 42 42 84 42 56 56

{14, 3} {8, 3-4} {6, 3-7} {7, 3} {14, 2-3} {3, 7} {6, 2-7} {4, 3-7} {8, 3} {6, 3-4} {7, 4} {14, 2-4} {4, 7} {8, 2-7} {4, 4-7} {8, 3-4} {6, 4} {6, 3-7}

14 14 10 56 140 24 108 80 112 98 42 126 24 108 66 98 84 80

21 24 21 84 168 84 168 168 168 168 84 168 84 168 168 168 168 168

16 14 32 2 2 2 2 2 2 4 2 2 2 2 2 2 1 2

24 24 42

{7, 7} {14, 2-7} {8, 4}

24 108 84

84 168 168

2 2 2

24 42 56

{14, 3-4} {8, 3-7} {6, 4-7}

98 80 66

168 168 168

4 4 4 continued on next page

58

Table 5: P SL(2, F7 ) - continued Genus 31 (2 components) 33 (5 components) 40 (6 components)

# Faces 42

Schl¨ afli Symbol {8, 4-7}

# Vertices 66

# Edges 168

# Components of This Type 2

24 56 24 42

{14, 3-7} {6, 7} {14, 4-7} {8, 7}

80 48 66 48

168 168 168 168

4 1 4 2

59

Table 6: Σ3 - Cell Complexes of 2 Total Components Genus 0 0

# Faces 2 3

Schl¨ afli Symbol {3, 2} {4, 2-3}

# Vertices 3 5

# Edges 3 6

# Components of This Type 1 1

Table 7: A4 ∼ = P SL(2, F3 ) - Cell Complexes of 5 Total Components Genus 0

# Faces 4 4 6

Schl¨ afli Symbol {3, 3} {6, 2-3} {4, 3}

# Vertices 4 10 8

# Edges 6 12 12

# Components of This Type 2 2 1

Table 8: Σ4 - Cell Complexes of 27 Total Components Genus 0 (26 components)

3

# Faces 2 2 3 4 4 6 6 8 8 12 6

Schl¨ afli Symbol {3, 2} {4, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 3} {8, 2-3} {3, 4} {6, 2-4} {4, 3-4} {8, 3-4}

# Vertices 3 4 5 4 10 8 20 6 18 14 14

60

# Edges 3 4 6 6 12 12 24 12 24 24 24

# Components of This Type 4 9 4 2 2 1 1 1 1 1 1

Table 9: A5 ∼ = P SL(2, F5 ) - Cell Complexes of 91 Total Components Genus 0 (79 components)

4 (5 components) 5 (2 components) 9 (5 components)

# Faces 2 2 3 4 4 5 6 12 12 20 20 30 12 12 30 20

Schl¨ afli Symbol {3, 2} {5, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 2-5} {4, 3} {5, 3} {10, 2-3} {3, 5} {6, 2-5} {4, 3-5} {5, 5} {10, 2-5} {4, 5} {6, 3-5}

# Vertices 3 5 5 4 10 7 8 20 50 12 42 32 12 42 24 32

# Edges 3 5 6 6 12 10 12 30 60 30 60 60 30 60 60 60

# Components of This Type 10 12 10 10 10 12 5 2 2 2 2 2 2 2 1 2

12 20

{10, 3-5} {6, 5}

32 24

60 60

4 1

61

Table 10: Σ5 - Cell Complexes of 284 Total Components Genus 0 (194 components)

1 (24 components) 3 (5 components) 4 (27 components)

5 (2 components) 6 (5 components)

# Faces 2 2 2 2 3 4 4 5 6 6 8 8 12 12 12 20 20 30 5 5 6

Schl¨ afli Symbol {3, 2} {4, 2} {5, 2} {6, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 2-5} {4, 3} {8, 2-3} {3, 4} {6, 2-4} {4, 3-4} {5, 3} {10, 2-3} {3, 5} {6, 2-5} {4, 3-5} {4, 4} {8, 2-4} {8, 3-4}

# Vertices 3 4 5 6 5 4 10 7 8 20 6 18 14 20 50 12 42 32 5 15 14

# Edges 3 4 5 6 6 6 12 10 12 24 12 24 24 30 60 30 60 60 10 20 24

# Components of This Type 20 45 12 10 40 10 10 12 5 5 5 5 5 2 2 2 2 2 12 12 5

4 5 12 12 24 24 30 30 60 20

{10, 4} {8, 4-5} {5, 5} {10, 2-5} {5, 4} {10, 2-4} {4, 5} {8, 2-5} {4, 4-5} {6, 3-5}

10 9 12 42 30 90 24 84 54 32

20 20 30 60 60 120 60 120 120 60

6 12 2 2 1 1 1 1 1 2

20 20 30 30 60

{6, 4} {12, 2-4} {4, 6} {8, 2-6} {4, 4-6}

30 90 20 80 50

60 120 60 120 120

1 1 1 1 1 continued on next page

62

Table 10: Σ5 - continued Genus 9 (9 components)

11 (3 components) 16 (6 components) 19 21 24 (6 components) 29

# Faces 12 20 20 24 24 60 20 20 30 20 30 40 30 20 20 24 30 20

Schl¨ afli Symbol {10, 3-5} {6, 5} {12, 2-5} {5, 6} {10, 2-6} {4, 5-6} {6, 6} {12, 2-6} {8, 3-4} {12, 3-4} {8, 3-6} {6, 4-6} {8, 4-5} {12, 3-6} {12, 4-5} {10, 4-6} {8, 5-6} {12, 5-6}

# Vertices 32 24 84 20 80 44 20 80 70 70 60 50 54 60 54 50 44 44

63

# Edges 60 60 120 60 120 120 60 120 120 120 120 120 120 120 120 120 120 120

# Components of This Type 4 1 1 1 1 1 1 1 1 2 2 2 1 1 2 2 2 1

Table 11: A6 - Cell Complexes of 1335 Total Components Genus 0 (909 components)

1 (80 components) 3 (30 components) 4 (120 components)

5 (24 components) 9 (60 components) 10 (20 components)

16 (6 components) 19 (8 components)

# Faces 2 2 2 3 4 4 5 6 6 8 8 12 12 12 20 20 30 9 9 6

Schl¨ afli Symbol {3, 2} {4, 2} {5, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 2-5} {4, 3} {8, 2-3} {3, 4} {6, 2-4} {4, 3-4} {5, 3} {10, 2-3} {3, 5} {6, 2-5} {4, 3-5} {4, 4} {8, 2-4} {8, 3-4}

# Vertices 3 4 5 5 4 10 7 8 20 6 18 14 20 50 12 42 32 9 27 14

# Edges 3 4 5 6 6 12 10 12 24 12 24 24 30 60 30 60 60 18 36 24

# Components of This Type 120 135 72 120 60 60 72 30 30 30 30 30 24 24 24 24 24 40 40 30

9 12 12 12 30 20

{8, 3-4} {5, 5} {6, 4} {10, 2-5} {4, 5} {6, 3-5}

21 12 18 42 24 32

36 30 36 60 60 60

40 24 20 24 12 24

12 20 72 72 90 90 180 90 120 72 72

{10, 3-5} {6, 5} {5, 4} {10, 2-4} {4, 5} {8, 2-5} {4, 4-5} {8, 3} {6, 3-4} {5, 5} {10, 2-5}

32 24 90 270 72 252 162 240 210 72 252

60 60 180 360 180 360 360 360 360 180 360

48 12 4 4 4 4 4 2 4 4 4 continued on next page

64

Table 11: A6 - continued Genus 25 (6 components) 40 (24 components) 46 (2 components) 49 (6 components) 55 (10 components) 64 (24 components) 73 (6 components)

# Faces 72 120 72 90 120 90

Schl¨ afli Symbol {10, 3} {6, 3-5} {10, 3-4} {8, 3-5} {6, 4-5} {8, 4}

# Vertices 240 192 210 192 162 180

# Edges 360 360 360 360 360 360

# Components of This Type 2 4 8 8 8 2

72 120 72 90 72 90 72

{10, 3-5} {6, 5} {10, 4} {8, 4-5} {10, 4-5} {8, 5} {10, 5}

192 144 180 162 162 144 144

360 360 360 360 360 360 360

4 2 2 8 16 8 6

65

Table 12: Σ6 - Cell Complexes of 4477 Total Components Genus 0 (2904 components)

1 (284 components)

3 (150 components) 4 (396 components)

# Faces 2 2 2 2 3 4 4 5 6 6 8 8 12 12 12 20 20 30 4 5 5 9 9 6 8 4 5 6 9 12 12 12 12 18 18 24 24 30 30 36 60

Schl¨ afli Symbol {3, 2} {4, 2} {5, 2} {6, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 2-5} {4, 3} {8, 2-3} {3, 4} {6, 2-4} {4, 3-4} {5, 3} {10, 2-3} {3, 5} {6, 2-5} {4, 3-5} {6, 3} {4, 4} {8, 2-4} {4, 4} {8, 2-4} {8, 3-4} {6, 4} {10, 4} {8, 4-5} {12, 2-6} {8, 3-4} {5, 5} {6, 4} {10, 2-5} {12, 2-4} {4, 6} {8, 2-6} {5, 4} {10, 2-4} {4, 5} {8, 2-5} {4, 4-6} {4, 4-5}

66

# Vertices 3 4 5 6 5 4 10 7 8 20 6 18 14 20 50 12 42 32 8 5 15 9 27 14 12 10 9 24 21 12 18 42 54 12 48 30 90 24 84 30 54

# Edges 3 4 5 6 6 6 12 10 12 24 12 24 24 30 60 30 60 60 12 10 20 18 36 24 24 20 20 36 36 30 36 60 72 36 72 60 120 60 120 72 120

# Components of This Type 240 540 72 240 720 120 240 72 120 120 60 120 120 24 24 24 24 24 60 72 72 40 40 120 30 36 72 40 40 24 20 24 20 20 20 12 12 12 12 20 12 continued on next page

Table 12: Σ6 - continued Genus 5 (24 components) 6 (60 components)

9 (108 components)

10 (20 components)

11 (36 components) 16 (138 components)

19 (20 components) 21 (12 components) 24 (72 components) 25 (6 components) 29 (12 components)

# Faces 20

Schl¨ afli Symbol {6, 3-5}

# Vertices 32

# Edges 60

# Components of This Type 24

20 20 30 30 60 12 20 20 24 24 60 72 72 90 90 180 20 20 30 12 18 20 30 40 90 120 30 72 72 20

{6, 4} {12, 2-4} {4, 6} {8, 2-6} {4, 4-6} {10, 3-5} {6, 5} {12, 2-5} {5, 6} {10, 2-6} {4, 5-6} {5, 4} {10, 2-4} {4, 5} {8, 2-5} {4, 4-5} {6, 6} {12, 2-6} {8, 3-4} {12, 4-6} {8, 6} {12, 3-4} {8, 3-6} {6, 4-6} {8, 3} {6, 3-4} {8, 4-5} {5, 5} {10, 2-5} {12, 3-6}

30 90 20 80 50 32 24 84 20 80 44 90 270 72 252 162 20 80 70 30 24 70 60 50 240 210 54 72 252 60

60 120 60 120 120 60 60 120 60 120 120 180 360 180 360 360 60 120 120 72 72 120 120 120 360 360 120 180 360 120

12 12 12 12 12 48 12 12 12 12 12 4 4 4 4 4 12 12 12 40 20 24 24 24 2 4 12 4 4 12

20 24 30 72 120 20

{12, 4-5} {10, 4-6} {8, 5-6} {10, 3} {6, 3-5} {12, 5-6}

54 50 44 240 192 44

120 120 120 360 360 120

24 24 24 2 4 12 continued on next page

67

Table 12: Σ6 - continued Genus 40 (24 components) 46 (2 components) 49 (14 components)

55 (10 components) 61 (15 components) 64 (24 components) 73 (6 components) 91 (9 components)

121 (42 components)

139 (24 components) 151 (21 components) 169 (44 components)

# Faces 72 90 120 90

Schl¨ afli Symbol {10, 3-4} {8, 3-5} {6, 4-5} {8, 4}

# Vertices 210 192 162 180

# Edges 360 360 360 360

# Components of This Type 8 8 8 2

72 120 120 144 144 360 72 90 120 120 360 72 90 72

{10, 3-5} {6, 5} {12, 2-5} {5, 6} {10, 2-6} {4, 5-6} {10, 4} {8, 4-5} {6, 6} {12, 2-6} {4, 6} {10, 4-5} {8, 5} {10, 5}

192 144 504 120 480 264 180 162 120 480 240 162 144 144

360 360 720 360 720 720 360 360 360 720 720 360 360 360

4 2 2 2 2 2 2 8 6 6 3 16 8 6

120 180 180 240 120 120 180 240 120 144 180 120 180 120 144

{12, 3-4} {8, 4} {8, 3-6} {6, 4-6} {12, 4} {12, 3-6} {8, 4-6} {6, 6} {12, 4-5} {10, 4-6} {8, 5-6} {12, 4-6} {8, 6} {12, 5-6} {10, 6}

420 360 360 300 360 360 300 240 324 300 264 300 240 264 240

720 720 720 720 720 720 720 720 720 720 720 720 720 720 720

2 3 2 2 8 14 16 4 8 8 8 14 7 30 14

68

Table 13: A7 - Cell Complexes of 16813 Total Components Genus 0 (8379 components)

1 (1514 components)

3 (1440 components)

# Faces 2 2 2 2 2 3 4 4 5 6 6 6 8 8 12 12 12 20 20 30 4 5 5 7 9 9 3 6 7 24 24 56 56 84

Schl¨ afli Symbol {3, 2} {4, 2} {5, 2} {6, 2} {12, 2} {4, 2-3} {3, 3} {6, 2-3} {4, 2-5} {4, 2-6} {4, 3} {8, 2-3} {3, 4} {6, 2-4} {4, 3-4} {5, 3} {10, 2-3} {3, 5} {6, 2-5} {4, 3-5} {6, 3} {4, 4} {8, 2-4} {6, 3} {4, 4} {8, 2-4} {14, 3} {8, 3-4} {6, 3-7} {7, 3} {14, 2-3} {3, 7} {6, 2-7} {4, 3-7}

69

# Vertices 3 4 5 6 12 5 4 10 7 8 8 20 6 18 14 20 50 12 42 32 8 5 15 14 9 27 14 14 10 56 140 24 108 80

# Edges 3 4 5 6 12 6 6 12 10 12 12 24 12 24 24 30 60 30 60 60 12 10 20 21 18 36 21 24 21 84 168 84 168 168

# Components of This Type 840 945 252 210 210 1260 420 840 252 420 420 420 420 420 420 126 126 126 126 126 210 252 252 240 280 280 240 420 480 60 60 60 60 60 continued on next page

Table 13: A7 - continued Genus 4 (1197 components)

5 (126 components) 6 (105 components)

8 (180 components) 9 (399 components)

10 (440 components)

# Faces 4 5 9 12 12 12 24 24 30 30 60 20

Schl¨ afli Symbol {10, 4} {8, 4-5} {8, 3-4} {5, 5} {6, 4} {10, 2-5} {5, 4} {10, 2-4} {4, 5} {8, 2-5} {4, 4-5} {6, 3-5}

# Vertices 10 9 21 12 18 42 30 90 24 84 54 32

# Edges 20 20 36 30 36 60 60 120 60 120 120 60

# Components of This Type 126 252 280 126 140 126 21 21 63 21 21 126

20 20 30 30 60 42 56 12 20 20 24 24 60 24 24 42 42 72 72 84 90 90 180

{6, 4} {12, 2-4} {4, 6} {8, 2-6} {4, 4-6} {8, 3} {6, 3-4} {10, 3-5} {6, 5} {12, 2-5} {5, 6} {10, 2-6} {4, 5-6} {7, 4} {14, 2-4} {4, 7} {8, 2-7} {5, 4} {10, 2-4} {4, 4-7} {4, 5} {8, 2-5} {4, 4-5}

30 90 20 80 50 112 98 32 24 84 20 80 44 42 126 24 108 90 270 66 72 252 162

60 120 60 120 120 168 168 60 60 120 60 120 120 84 168 84 168 180 360 168 180 360 360

21 21 21 21 21 60 120 252 63 21 21 21 21 60 60 60 60 28 28 60 28 28 28 continued on next page

70

Table 13: A7 - continued Genus 11 (63 components) 15 (90 components) 16 (168 components)

17 (60 components) 19 (197 components)

21 (21 components) 22 (60 components) 24 (486 components)

25 (42 components) 29 (21 components) 31 (60 components) 33 (150 components) 40 (348 components)

# Faces 20 20 30 42 56 20 30 40 90 120 56

Schl¨ afli Symbol {6, 6} {12, 2-6} {8, 3-4} {8, 3-4} {6, 4} {12, 3-4} {8, 3-6} {6, 4-6} {8, 3} {6, 3-4} {6, 3-7}

# Vertices 20 80 70 98 84 70 60 50 240 210 80

# Edges 60 120 120 168 168 120 120 120 360 360 168

# Components of This Type 21 21 21 60 30 42 42 42 14 28 60

24 24 30 72 72 20

{7, 7} {14, 2-7} {8, 4-5} {5, 5} {10, 2-5} {12, 3-6}

24 108 54 72 252 60

84 168 120 180 360 120

60 60 21 28 28 21

42

{8, 4}

84

168

60

20 24 24 30 42 56 72 120 20

{12, 4-5} {10, 4-6} {14, 3-4} {8, 5-6} {8, 3-7} {6, 4-7} {10, 3} {6, 3-5} {12, 5-6}

54 50 98 44 80 66 240 192 44

120 120 168 120 168 168 360 360 120

42 42 120 42 120 120 14 28 21

42

{8, 4-7}

66

168

60

24 56 24 42 72 90 120

{14, 3-7} {6, 7} {14, 4-7} {8, 7} {10, 3-4} {8, 3-5} {6, 4-5}

80 48 66 48 210 192 162

168 168 168 168 360 360 360

120 30 120 60 56 56 56 continued on next page

71

Table 13: A7 - continued Genus 46 (14 components) 49 (42 components) 55 (70 components) 64 (168 components) 73 (42 components) 136 (20 components)

169 (3 components) 199 (20 components)

211 (15 components)

241 (26 components)

271 (13 components) 274 (30 components)

# Faces 90

Schl¨ afli Symbol {8, 4}

# Vertices 180

# Edges 360

# Components of This Type 14

72 120 72 90 72 90 72

{10, 3-5} {6, 5} {10, 4} {8, 4-5} {10, 4-5} {8, 5} {10, 5}

192 144 180 162 162 144 144

360 360 360 360 360 360 360

28 14 14 56 112 56 42

360 360 630 630 1260 504 840 360 360 504 504 1260 420 630 840 840 360 360 360 420 420 840 1260 360 360 1260 504 630 840

{7, 4} {14, 2-4} {4, 7} {8, 2-7} {4, 4-7} {10, 3} {6, 3-5} {7, 5} {14, 2-5} {5, 7} {10, 2-7} {4, 5-7} {12, 3} {8, 3-4} {6, 4} {6, 6-3} {7, 6} {14, 3} {14, 2-6} {6, 7} {12, 2-7} {6, 3-7} {4, 6-7} {7, 7} {14, 2-7} {4, 7} {10, 3-4} {8, 3-5} {6, 4-5}

630 1890 360 1620 990 1680 1344 504 1764 360 1620 864 1680 1470 1260 1260 420 1680 1680 360 1620 1200 780 360 1620 720 1470 1344 1134

1260 2520 1260 2520 2520 2520 2520 1260 2520 1260 2520 2520 2520 2520 2520 2520 1260 2520 2520 1260 2520 2520 2520 1260 2520 2520 2520 2520 2520

4 4 4 4 4 1 2 4 4 4 4 4 1 8 4 2 4 2 4 4 4 4 4 6 6 1 10 10 10 continued on next page

72

Table 13: A7 - continued Genus 316 (30 components)

337 (9 components) 346 (48 components) 379 (51 components)

409 (54 components) 421 (21 components)

442 (12 components) 451 (88 components)

481 (37 components) 484 (24 components) 505 514 (108 components) 526 (6 components) 547 (12 components)

# Faces 420 630 630 840 504 840 360 630 840 420 504 504 630 840 360 504 840 420 420 630 840 504 630 360 360 420 630 840 360 840 420 504 630 504 360 504 630 420 630 420 504

Schl¨ afli Symbol {12, 3-4} {8, 4} {8, 3-6} {6, 4-6} {10, 3-5} {6, 5} {14, 3-4} {8, 3-7} {6, 4-7} {12, 3-5} {10, 4} {10, 3-6} {8, 4-5} {6, 5-6} {14, 3-5} {10, 3-7} {6, 5-7} {12, 4} {12, 3-6} {8, 4-6} {6, 6} {10, 4-5} {8, 5} {14, 4} {14, 3-6} {12, 3-7} {8, 4-7} {6, 6-7} {14, 3-7} {6, 7} {12, 4-5} {10, 4-6} {8, 5-6} {10, 5} {14, 4-5} {10, 4-7} {8, 5-7} {12, 4-6} {8, 6} {12, 5} {10, 5-6}

73

# Vertices 1470 1260 1260 1050 1344 1008 1470 1200 990 1344 1260 1260 1134 924 1344 1200 864 1260 1260 1050 840 1134 1008 1260 1260 1200 990 780 1200 720 1134 1050 924 1008 1134 990 864 1050 840 1008 924

# Edges 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520

# Components of This Type 6 12 6 6 6 3 16 16 16 6 11 6 22 6 18 18 18 6 2 12 1 8 4 20 8 8 44 8 26 11 8 8 8 1 36 36 36 4 2 4 8

Table 13: A7 - continued Genus 556 (48 components) 577 (34 components) 586 (60 components) 589 (3 components) 619 (42 components) 631 649 (70 components) 661 (4 components) 691 (21 components) 721 (20 components)

# Faces 360 420 630 360 504 360 630 420 504 360 420 504 420 360 504 420

Schl¨ afli Symbol {14, 4-6} {12, 4-7} {8, 6-7} {14, 5} {10, 5-7} {14, 4-7} {8, 7} {12, 5-6} {10, 6} {14, 5-6} {12, 5-7} {10, 6-7} {12, 6} {14, 5-7} {10, 7} {12, 6-7}

# Vertices 1050 990 780 1008 864 990 720 924 840 924 864 780 840 864 720 780

# Edges 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520 2520

# Components of This Type 16 16 16 10 24 40 20 2 1 14 14 14 1 48 22 4

360 420 360

{14, 6-7} {12, 7} {14, 7}

780 720 720

2520 2520 2520

14 7 20

74