An Asphericity Conjecture and Kaplansky Problem on Zero Divisors

Journal of Algebra 216, 13᎐19 Ž1999. Article ID jabr.1998.7756, available online at http:rrwww.idealibrary.com on

An Asphericity Conjecture and Kaplansky Problem on Zero Divisors S. V. Ivanov U Department of Mathematics, Uni¨ ersity of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801 E-mail: [email protected] Communicated by Efim Zelmano¨ Received April 8, 1998

Suppose a group representation H s ² A 5 R : is aspherical, x f A, W Ž A j x . is a word in alphabet Ž A j x . " 1 with nonzero sum of exponents on x, and the group H naturally embeds in G s ² A j x 5 R j W Ž A j x .:. It is conjectured that the presentation G s ² A j x 5 R j W Ž A j x .: is aspherical if and only if G is torsion free. It is proven that if this conecture is false and G s ² A j x 5 R j W Ž A j x .: is a counterexample, then the integral group ring ⺪Ž G . of torsion free group G will contain zero divisors. Some special cases when this conjecture holds are also indicated. 䊚 1999 Academic Press

Let ² A 5 R : be a group presentation, where A is an alphabet and R is a set of defining relators Žwhich are words in A " 1 s A j Ay1 .. The group G given by this presentation is the quotient F Ž A .rN Ž R ., where F Ž A . is the free group over the alphabet A and N Ž R . is the normal closure of R in F Ž A .. The quotient N Ž R .rN X Ž R ., where N X Ž R . is the commutator subgroup of N Ž R ., can be turned into a Žleft. G-module as follows: If ␣ : F Ž A . ª G and ␤ : N Ž R . ª N Ž R .rN X Ž R . are natural epimorphisms, and W g F Ž A ., S g N Ž R ., then ␤

W ␣ ⭈ S ␤ s Ž WSWy1 . . * Supported in part by Alfred P. Sloan Research Fellowship and NSF Grant DMS 95-01056. 13 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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S. V. IVANOV

Clearly, this G-action extends to an action of the integral group ring ⺪Ž G . of G over M Ž G . s N Ž R .rN X Ž R . by setting ␤

Ž W1␣ " W2␣ . ⭈ S ␤ s Ž W1 SW1y1W2 S " 1Wy1 2 . . This ⺪Ž G .-module M Ž G . is called the relation module of G s ² A 5 R :. A group presentation is called aspherical if its relation module is freely generated by images R ␤ of relators R g R. If K G is a 2-complex associated with G s ² A 5 R : in standard way Ž K G has a single 0-cell and ␲ 1Ž K G . s G ., then G is aspherical if and only if so is K G Žsee wGRx; recall that a 2-complex K is called aspherical if ␲ 2 Ž K . s 0.. The Whitehead asphericity conjecture Žoriginally stated as a question in wWx. claims that any subcomplex of an aspherical 2-complex is also aspherical Žfor some reductions see wH2, Lf, I1, I2, I3x.. In group-theoretic terms this means that if ² A 5 R : is an aspherical presentation then for any S ; R the presentation ² A 5 S : is also aspherical Žsee wGR, Hb, Px.. Consider a special type of presentations, G s ² A j x 5 RŽ A . j W Ž A j x . :,

Ž 1.

where x f A, all relators in RŽ A . s R are words in A " 1 s A j Ay1, and W Ž A j x . is a word in Ž A j x . " 1 with nonzero sum of exponents on x. Assuming that Ž1. is aspherical, one can easily reduce the problem on asphericity of subpresentation ² A 5 R : to whether or not the group H s ² A 5 R : embeds in the group G given by Ž1., that is, whether the equation W Ž A j x . s 1 is solvable over H. This, however, is another difficult open problem Žthe so-called Kervaire problem about the solvability of equations over groups. and an affirmative solution is known only in some special cases, for example: if all occurrences of x " 1 in W have positive Žor negative. exponents ŽLevin wLvx.; if the sum of exponents on x in W is "1 and H is torsion free ŽKlyachko wKx.; and if H is locally indicable, that is, every nontrivial finitely generated subgroup of H has an infinite cyclic epimorphic image ŽHowie wH1x, see also Brodskii wBx.. The following seems worth mentioning and is immediate from the foregoing reduction and Klyachko’s result. PROPOSITION. If Ž1. is a balanced presentation of the tri¨ ial group Ž and hence aspherical ., then its subpresentation H s ² A 5 RŽ A .: is aspherical if and only if H is a torsion free group. In this paper, we will turn tables around to indicate an interesting connection between the asphericity of presentation Ž1., torsion in the group G, and the Kaplansky problem on zero divisors which asks whether the group ring of a torsion free group over an integral domain can have zero divisors.

AN ASPHERICITY CONJECTURE

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First let us state the main Conjecture that, like the Whitehead asphericity conjecture, is actually a problem more convenient to state in the affirmative form. Conjecture. Suppose a group presentation H s ² A 5 R : is aspherical, x f A, W Ž A j x . is a word in Ž A j x . " 1 with nonzero sum of exponents on x, and the group H naturally embeds in G s ² A j x 5 R j W Ž A j x . :.

Ž 2.

Then presentation Ž2. is aspherical if and only if the group G is torsion free. THEOREM 1. If the Conjecture fails and G s ² A j x 5 R j W Ž A j x .: is a counterexample to it, then the group G is a torsion free group whose integral group ring ⺪Ž G . contains zero di¨ isors. In addition, if W Ž A j x . has n occurrences of x " 1, then ⺪Ž G . contains a zero di¨ isor Z with
Ž 2.

provides a counterexample. This means Žsee wLS, Olx. the existence of a spherical diagram ⌬ over Ž2. such that there is no involution i: ⌸ ª ⌸ i Žmeaning Ž ⌸ i . i s ⌸ . on the set of all cells in ⌬ with the following properties Žlike in wLS, Olx, by ␾ we denote the labelling function from the set of oriented edges of ⌬ to A " 1 ; it is convenient to consider the boundaries ⭸ ⌸ of cells ⌸ in ⌬ positively oriented.: ŽP1. If ␾ Ž ⭸ ⌸ . s R then ␾ Ž ⭸ ⌸ i . s Ry1 . ŽP2. If o g ⭸ ⌸, o i g ⭸ ⌸ i are vertices such that the label ␾ Ž ⭸ ⌸ < o . of ⭸ ⌸ starting at o is identical with ␾ Ž ⭸ ⌸ i < o i .y1 and p s o y o i is a path in ⌬ going from o to o i, then ␾ Ž p . s 1 in G.

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S. V. IVANOV

Let ⌸ 1 , . . . , ⌸ m be all cells in ⌬ with ␾ Ž ⭸ ⌸ k . s W ␦ k , where ␦ k g  "14 , k s 1, . . . , m, and W ' U1 x ␧ 1 . . . Un x ␧ n , where ␧ 1 , . . . , ␧ n g  "14 , U1 , . . . , Un are words in A " 1. Pick a vertex ¨ in ⌬ and let the vertices o1 g ⭸ ⌸ 1 , . . . , om g ⭸ ⌸ m be such that

␾ Ž ⭸ ⌸ k < o k . ' U1 x ␧ 1 . . . Un x ␧ n for all k. Consider some paths p1 s ¨ y o1 , . . . , pm s ¨ y om that go from ¨ to o1 , . . . , om , respectively. Denote the labels ␾ Ž p1 ., . . . , ␾ Ž pm . by V1 , . . . , Vm , respectively, and consider the element

␦ 1V1␣ q ␦ 2 V2␣ q ⭈⭈⭈ q␦mVm␣

Ž 3.

in ⺪Ž G ., where ␣ : F Ž A j x . ª G is the natural epimorphism. Yet we construct a sequence of initial subwords of the word W as follows Tl s U1 . . . Ul provided ␧ l s 1 and Tl s U1 . . . Ul x ␧ l if ␧ l s y1. Consider another element

␧ 1T1␣ q ␧ 2 T2␣ q ⭈⭈⭈ q␧ nTn␣

Ž 4.

in ⺪Ž G .. First let us show that

Ž ␦ 1V1␣ q ␦ 2 V2␣ q ⭈⭈⭈ q␦mVm␣ . Ž ␧ 1T1␣ q ␧ 2 T2␣ q ⭈⭈⭈ q␧ nTn␣ . s 0. Ž 5 . Let ⭸ ⌸ k␦ k < o k s u1Ž k . e1Ž k . . . . u nŽ k . e nŽ k ., where e1Ž k ., . . . , e nŽ k . are the edges of ⭸ ⌸ ␦k k labelled by x ␧ 1 , . . . , x ␧ n . By t l Ž k . s ok y ol Ž k . denote the arc of ⭸ ⌸ ␦k k that starts at ok and ends at ol Ž k . so that

␾ Ž t l Ž k . . s Tl . It follows from the definition of the words Tl that the terminal vertex ol Ž k . of t l Ž k . is always the initial vertex of e l Ž k . ␻ , where ␻ s "1 is chosen so that ␾ Ž e l Ž k . ␻ . s x. Let us make another observation: Suppose e is an edge of Žpositively oriented. ⭸ ⌸ k and e s e k Ž l . " 1. Then ␾ Ž e . s x provided ␦ k ␧ l s 1 and ␾ Ž e . s xy1 if ␦ k ␧ l s y1 Žto see this it suffices to consider cases ␦ k s "1.. Now consider all possible products Vk Tl . Each of them can be interpreted as the label ␾ Ž qk l . of the path qk l s pk t l Ž k . that goes from ¨ to ol Ž k .. Let j: e ª ey1 be an involution defined on the set X Ž1. of all oriented edges e in ⌬ with ␾ Ž e . s x " 1. Clearly, if e g X Ž1. then there is a cell ⌸ k 1 such that e g ⭸ ⌸ k 1 and so e s e l 1Ž k 1 . " 1. Let ey1 g ⭸ ⌸ k 2 and ey1 s e l 2Ž k 2 . " 1. By the above remarks, we have that ol 1Ž k 1 . s ol 2Ž k 2 . and ␦ k 1 ␧ l 1 s y␦ k 2 ␧ l 2 for ␾ Ž ey1 . s ␾ Ž e .y1 . Therefore, ␾ Ž qk 1 l 1 . s ␾ Ž qk 2 l 2 . in G

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AN ASPHERICITY CONJECTURE

and hence

␦ k 1 ␧ l 1Vk␣1 Tl␣1 q ␦ k 2 ␧ l 2Vk␣2 Tl␣2 s 0. Thus the involution j induces an involution j ⌺ on the set of all terms in Ý k, l ␦ k ␧ l Vk␣ Tl␣ with the property that ␦ k ␧ l Vk␣ Tl␣ q j ⌺ Ž ␦ k ␧ l Vk␣ Tl␣ . s 0 and equality Ž5. is proven. Let us show that elements Ž3. ᎐ Ž4. are not zero in ⺪Ž G .. By assumption,

Ý ␧l / 0 l

and hence the image of Ž4. under ⺪Ž G . ª ⺪ is not zero as required. Since the support of element Ž4. contains at most n elements, it suffices to prove that Ž3. cannot be zero. Arguing on the contrary, assume that

␦ 1V1␦ 1 q . . . ␦mVm␦ m s 0.

Ž 6. X

This obviously implies the existence of an involution iX : ⌸ ª ⌸ i on the set of cells ⌸ 1 , . . . , ⌸ m with properties ŽP1. ᎐ ŽP2.. By taking the cells ⌸ 1 , . . . , ⌸ m out of ⌬, we create m holes in ⌬ and X then we attach ⭸ ⌸y1 to ⭸ ⌸ ki along the edges with identical labels. Up to k arbitrarily small deformations Žor, alternatively, by introducing 0-cells in ⌬ and 0-refinements of ⌬ as described in wOlx. we will get a diagram ⌬ 0 on an orientable surface ⌿ of genus mr2 tiled with cells corresponding to relators R g R of H s ² A 5 R : and x-annuli whose boundary components are labelled by empty words. In addition, it follows from construction of ⌬ 0 that for every closed path p in ⌬ 0 it is true that ␾ Ž p . s 1 in G. By taking x-annuli out of ⌿ we will get several disconnected components ⌿1 , . . . , ⌿t which themselves are orientable surfaces with holes Žwhose boundaries are former boundary components of the removed x-annuli labelled by empty words.. Contracting the holes into points Žor, alternatively, using 0-cells to fill the holes in., we will obtain diagrams ⌬ 1 , . . . , ⌬ t over H s ² A 5 R : on orientable surfaces ⌿1X , . . . , ⌿tX , respectively, with no holes. Clearly, there are no edges labelled by x " 1 in ⌬ 1 , . . . ,⌬ t and if p is a closed path in any of ⌬ 1 , . . . , ⌬ t then ␾ Ž p . s 1 in G. Since H embeds in G, we also have ␾ Ž p . s 1 in H. Let ⌿1X have genus g. By making g cuts along suitable curves on the surface ⌿1X , we turn D 1 into a spherical diagram ⌬ 1Ž2 g . with 2 g holes L1 , . . . , L2 g . It is clear that there is an involution i: L ª Li on the set L s  L1 , . . . , L2 g 4 of holes in ⌬ 1Ž2 g . with properties ŽP1. ᎐ ŽP2. Žin which ␲ , ⭸␲ are replaced with L, ⭸ L.. Since ␾ Ž ⭸ L r . s 1 in H, there is a disk diagram ⌫r over H s ² A 5 R : with ␾ Ž ⭸ ⌫r . s ␾ Ž ⭸ L r .. By filling L r and Lir with ⌫r and a mirror copy ⌫r

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S. V. IVANOV

of ⌫r , respectively, we will get a spherical diagram ⌬ 1Ž0. from ⌬ 1Ž2 g .. Note that the set of all new cells of ⌬ 1Ž0. Žcontained in all ⌫r , ⌫r . can be obviously equipped with an involution j having properties ŽP1. ᎐ ŽP2.. Since H s ² A 5 R : is aspherical and so ⌬ 1Ž0. represents a trivial relation in the relation module of H, one can extend j to an involution j1U with properties ŽP1. ᎐ ŽP2. on the set of all cells in ⌬ 1Ž0.. Clearly, the restriction j1 of j1U on the set of cells in ⌬ 1Ž2 g . and so in ⌬ 1 also has properties ŽP1. ᎐ ŽP2.. Repeating this argument for all other diagrams ⌬ 2 , . . . , ⌬ t , we will obtain similar involutions j2 , . . . , jt . Their existence along with the involution iX on the set of cells ⌸ 1 , . . . , ⌸ m implies the existence of an involution i with properties ŽP1. ᎐ ŽP2. on the set of all cells in the original diagram ⌬. This contradiction disproves equality Ž6. and completes the proof of Theorem 1. Let us prove Theorem 2. Ža. This follows directly from Howie’s results wH1x. Žb. Without loss of generality we may assume that if x " 1Ux . 1 is a subword of W " 1 , then U / 1 in H. Since H is aspherical and hence torsion free, it can be readily seen from Klyachko’s proof wKx that any reduced spherical diagram over G has no cells corresponding to relator W. Hence, asphericity of G follows from that of H and Conjecture trivially holds. Žc. As above, we can assume that if x " 1Ux . 1 is a subword of W " 1, then U / 1 in H. If the number of occurrences of x " 1 in W is at most 2 then the Conjecture holds in view of Theorem 1 Žfor no zero divisor Z with
W. A. Bogley and S. J. Pride, Aspherical relative presentations, Proc. Edinburg Math. Soc. 35 Ž1992., 1᎐39. wBx S. D. Brodskii, Equations over groups and groups with a single defining relator, Uspekhi Mat. Nauk 35 Ž1980., 183. wGRx M. Gutierrez and J. G. Ratcliffe, On the second homotopy group, Quart. J. Math. Oxford Ž2. 32 Ž1981., 45᎐55.

AN ASPHERICITY CONJECTURE

wH1x wH2x wHbx wI1x wI2x wI3x wKx wLvx wLfx wLSx wOlx wPx wSx wWx

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