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Discrete Mathematics 67 (1987) 81-87 North-Holland

CONTEXT-FREE L A N G U A G E S A N D R A N D O M WALKS ON G R O U P S Wolfgang WOESS Institut fiir Mathematik und Angewandte Geometrie, Montanuniversitiit, A-8700 Leoben, Austria Received 13 March 1986 The Green function of an arbitrary, finitely supported random walk on a discrete group with context-free word problem is algebraic. It is shown how this theorem can be deduced from basic results of formal language theory. Context-free groups are precisely the finite extensions of free groups.

1. Introduction Let F be a finitely generated discrete group with unit element e and let/z be a probability measure on F, that is, #(x)i> 0 and ~ r # ( X ) = 1. We assume that # has finite support s u p p ( # ) = {x e r [ # ( x ) > 0}. The (right) random walk on F with law # is the Markov chain An, n - 0 , 1, 2 , . . . , with state space F and one-step transition probabilities Pr[Xn+l = y [An = x] = #(x-ly). The n-step transition probabilities are then given by Pr[X, = y [Xo = x] = U(")(x-ly), where #(") denotes nth convolution power, #(o)= 6e. See [12, 17,9, 11] for fundamental results on random walks on groups. Consider the collection ~3(z) = (Gx(z))x~r of complex functions defined by power series

Gx(z)=

U(")(x)z ", z

e C.

n--0

We call ~(z) the Green function or resolvent of #; functional analists usually consider ( l / z ) ~ ( l / z ) instead. A group is called context-free if for some finite set A of generators, the set of all formal words over A 0 A-1 which reduce to the group identity in F constitutes a context-free language (see Section 2 for precise explanations). Principal purpose of this note is to show how the following theorem can be deduced from basic results of formal language theory.

Theorem 1. If F is a context-free group, then ~(z) is an algebraic function of z and the coefficients of #: If A = {ax, • • . , aN} is a finite subset of F, then for each 0012-365X/87/$3.50 ~ 1987, Elsevier Science Publishers B.V. (North-Holland)

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x e F there is a nonconstant polynomial Px(3,1, . . . , AN, ~l) of complex variables with integer coefficients, such that Px(#(aOz, . . . , #(aN)Z, G~(z))=--O for any probability measure lz with supp(#) ~ A. An algebraic classification of all context-free groups is essentially given by [14], in combination with a recent result [5]. Theorem 2. A group is context-free if and only if it contains a finitely generated

free subgroup of finite index. For the case of a free group, long direct proofs of the algebricity of ~3(z) have been given in [1, 19]. Explicit calculations are known for "nearest neighbour" random walks [8, 1, 19, 21, 3]. In Section 2, concepts of formal language theory are explained and the above theorems are proved. In Section 3, as an application, a local limit theorem is deduced. It should be pointed out that to a large extent, this note has the spirit of a survey paper, revealing connections between different mathematical areas. Particular emphasis is laid on the explanation of those facts from formal language theory which a reader may need who is not familiar with this field.

2. Context-free groups

For the following concepts from formal language theory, confer [4, 10, 13] as basic references. Let X be a finite set. By X* we denote the free monoid over X, that is, the set of all finite words (strings) over X including the empty word 0, with concatenation of words as product rule. A formal language over X is an arbitrary subset L of X*. A context-free (CF) grammar is a quadruple ~ = (V, X, P, S), where V and X are finite disjoint sets (variables and terminal alphabet), S e V is the start symbol, and P ~ V x ( V U X)* is a finite set of production rules. We write T--->u if (T, u ) e P . For v, w e ( V U X ) * , v ~ w indicates that we can write v =vlTv2 and w = vluv2, where T--> u, Vle ( V t3X)* and v2 eX*. A rightmost derivation is a sequence v = w o, w l , . . . , Wk=W in ( V U X ) * , such that Wi-l==>wi; we then write v ~ w . The language over X generated by ~ is the set L(~g)= {w X * I S ~ w}. A context-free language is a language which can be generated by some CF grammar ca. For a given grammar ~, the ambig, uity degree dr(w) of w e X* is the number of different rightmost derivations T ~ w. If ds(w) = 1 for all w in L(qg), then ~ and L(qg) are called unambiguous. A grammar is called proper if it contains no production rules of the form T~--->T2, T~--->0 (T/e V) and

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each variable occurs in a production of some rightmost derivation S w (w e X*) (thus, in our definition of a proper grammar, we include that it is 'reduced' in the sense of [13]). In this case, dr(w) is finite for all w e X* and T ~ V. If L 6: {0} is an (unambiguous) CF language, then L - {0} can always be generated by an (unambiguous) proper grammar (see [10, Section 4.3]). Given a finite set A of generators of the group F, we associate with it a set A of "letters" of the same cardinality and a bijection of 0 :A--->A. Furthermore, we choose a set A -1 of "formal inverses", disjoint from A, with a bijection a ~-->a-1 from A onto A -1. We write ( a - l ) -~ = a. Let XA =A UA -~. Then 0 extends to a unique homomorphism from XA onto F, such that O(a-1) = O(a)-1 for each a in A. The word problem of F w r t A is the language WA(F) = {w eXit [ O(w) = e}. If this is a CF language for some generating set A, then F is called a context-free group. In [14], the following result is proved:

If WA(F) is context-free for some generating set A, then this is true for all finite generating sets of F. (1) A CF word problem for the free group is given by the Dyck language, see [4]. A formal power series (FPS) over Z with noncommuting indeterminates in X is a mapping f :X*---~ 2~, written in the form

f = ~ f(w)w. w~X*

The set 7/[[X]] of all these FPS carries a ring structure with usual commutative addition and the product

A polynomial is an element of Z[[X]] with finite support {w e X* If(w) 6: 0}. The ring of integer polynomials in X is denoted Z[X]. Now let c£ be a proper CF grammar. For each T in V, consider the FPS

&= WEX* E dr(w)w

in Z[[X]]

and the polynomial P r =pr(X; I1) defined by pr = ~

u

i n Z [ [ X U I1]].

T---~u

Key to our proof of Theorem 1 will be the following result [4, 13]:

fr =pr(X;fu, U ~ I1") for each T e V.

(2)

Here, the term on the right denotes the d e m e n t of Z[[X]] obtained as follows: each variable U which occurs in P r is replaced by the corresponding FPS fv, the resulting products and sums are then evaluated in Z[[X]]. In other words, we have a nontrivial system of algebraic equations for the FPS fr, T ~ V.

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Proof of Theorem 1. The subgroup of F generated by A is also CF by [14, Lemma 2] or by Theorem 2. Hence we may assume that A generates F. In view of Theorem 2 (an outline of its proof will be given below), it also follows from [14, Lemmas 2 and 3] that for a CF group F, any word problem WA(F) is deterministic. Without giving the automata-theoretic definition of this property (see [10, p. 139]), we use the fact that this in turn implies unambiguity of the word problem [10, p. 158]. Now let x e F. If x is not contained in A n for some n, then Gx(z) - 0 and we have nothing to prove. Otherwise we can choose Wx ~ ( A - l ) * such that O(wx)= x -1 (wx = 0 ifx = e). Consider the language LA(F, x) = {w e A * I O(w) =x}. Then we have Wx LA(F, x) = WA(F) fq wxA. As wxA is a prototype of a regular language (a language accepted by a finite automaton, see [10, Section 2.2]), wxLA(F, x) is an unambiguous CF language by [10, Theorem 6.4.1]. Hence, as Wx = 0 or Wx ~ A , LA(F, x) is also an unambiguous CF language. Now, there is an unambiguous proper CF grammar ~ - (V, A, P, S) that generates LA(F, x) - {0}; A = {al, • .. , aN}, V = {S = T 1 , . . . , TM}. Consider the system (2) (with terminal set A instead of X). Replace all ai and Tj with complex variables 3.i and rb, respectively. Then the formal polynomials P rj are transformed into ordinary polynomials p j - p j ( 3 . ~ , . . . , ) . N ; r / 1 , . . . , r/M) with integer coefficients, and the f ~ become ordinary power series fj = f j ( ) , l , . . . , AN) in commuting complex variables, analytic near the origin. (Note that, ~ being proper and unambiguous, dr(w) <~ 1 for each T e V, w ~ A*. Thus the fj converge at least for I~./I< 1/N.) In particular, (2) is transformed into a system of nontrivial ordinary algebraic equations

=pj(zl,..., zN; with solutions

(3)

ffj=~(3q,...,3.N), j = I , . . . , M . The particular form of this system allows the application of "elimination theory" to obtain an equation Qx(Al,. • . , )-N, 7/1) = 0

with solution

)/1 =~(~q, • • . , )-N),

where Qx is a nonconstant polynomial in complex variables ~ q , . . . , )-N, r/1 with integer coefficients: see [13, Section 16], where the algorithm is described in detail, see also [20, Section 31]. Now define P x ( A l , . . . , ;~N, 71)= Q , ( ~ . , , . . . , AN, T/- 6~(x)). As f~ =fs =

~,

weLA(r,x)--{O}

W

by unambiguity, the substitutions Z~ = p(ai)z (ai= O(a~)) transform ]1 into a power series in one complex variable z, whose coefficient of z n is ~.

l~(ai,) " " " lz(ai.)= /~(")(x),

n t> 1.

ail" "'din ~'X

That is, ~(lz(al)z, . . . , l~(aN)z) = G , ( z ) - 6,(x), and the proof is completed.

[]

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Proof of Theorem 2. We give a brief outline of the proof. Confer [14] for all unexplained notions. The main result of [14] is the following:

A finitely generated group has a free subgroup of finite index if and only if it is context-free and accessible. Here, accessible means that F satisfies a certain finite-chain condition regarding decompositions in the sense of [18]. Without going into detail, we use a recent theorem [5]:

All finitely presented groups are accessible. A group is finitely presented, if there is a finite generating set A and a finite subset R of WA(F), such that the sets of pairs R x {0} and WA(F) x {0} generate the same congruence relation in the free monoid XA. Thus what remains is to show that every CF group is finitely presented. This follows from the "triangulation theorem" [14, Theorem I]: choose a finite generating set A of F not containing e. The group being CF, its Cayley graph wrtA has the property that every closed path admits a triangulation with labels on the triangle sides which are words in XA not exceeding length K < ~. Thus only finitely many different words in XA can be obtained by concatenating the labels around each of the triangles which may occur in some triangulation. These words together with the "trivial relations" aa -1, a eXA, constitute a set R for a finite presentation of F. []

3. Application and remarks As an application of Theorem 1, one can prove a local limit theorem. We say that # is irreducible and aperiodic if supp(#) generates F as a semigroup and is not contained in a proper coset of some normal subgroup of F, respectively. These definitions correspond to the usual notions of irreducibility and aperiodicity of the Markov chain (X~). In this case, the radius of convergence r(#) of the power series Gx(z) does not depend on x e F, 1 ~
for all x e F.

(4)

n---~O0

Theorem 3. If F is infinite and context-free and # is irreducible and aperiodic, then .

. n-q

as

where v is a positive function on F and q is a positive rational number not depending on x ~ F.

Proof. By Theorem 1, there are finitely many singularities of Gx(z) on the circle of convergence Izl = r(•), which are all algebraic and have rational weight. Let 1 - q ( x ) be the maximal weight occurring at these singularities. Applying

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Darboux's Theorem (see [2, Theorem 4]), the combination of (4) with an observation in [2, p. 499] yields that z = r o t ) is the only singularity on the circle of convergence which has weight 1 - q ( x ) . Now Darboux's Theorem implies

lz(n)(x)~v(x)'r(Iz)-n'n -q(x) as n---~ ~, where v(x)>0. By irreducibility, there is some k such that Iz~k)(x)= C > 0 . Hence, l~n+k)(x) >-Cla~n)(e) for each n, and !a~)(x)/t~)(e) >1Cla~)(x)/la~+k)(x), which is asymptotically bounded below according to (4). Similarly, la~n)(x)/l~)(e) is bounded above, as n---~ 0o. Hence it must be q(x)= q(e)=q for all x e F. As F is infinite, r(~)" •/~")(x) tends to zero [7] and q must be positive. [] If F is a finite extension of 7/, then q = ½ and r(/z) may assume value one. In all other cases, F is nonamenable, r(#) > 1 and q > 1 [12], and it is conjectured that q = a2 always. This is true in all known cases [16, 15, 8, 21]. (In a private communication to the author, T. Steger has in fact announced a proof, that q = 3 always on free groups.) Suppose we have found free generators of a free subgroup 0: of finite index of F and a set of representatives of the I:-cosets in F. Then there are algorithms to find a proper context-free grammar for any of the languages LA(F, x) - {0} defined in the proof of Theorem 1. Thus system (3) can be found, and the polynomial Px can be constructed via the algorithm described in [13, Section 16]. However, for numerical computation of Gx(z) (at least for small z) it may be more convenient to apply the fixed-point method to system (3) (compare [4]).

Conjecture. Theorem 1 determines the class of context-free groups. Acknowledgments The author acknowledges stimulations that came from Peter Gerl, Chris Godsil and Alessandro Fig~-Talamanca, and a helpful conversation with W. Kuich, who kindly provided the author with a preprint of Section 16 of his book with A. Salomaa [13].

References [1] K. Aomoto, Spectral theory on a free group and algebraic curves, J. Fac. Sci. Univ. Tokyo, Sec. IA, 31 (1984) 297-317. [2] E.A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974) 485-515. [3] D.I. Cartwright and P.M. Soar&, Random walks on free products, quotients and amalgams, Nagoya Math. J., 102 (1986) 163-180. [4] N. Chomsky and M.P. Schiitzenberger, The algebraic theory of context-free languages, in: P. Brattort and D. Hirschberg, eds., Computer Programming and Formal Systems (North-Holland, Amsterdam, 1963) 118-161.

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[5] M.J. Dunwoody, The accessibility of finitely presented groups, Inventiones Math. 81 (1985) 449-457. [6] P. Gerl, Wahrscheinlichkeitsmage auf diskreten Gruppen, Archiv Math. 31 (1978) 611-619. [7] P. Gerl, Ein Konvergenzsatz fiir Faltungspotenzen, in: Probabability Measures on Groups, Lect. Notes in Math. 706 (Springer, Berlin, 1979) 120-125. [8] P. Gerl and W. Woess, Local limits and harmonic functions for nonisotropic random walks on free groups, Probab. Th. Rel. Fields 71 (1986) 341-355. [9] Y. Guivarc'h, M. Keane and B. Roynette, Marches Al6atoires sur les Groupes de Lie, Lect. Notes in Math. 624 (Springer, Berlin, 1977). [10] M.A. Harrison, Introduction to Formal Language Theory (Addison-Wesley, London, 1978). [11] V.A. Kaimanovich and A.M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983) 457-490. [12] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92, (1959) 336-354. [13] W. Kuich and A. Salomaa, Semirings, Automata, Languages (Springer, Berlin, 1985). [14] D.E. Muller and P. E. Schupp, Groups, the theory of ends, and context-free languages, J. Comp. Syst. Sci. 26 (1983) 295-310. [15] M.A. Picardello, Spherical functions and local limit theorems on free groups, Ann. Math. Put. Appl. 33 (1983) 177-191. [16] S. Sawyer, Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete 42 (1978) 279-292. [17] F. Spitzer, Principles of Random Walk (Van Nostrand, Princeton, 1964). [1[] J. Stallings, Group Theory and Three-dimensional Manifolds (Yale Univ. Press, New Haven, 1971). [19] T. Steger, Harmonic analysis for an anisotropic random walk on a homogeneous tree, Ph.D. thesis, Washington Univ., St. Louis, 1985. [20] B.L. van der Waerden, Einfiihrung in die algebraische Geometrie (Springer, Berlin, 1939). [21] W. Woess, Nearest neighbour random walks on free products of discrete groups, Boll. Un. Mat. It., 5-B (1986) 961-982.