Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
On Growth in Group Theory Rostislav I. Grigorchuk Moscow Institute of Railway Transportation Engineers ul. Obraszowa 15, Moscow, USSR 1. General The concept of growth appeared in group theory in the mid-fifties and now it plays an increasing role in this theory. Among different notions concerning the idea of growth in group theory the most important and successful is the notion of growth of a finitely generated (f.g.) group. We start with the main definition, then enumerate the results concerning this definition and at the end we shall touch some other aspects of growth in group theory. 1.1 The Main Definition Let G be a finitely generated (f.g.) group with generator system A = {al9..., am}, and let d(g) be the length of the element g e G with respect to the system A, in other words the minimal number k such that g can be represented in the form g= all...a%, fi/ = ± 1, j=l,...,k. The growth function of the group G with respect to the system A is the function y(n) = card{# e G; ô(g) < n}. As y(n) depends on the generator system it is convenient to introduce an equivalence relatiop on the set of growth functions: yM ~ y2(n)oiC(yM < y2(Cn)&y2(n) < 7l(Cn)) and the preordering relation: yi(n)<y2(n)olC(yi(n)<y2(Cn)). The equivalence class [y/(n)] is an invariant of the group G. We shall call it the growth degree of the group G, Obviously the growth degrees of a group and of any of its subgroup of finite index coincide. The partially ordered (with respect to <) set 2B of the growth degrees of f.g. groups will be the main object of our consideration. Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990 © The Mathematical Society of Japan, 1991 326 Rostislav I. Grigorchuk 1.2 Examples If G = Zd is the free abelian group of rank d, then y(n) ~ nd. If G = Fm is an (absolute) free group of rank m > 2 then y(n)coen. Thus the growth of a group can be polynomial of any degree d e Z+ and can be exponential. 1.3 Connection with Geometrical Growth The notion of the growth function was introduced by A.S. Schwarze [1] and independently by J. Milnor [2]. This notion is a combinatorial form of the geometrical growth notion explored by VA. Efremovich [3]. Let M be a Riemannian manifold. Its growth at infinity is characterized by the growth when r -> oo of the function v(r) = Vol(Bx(r)), expressing the volume of the geodesic ball of radius r with center at the point xe M. The examples of the Euclidean space R d and hyperbolic space H d show that this function in the first, case grows as the polynomial rd and in the second case as the exponential function er. On the other hand the growth function yG(n) of the f.g. group G expresses the volume of the ball Bx(n) of radius n with center at the unit element 1 e G if the group G is supplied with the metric d(g, h) = d(g~1h) and Haar measure. The connection between geometric and algebraic growths can be also illustrated by A.S. Schwarzc's theorem [1]: if the manifold M is a universal covering manifold of the compact Riemannian manifold M then "MW ~ y*l(M)(r)In other words the growth of the covering manifold M is the same as the growth of the fundamental group nx(M) of the manifold M. 2. Milnor's Problems 2.1 Some Results Intensive investigation of the growth functions of f.g. groups began after J. Milnor's paper [2] was published. A series of important results was obtained within a short time interval. Here are some of them, 2.1.1 A f.g. nilpotent group G has polynomial growth (Wolf [4]) like hd, where d= YJkvmkQ(Gk/Gk+1) k and {GjJ is the lower central series of the group G (Bass [5]). 2.1.2 A f.g. solvable group has exponential growth if it is not virtually nilpotent (Wolf [4], Milnor [6]). (Virtually nilpotent means: contains a nilpotent subgroup of finite index). On Growth in Group Theory 327 2.1.3 A f.g. linear group has exponential growth if it is not virtually nilpotent (Tits [7]). 2.1.4 A free periodic Burnside group B(m,n) = (au...,am\gn = l,geG} has exponential growth when m > 2 and n > 665 is odd (Adyan [8]). In 1968 J. Milnor [9] proposed a number of problems concerning growth of groups. 1) "Is the growth function y(n) necessarily equivalent either to a power of n or to the exponential function TT 2) In particular, is the growth exponent dl=lim!ogyW »-oo (1) l o g 77 always either a well-defined integer or infinitelyl For which groups is d < oo? (A possible conjecture would be that d < oo if and only if G contains a nilpotent subgroup of finite index)". (Division of Milnor's problem into Parts 1) and 2) has been made by me). As a conjecture Problem 1) was formulated by Wolf [4] and Bass [5]. Conjecture: "A f.g. group of non-exponential growth is virtually nilpotent". 2.2 Two Definitions A group whose growth function is not majorized by any function nd and is not equivalent to the function e" is called an intermediate growth group. And a group G is called a group of subexponential growth if y(n) •< en (strong inequality), in other words, if lim */yfr) = 1 II-+00 (the limit exists due to the semi-multiplicativity of the function y(n)). 3. Answer to Milnor's Problem 1) Milnor's problem 1) was solved negatively in [10,11, 12]. s 3.1 On the Construction of Intermediate Growth Groups The simplest example of such a group is the group /"from the paper [13] which was constructed as a simple example of a f.g. infinite periodic group. T is defined as the group of transformations of the interval [0,1] from which rational points of the formfc/2",n — 1, 2, ..., 0 < k < 2" are removed. This group is generated by four transformations a, b, c, d, where a is the permutation of the halves of the interval [0, 1] and b, c, d, are defined with the help of infinite periodic words 328 Rostislav I. Grigorchuk P P a\ 1 0 1 0 1 d| c | P T P ... 1 _ | _ | 0 Ì U 2 3 7 4 1 g ... . . 1 P 1 1 I I J...1 b\ T A U i 1 T P P . . . 1 2 . 1 1 I 7 4 g ... 1 1 | l Fig. 1. (P denotes the permutation of the halves of the interval, T is the identity transformation). PPTPPT... PTPPTP... TPP TPP ... as in Figure 1. We constructed uncountably many intermediate growth groups on the basis of this example (see [10-12]). 3.2 The Main Idea of the Growth Function yr(n) Upper Estimation The group T contains a subgroup H of finite index, which allows an embedding W in the direct product of eight copies of the group T: H^H<rxrxrxrxrxrxrxr, and H is a subgroup offiniteindex in 7"8. This embedding has the following property: if g e H and ^(0) = (0i,02, •••>0 8 )ei? then where k is some constant less than 1. Hence y(«)<cSr( W l )...yK) (2) (C is some constant), where the summation in (2) is taken over those sets of nonnegative integers nl9..., n8 for which the inequality 8 Y nt < kn i=l is valid. If we assume that lim 7y(n) = X > 1 On Growth in Group Theory 329 then we shall obtain a contradiction k < Xk from (2). So r has sub-exponential growth. The fact that the function y(n) grows faster than any polynomial results, for example, from the following consideration. The group r is commensurable with its square, in other words the groups T and r x T contain isomorphic subgroups of finite index. Hence y(n) satisfies the so-called ^-condition y(n) ~ y2(n), which results in the lower estimate by some function of the type e"p, where ß > 0. However, the lower estimate of the function yr(n) can be obtained directly. It is proved more precisely in [11] that e^x < yr(n) < en\ where a — log 32 31. This obviously gives a negative answer to Milnor's problem 1). On the basis of a construction which generalizes this example we proved the following theorem (see [11, 14]). Theorem 1. The set of growth degrees of f.g. groups has the cardinality of the continuum. It contains a chain of the cardinality of the continuum, and an antichain of the cardinality of the continuum. I would like to stress that we are speaking precisely about intermediate degrees of groups in this theorem. Another example of a group of intermediate growth was constructed by J. Fabrykowski and N. Gupta [15]. 3.3 Some Properties of Intermediate Growth Groups Intermediate growth groups can be constructed both in the class of p-groups (p is any given prime) and in the class of torsion free groups [11, 12]. The intermediate growth groups known up to this moment belong to the class of residually finite groups. Moreover, the groups from the papers [10-12] are residually-p-finite. The periodic groups constructed in [11,12] have the property that all their proper factor groups are finite. At the same time in [16] for any prime p, & f.g. p-group of intermediate growth having continuum nonisomorphic factor groups was constructed. The torsion free group of intermediate growth constructed in [12] admits an invariant linear ordering. The intermediate growth groups from [11, 12] are not finitely presentable. There exist recursively presentable (by generating elements and defining relations) intermediate growth groups having solvable word problem. At the same time there are analogous groups with unsolvable word problem [11, 14]. 4. On Groups with Polynomial Growth An exhaustive answer to Part 2) of Milnor's problem was given in the papers [17, 18] and Milnor's conjecture suggested in this part was confirmed. Note that a 330 Rostislav I. Grigorchuk more general result was obtained in [18], although this paper is mainly devoted to proving Gromov's theorem by methods of nonstandard analysis. 4.1 On Gromov's Theorem Gromov's theorem proved in [17] gives the description of f.g. groups whose growth functions admit polynomial estimate y(n)<nd, (3) (d is some constant). This theorem states that a group G is virtually nilpotent if the estimate (3) is valid for its growth function. Hence the estimate (3) results in an equivalence y(n) ~ nd for a suitable deZ+. The proof of this theorem is based on geometric considerations. The left invariant metric d(g, h) = d(g~1h), g,hsGis built on the group G and the sequence Xn = (G,±d), neN of metric spaces is considered* Gromov proved that from this sequence one can extract some subsequence which converges in an exactly defined way to some metric space X^ if the condition (3) is valid. A homomorphism G -> Isomf-X^) from the group G into the isometry group of X^ arises and Isorn^^) is a Lie group with a finite number of connected components. Some simple algebraic considerations complete the proof of this theorem. 4.2 On L. van den Dries and A. Wilkie's Theorem The main result of [18] is: if the inequality y(n) < Cnd is valid on some infinite subset N 0 ç N of the set of natural numbers then the group G is virtually nilpotent. From this statement if follows that the limit (1) (finite or infinite) always exists, and also d < oo if and only if the group G is virtually nilpotent. 5. On the Growth of Cancellation Semigroups The notion of the growth function of a f.g. semigroup is defined as in the case of a group. The growth of a semigroup can be very strange even in the case when the condition (3) holds true. But if we consider the class of semigroups with left and right cancellation laws then in this case the statement similar to Gromov's theorem is valid. 5.1 The Nilpotency of Semigroups Due to A.I. Malcev Let x, y, Çl9 Ç2, ..., Çn, ... be symbols denoting variables running through a semigroup S. Let us denote X0 = x, Y0 = y and after that by induction On Growth in Group Theory X„+i = Xnç„Yn, 331 Y)l+1 = YnçnXtr The semigroup S whose elements satisfy the identity Xn = Yn for some n is called nilpotent. A.I. Malcev proved [19] that the classical nilpotency identity for groups is equivalent to the semigroup identity X„ = Yn for corresponding n. 5.2 On Cancellation Semigroup of Polynomial Growth Let S0 £ S be a subsemigroup. We shall say that S0 has finite index in S if there exists a finite subset K ^ S such that for any se S there exists k e K for which ske S0. Theorem 2. A f.g. cancellation semigroup S has polynomial growth (Condition (3)) if and only if S contains a nilpotent subsemigroup of finite index. The proof of this theorem is based on the remark that for a cancellation semigroup of subexponential growth the Ore condition is satisfied and this semigroup possesses a group Gs = S - 1 S of left quotients. The estimate (3) on the growth function of the semigroup S makes it possible to give a polynomial type estimate on the growth function of the group Gs after detailed analysis. Then Gromov's theorem can be applied. 5.3 Some Remarks In [20] it was proved that if a semigroup S with cancellation has polynomial growth, then the group Gs of left quotients also has polynomial growth of the same degree. It would be interesting to construct examples of semigroups with cancellation of subexponential growth for which there is a jump of growth degree in the diagram S^GS. If one could construct a semigroup S with cancellations of subexponential growth such that Gs has exponential growth, then Problem N12 from [21] will be positively solved. 6. On Lacunae in the Set of Growth Degrees of Residuali} Nilpotent Groups Recall that a group G is residually-p if for any nonunit element g e G there exists a finite p-group K and a homomorphism cp:G -> K such that cp(g) ^ 1. Theorem 3 [22]. Let p be any prime, let the f.g. group G be residually-p and its growth function satisfy the estimate (3). Then G is virtually nilpotent and so has polynomial growth. 6.1 Proof of Theorem 3 Let Fp be a finite prime field, F p [G] the group algebra, A < F p [G] the augmentation ideal, in other words the ideal generated by elements of the form g - 1, g e G, gr(G) 332 Rostislav I. Grigorchuk Polynomial degrees Lacunae 1 n n2 ... nd ... Chain e^1 . en exp. degree Antichain Fig. 2 the associative graded algebra defined by means of the powers of the augmentation ideal gr(G)=@A„ = ©A"/A»+\ n=0 n=0 let fGtp(t) be the Hilbert-Poincaré :aré series of the algeb: algebra gr(G): where an(G) = dim F An. It follows from Lazard's Theorem 3.11 [23] that either the sequence an(G) has polynomial growth and then the p-completion G is p-analytic or an(G)>e^. Besides if y (ri) is the growth function of the group G with respect to any system of generators then the inequality an(G) < y(n), n = 1, 2 , . . . is valid. Using Lazard's result and Condition (3) we obtain that the growth of coefficients an(G) when n -» oo is polynomial and so the p-completion G is analytic. It follows from Tit's theorem [7] that either G contains a free subgroup with two generators or G is a virtually solvable group. But the first case is impossible due to limitation on the growth, and in the second case, we can conclude that G is virtually nilpotent due to results of [4, 6]. Recently A. Lubotzky and A. Mann [24] have pointed out that in Theorem 3 the assumption that G is residually-p can be changed by the assumption that G is residually nilpotent. 6.2 On the Scale of Growth Degrees of Residually-/? groups Let p be any prime and denote by Wp the set of growth degrees of f.g. residually-p groups. Theorem 1 formulated above for the whole class of groups is also true for every set 2Bp. So, due to Theorem 3 the structure of the partially ordered set 2Bp can be presented in the form as in Figure 2. 7. On the Generating Series of Growth Function Many questions of a theoretical and applied character require more detailed investigation of the asymptotic behavior of the growth function y (ri) than up to the On Growth in Group Theory 333 equivalence ~, defined above. For this it is sometimes useful to connect the generating series m = t y(r>)t" N= 0 to the function y(n). 7.1 Some Cases When the Function T(t) is Rational for any System of Generators i) The group is virtually abelian (Benson [25]) ii) The group is a cocompact group of isometries of the hyperbolic space WLd (Cannon [26]) iii) The group is hyperbolic (Gromov [27]). For many nilpotent groups the series the r(t) also represents a rational function (see for example [28]). At the same time the conjecture that for any f.g. nilpotent group and any of its finite generating set the function T(t) is rational proved to be false. Namely F. Grunewald proved that for the nilpotent group G = <fll5 a2, bu b2, z\\_au foj = \_a2,fc2]= z,ze Z(G), f<*u a 2 ] = [Pu M = [ßu hl = \ßi> M = O (Z(G) is the centre of the group G and the function T(t) defined using the system of generators {al9 a2, bx, b2, z}), the generating series r(t) is not a rational function. 7.2 Information on the Growth Functions of Nilpotent Groups Let G be a nilpotent group, y (ri) the growth function of G with respect to any system of generators, d the power of polynomial growth of G. Then a) there exists the limit lim^ =C n -»oo n (Pancu [29]), b) the estimate y(ri) = Cnd + 0(nd~112) is valid (Grunewald, unpublished). It is interesting to find new examples of rationality of the growth function T(i) and also to obtain another information about the generating series of growth functions. 8. Some Problems and Conjectures The theory of growth degrees of groups has been developed for more than three decades and has already accumulated a lot of unsolved problems. Let me formulate some of them. 334 Rostislav I. Grigorchuk 8.1 (A well known problem.) Is there a f.g. group of intermediate growth with a finite set of defining relations? 8.2 Is it true that if the growth function of a f.g. group G grows slower than the function e^n, then G is virtually nilpotent? 8.3 Does there exist a f.g. group whose growth function is equivalent to the function 8.4 Is it true that any group of subexponential growth is residually finite? 8.5 To find the asymptotic behaviour of the growth function of the group T from [13] (conjecture is that y (ri) ~ e*" in this case). 8.6 Is it true that for any f.g. nilpotent group the function T(t) is meromorphic or even algebraic? (F. Grunewald). Let G = (A\r= l(reÄ)) be f.g. and 3 be the Cayley graph of G with respect to the system of generators A. A spanning tree T s ^ i s regular if it is defined by a finite automaton. A spanning tree T is minimal if every word in T is the shortest. 8.7 Conjecture. G has a rational growth function if and only if 3 has a minimal regular spanning tree (Machi, Schupp). 9. Other Aspects of Growth in Groups 9.1 Cogrowth Let a f.g. group G be realized as a factor group FJH of the free group Fm of rank m, H < Fm2L normal subgroup, the elements of which will be further considered as reduced words over the basis of Fm. Denote by h(ri) the number of words of length <n in H. Obviously h(ri) <, 2m(2m - l) w_1 . Let 00 aH = limsup y/h(n), ii-»oo J^(t) = £ h(ri)tn. n=0 The value a H which appeared in [30] is called the. growth exponent of the normal subgroup H and belongs to the interval (y/2m — 1, 2m — 1]. In [31] the following amenability criterion was proved. Theorem 4. The group G = Fm/H is amenable if and only if ocH = 2m — 1. The function J^(t) and growth exponent aH can be defined also for any subgroup H < Fm. In [31] it was proved that if if is a f.g. group then J f (t) is a rational function which can be effectively calculated if the system of generators of H is known. On Growth in Group Theory 335 9.2 Subgroup Growth Let G be a f.g. group, and let Qn(G) be the (finite) number of its subgroups of index n. The problem of investigation of the asymptotic behavior of the sequence g(n) when n -» oo has been drawing more and more interest. For example in [32] F. Grunewald, D. Segal and G. Smith investigated in detail the Dirichlet series EG(s) = £ft,(G)n-' of the sequence gn(G) for the case when G is nilpotent. On the other hand A. Lubotzky and A. Mann [24] showed that if G is a residually nilpotent f.g. group then Qn(G) has polynomial growth if and only if G is solvable offiniterank. The general case of a residually finite group is still open. See [24] for other references and comments. 9.3 On the Growth of Graded Algebras Associated to Groups During investigations of groups different graded algebras associated to these groups arise. For example the associative graded algebra gr G was defined above. Studying the growth of algebras and in particular graded algebras associated to groups is important from different points of view and is connected with different applied aspects of algebra [33-37]. 9.4 On the Asymptotic Behavior of Random Walks on Groups Investigation of random walks on groups was started by H. Kesten in [38]. It is interesting to investigate the asymptotic behavior when n -^ oo of the probability Plf i(/,) of returning to the unit element after n steps [38, 39], to calculate the entropy of random walks [40], to describe Martin's and Poisson's boundaries [41, 42] and so on. 10. Applications 10.1 To the Theory of Invariant Means The definition of an invariant means and the corresponding notion of amenable groups belong to J. von Neumann [43]. Finite and abelian groups are amenable. The class of amenable groups is closed under operations of extension and inductive limit. Let us denote the class of amenable groups by AG, the class of groups without free subgroups with two generators by NF, and the class of elementary amenable groups constructed by J. von Neumann by EG, i.e. the minimal class of groups containing finite groups, abelian groups and closed under the operations of extension and inductive limit. The following imbeddings EG £ AG £ NF hold true. 336 Rostislav L Grigorchuk In 1957 M. Day [44] proposed the following two problems: (1) Is it true that AG = NF? (2) Is it true that AG = EG? Some mathematicians ascribe Problem (1) to J. von Neumann but there is no written confirmation of the fact that it was he who suggested this problem. In 1979 to disprove the conjecture AG = NF I suggested A.Yu. Olshanskii to apply the amenability criterion (Theorem 4) to the groups constructed by him at that moment for the solution of some famous problems of group theory. This suggestion was realized in the short paper [45]. A little later S.I. Adyan [46] also applied the amenability criterion to prove that free periodic Burnside groups B(m,ri)are nonamenable when m > 2 and n > 665 is odd (the conjecture that these groups are nonamenable was expressed by S.I. Adyan in 1975). The negative answer to the problem (2) was given in [11]. Theorem 5. There exist uncountably many amenable groups not belonging to the class EG. Hence the class AG of amenable groups is much more than the class EG of elementary amenable groups. Theorem 5 is an easy corollary to the existence of a continuum of nonisomorphic intermediate growth groups. Using intermediate growth groups we managed [48] to disprove one of Rosenblatt's conjectures [47] (see also [21], Problem 11) about the so-called superamenable groups. 10.2 To Riemannian Geometry The results concerning the growth functions of groups are widely used in the classification theory of Riemannian manifolds up to quasi-isometries, in the theory of foliations, in the investigation of Laplace operator on Riemannian manifolds (see [49, 50]),.... 10.3 Other Applications Information concerning the growth functions of groups is used in many, sometimes unexpected branches of mathematics: in the theory of random walks [38, 39], ergodic theory [50, 52], the theory of finite automata [20] and so on. In my lecture I did not touch upon many other aspects of applications of the notion of growth in group theory, many of which at this moment are only in their initial stage of development. References 1. Schwarze, A.S.: A volume invariant of coverings. Dokl. Ak. Nauk USSR 105 (1955) 32-34 2. Milnor, J.: A note on curvature and fundamental group. J. Diff. Geom. 2 (1968) 1-7 On Growth in Group Theory 337 3. Efremovic, V.A.: The proximity geometry of Riemannian manifolds, Uspekhi Math. Nauk 8 (1953) 189 (Russian) 4. Wolf, J.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Diff. Geom. 2 (1968) 424-446 5. Bass, H.: The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. 25 (1972) 603-614 6. Milnor, J.: Growth in finitely generated solvable groups. J, Diff. Geom. 2 ( 1968) 447-449 7. Tits, J.: Free subgroups in linear groups. J. Algebra 20 (1972) 250-270 8. Adyan S.I.: The Burnside problem and identities in groups. Nauka, Moscow 1975 (Russian) [English transi.: Proc. Steklov Inst. Math. (1970) 142] 9. Milnor, J.: Problem 5603. Amer. Math. Monthly 75 (1968) 685-686 10. Grigorchuk, R.I.: On Milnor's problem of group growth. Dokl. Ak. Nauk SSSR 271 (1983) 31-33 (Russian) [English transi: Soviet Math. Dokl. 28 (1983) 23-26] 11. Grigorchuk, R.I.: The growth degrees of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR. Ser. Math. 48 (1984) 939-985 (Russian) [English transi.: Math. SSSR Izv. 25 (1985)] 12. Grigorchuk, R.I.: On the growth degrees of p-groups and torsion-free groups. Math. Sbornik 126 (1985) 194-214 (Russian) [English transi.: Math. USSR Sbornik 54 (1986) 185-205] 13. Grigorchuk, R.I.: On the Burnside problem for periodic groups. Funct. Anal. Prilozen. 14 (1980) 53-54 [English transi.: Funct. Anal. Appi. 14, 41-43] 14. Grigorchuk, R.I.: A groups with intermediate growth function and its applications. Second degree doctoral thesis, Moscow 1985 15. Fabrykowski, J., Gupta, N.: On groups with subexponential growth functions. J. Ind. Math. Soc. 49 (1985) 249-256 16. Grigorchuk, R.I.: The construction of intermediate growth groups having continuum factor groups. Algebra and Logic 23 (1984) 383-394 17. Gromov, M.: Groups of polynomial growth and expanding maps. Pubi. Math. IHES 53 (1981) 53-73 18. Van den Dries, L., Wilkie, A. J.: Gromov's theorem on groups of polynomial growth and elementary logic. J. Alg. 89 (1984) 349-374 19. Malcev, A.I.: Nilpotent semigroups. Uchen. Zap. Ivanovsk. Ped. In-ta 4 (1953) 107-111 (Russian) 20. Grigorchuk R.I.: On the cancellation semigroup of polynomial growth. Matem. Zametki 43 (1988) 305-319 (Russian) 21. Wagon, S.: The Banach-Tarski Paradox. Encyclopedia of mathematics and its applications 24. Cambridge University Press 1985 22. Grigorchuk, R.I.: On the Hilbert-Pouncare series of the graded algebras associated to groups. Mat. Sbornik 180 (1989) 307-225 23. Lazard, M.: Groupes analytiques p-adiques. Pubi. Math. IHES 26 (1965) 389-603 24. Lubotzky, A., Mann, A.: On groups of polynomial subgroup Growth. Preprint 1990, pp. 1-22 25. Benson, M.: Growth series of finite extentions of Zd are rational. Invent, math. 73 (1983) 251-269 26. Cannon, J.: The combinatorial structure of cocompact discrete hyperbolic groups. Geometricae Dedicata 16 (1984) 123-148 27. Gromov, M.: Hyperbolic groups. Essays in group theory. MSRI Publications 8 (ed. S.M. Gersten). Springer, Berlin Heidelberg New York 1987, pp. 75-263 28. Shapiro, M.: A geometrical approach to the almost convexity and growth of some nilpotent groups. Preprint 1989 29. Pancu, P.; Croessance des boules et des geodeseques fermées dans les nilvarietes. Ergodic theory and dynamic systems, 1983, pp. 415-445 338 Rostislav I. Grigorchuk 30. Grigorchuk, R.I.: Symmetrical random walks on discrete groups. Uspekhi Math. Nauk XXXII (1977) 217-218 (Russian) 31. Grigorchuk, R.I.: Symmetrical random, walks on discrete groups. In: Multicomponent random systems (ed. R.L. Dobrushin, Ya.G. Sinai). Nauka, Moscow 1978, pp. 132-152. [English transi.: Advances in probability and related topics, vol. 6. Marcel Dekker 1980, pp. 285-325] 32. Grunewald, F.J., Segal, D., Smith, G.C.: Subgroups of finite index in nilpotent groups. Invent, math. 93 (1988) 185-233 33. Gelfand, I.M., Kirillov, A.A.: Sur les corps lie's aux algebres enveloppantes des algebres de Lie. Pubi. Math. IHES 31 (1966) 5-19 34. Krause, G.R., Lenagan, T.H.: Growth of algebras and Gelfand-Kirillov dimention. Pitman Advanced Publishing Program, Boston London Melbourne 1985 35. Babenco, LC: The problems of growth and rationality in Algebra and Topology. Uspekhi Math. Nauk 41 (1989) 95-142 36. Grigorchuk, R.L: On the topological and metrical types of a regular covering surfaces. Izv. Akad. Nauk SSSR. 54 (1990) 498-536 (Russian) 37. Bereznyi, A.E.: Discrete sub-exponential groups Zap. Nauch Sem. LOMI, 123 (1983) 155-166 38. Kesten, H.: Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959) 336-354 39. Kesten, H.: Full Banach mean values on countable groups. Math. Scand. 7 (1959) 146-156 40. Kaimanovich, VA., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11 (1983) 457-490 41. Dynkin, E.B., Maljutov, M.B.: Random walks on groups with a finite number of generators. Soviet Math. Dokl. 2 (1961) 339-402 42. Furstenberg, H.: Random walks and Discrete subgroup of Lie groups. Adv. Probab. Related Topics 1, 3-63. Dekker, New York (1971) 43. Von Neumann, J.: Zur allgemeinen Theorie des Masses. Fund. Math. 13 (1929) 73-116 44. Day, M.: Amenable semigroups. 111. J. Math. 1 (1957) 509-544 45. Olshanskii, A.Yu.: On the problem of the existence of an invariant mean on a group. Uspekhi Mat. Nauk 35 (1980) 1165-1166 46. Adyan, S.I.: Random walks on free periodic groups. Izv. Akad. Nauk SSSR, Ser. Mat. 46 (1982) 1139-1149 (Russian) [English transi.: Math. USSR Izv. 21 (1983)] 47. Rosenblatt, J.: Invariant measures and growth conditions. Trans. Amer. Math. Soc. 193 (1974) 33-53 48. Grigorchuk, R.L: Supramenability and problem of embedding a free semigroups. Funkt. Anal. Priloz. 21 (1987) 74-75 (Russian) 49. Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comm. Math. Helv. 56 (1981) 581-598 50. Varopoulos, N : Brownian motion and transient groups. Ann. Inst. Fourier (Grenoble) 34 (1984) 243-269