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On Growth of Grigorchuk Groups On growth of Grigorchuk groups Roman Muchnik, Igor Pak March 16, 1999 Abstract We present an analytic technique for estimating the growth for groups of intermediate growth. We apply our technique to Grigorchuk groups, which are the only known examples of such groups. Our esti- mates generalize and improve various bounds by Grigorchuk, Bartholdi and others. 2000 Mathematics Subject Classification: 20E08, 20E69, 68R15 1 Introduction In a pioneer paper [3] R. Grigorchuk discovered a family of groups of inter- mediate growth, which gave a counterexample to Milnor's Conjecture (see [3, 7, 13]). The groups are defined as groups of Lebesgue-measure-preserving transformations on the unit interval, but can be also defined as groups act- ing on binary trees, by finite automata, etc. While Grigorchuk was able to find both lower and upper bounds on growth, there is a wide gap between them, and more progress is desired. In this paper we present a unified approach to the problem of estimating the growth. We introduce an analytic result we call Growth Theorem, which lies in the heart of our computations. This reduces the problem to combi- natorics of words which is a natural language in this setting. We proceed to obtain both upper and lower bounds in several cases. This technique simplifies and improves the previous bounds obtained by various ad hoc ap- proaches (see [2, 4, 5]). We believe that our Growth Theorem can be also applied to other classes of groups. Let G be an infinite group generated by a finite set S, S = S−1, and let Γ be the corresponding Cayley graph. Let B(n) be the set of elements g 2 G at a distance ≤ n in graph Γ. The growth function of G with respect to the set of generators S is defined as γG(n) = jB(n)j. 1 We say that a function f : N ! R is dominated by a function g : N ! R, denote by f 4 g, if there is a constant C > 0 such that f(n) ≤ g(C · n) for all n 2 N. Two functions f; g : N ! R are called equivalent, denoted by f s g, if f 4 g and g 4 f. It is known that for any two finite sets of generators S1, S2 of a group G, the corresponding two growth functions are equivalent (see e.g. [10, 13]). Note also that if jSj = k, then γ(n) ≤ kn. Growth of group G is called exponential if γ(n) s en. Otherwise the growth is said to be subexponential. For example, all non{amenable groups1 have an exponential growth, but not vice versa (see [1, 6]). Growth of group G is called polynomial if γ(n) s nc for some c > 0. The celebrated result of Gromov implies that c must be an integer, and G is almost nilpotent. See [8, 6] for details and references. c If γG(n) < n for all c, the growth of G is said to be superpolynomial. If the growth is subexponential and superpolynomial, it is called intermediate. This is a very interesting, but hardly understood class of groups. Let ! be an infinite sequence of elements in the set f0; 1; 2g. Grigorchuk group G! is a infinite profinite 2-group whose construction depends on ! (see [4, 7]). Groups G! are generated by 4 involutions, while the structure and even the growth is different for different !. We postpone definition of G! till section 3. Since the original publication, much has been discovered regarding the Grigorchuk groups. Recent advancements include improved upper and lower bounds, solution of the word problem, abstract presentation, bond perco- lation, etc. (see [5, 9, 11, 12]). We refer to review articles [7, 9] for the references. In this paper we present a new technique to estimate the growth of the Grigorchuk groups. First, we present a simple proof of the lower bound p γ(n) < e n for Grigorchuk group G! corresponding to non-flat sequences (see section 4). Using a different approach Grigorchuk showed in [5] that the result holds in greater generality. While neither our bound nor the idea of the proof is new, we believe that the technique may be proved useful in the future. In section 5 we present an upper bound on the growth of Grigorchuk group G! such that every interval of ! of length k contains each element 0, 1Non{amenable group G can be defined as a group whose Cayley graphs have positive Cheeger constant h > 0, where j@Xj h = inf X2G jXj 2 1, 2 at least once. We prove that the growth γ(n) in this case satisfies γ(n) 4 exp(nα); k k−1 where α = log2/ν 2 and ν + ν + ν = 2. In a special case when k = 3 we obtain the recent result of Bartholdi (see [2]). Interestingly, he gives the exactly same estimate while using a totally different approach. We would like to remark that one can try to improve our bounds if more information is known about frequencies of generators ρb, ρc, ρd (see Section 5.) We conclude with a improved bound for Grigorchuk p-groups. Without going through the combinatorial estimates, we apply our Growth Theorem to inequalities proved by Grigorchuk to obtain sharper upper bounds on growth. Acknowledgements. We are grateful to Efim Zelmanov for introduction to the subject and encouragement. We also thank Zydrunas Gimbutas, L´aszl´oLov´asz,Gregory Margulis and Alexander Retakh for helpful conversations. Special thanks to Laurent Bartholdi, Rostislav Grigorchuk, Alexander Retakh and Tatiana Smirnova{Nagnibeda for reading the first draft of the paper. The second author was supported by the NSF Postdoctoral Research Fellowship. 3 2 Growth Theorem In this section we present analytic estimates on growth of functions restricted by the recurrence inequalities. We refer to the following result as Growth Theorem. Theorem 2.1 Let B : N ! N be an increasing integer function with n B(n) ! 1 as n ! 1 and B(n) 4 e . Let sj < 1, cj be fixed constants and ≤ ≤ (j) N ! N ≤ ≤ ≤ ≤ let mj be fixed integers, 1 j l. Let πi : , 1 i mj, 1 j l be integer functions such that Xmj (j) ≤ πi (n) sjn + cj i=1 In addition, let Fj : N ! R+, 1 ≤ j ≤ l be positive functions such that log(F (n)) j ! 0 as n ! 1 for all ϵ > 0: nϵ Assume that for all n 2 N, at least one inequality is satisfied in the following system: ( Ymj ( ) ≤ · (j) B(n) Fj(n) B πi (n) ; where j = 1; : : : ; l: i=1 Then B(n) 4 exp(nα), where log(m ) α = max j : 1≤j≤l log(mj) − log(sj) The proof requires the following technical result. Lemma 2.2 For any 0 < s < 1, c > 0, ϵ > 0 and sequence (t1; : : : ; tn) of positive numbers such that c + 1 < t < t < ··· < t 1 − s 1 2 n and tk ≤ s · tk+1 + c we have ( ) Yn 1 ≤ 1 (1 + ϵ ) exp ϵ : tk 1 − s k=1 4 Proof of Lemma 2.2 Observe that log(1 + x) < x for all x > 0. This gives us ( ) Xn Xn 1 ≤ 1 log 1 + ϵ ϵ tk tk k=1 k=1 Define a sequence (j1; : : : ; jn) as follows: ji = ti −c=(1−s), 1 ≤ i ≤ n. Then ( ) c c c j = t − ≤ s · t + c − = s t − = s · j i i 1 − s i+1 1 − s i+1 1 − s i+1 i−1 Therefore j1 ≤ s j2 ≤ · · · ≤ s ji. By definition we have j1 = t1−c=(1−s) > 1−i 1 and ji ≥ s , 1 ≤ i ≤ n. From here for all ϵ > 0 we obtain 1 Xn 1 Xn 1 Xn X 1 ≤ ≤ (sϵ)i−1 < (sϵ)i−1 = t ϵ j ϵ 1 − sϵ i=1 i i=1 i i=1 i=1 We conclude ( ) ( )! ! ( ) Yn 1 Xn 1 Xn 1 1 1 + = exp log 1 + ≤ exp < exp t ϵ t ϵ t ϵ 1 − sϵ i=1 i i=1 i i=1 i This proves the lemma. Proof of Theorem 2.1 Let f(n) = log B(n). Then f(n) satisfies at least one of the inequalities in the following system ( Xmj ≤ (j) f(n) f(πi (n)) + log Fj(n) ; j = 1; : : : l i=1 ≤ υ 1 f(n) Let 0 < υ 1 be such that supn f(n)= n = . Let h(n) = nυ . 1 Clearly, supn h(n) = . Define bh(n) = maxfh(k) : 1 ≤ k ≤ ng. Clearly bh(n) is non-decreasing and bh(n) = h(n) for infinitely many n. Observe that bh(t) = h(t) if and only if bh(t) > bh(t − 1). Call an integer t of type (I) if bh(t) > bh(t − 1) and of type (II) if bh(t) = bh(t − 1). Clearly for any integer m of type (II) there exist an integer n ≤ m of type (I) such that bh(n) = bh(m). Take a large integer a of type (I). Then f(a) = bh(a)aυ. Assume that for a inequality j holds. Xmj υb υ ≤ (j) a h(a) = a h(a) = f(a) f(πi (a)) + log Fj(a) i=1 5 mj h i X υ ≤ (j) (j) πi (a) h(πi (a)) + log Fj(a) i=1 mj h i X υ ≤ (j) b (j) πi (a) h(πi (a)) + log Fj(a) i=1 Xmj h i b (j) (j) υ ≤ max(h(π (a))) · π (a) + log Fj(a): i i i i=1 ( ) Since the max bh(π(j)(a)) is taken over a finite set, there exist an integer i ( ) b b (j) b of type (I) such that h(b) = max h(πi (a)) and mj h i X υ υb ≤ b · (j) a h(a) h(b) πi (a) + log Fj(a): i=1 υb Trivially b ≤ sja + cj.
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