On Growth of Grigorchuk Groups

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 estimates generalize and improve various bounds by Grigorchuk, Bartholdi and others.

Milnor's problem on the growth of groups and its consequences

“…Another technical tool was explored by R. Muchnik and I. Pak [MP01a] to get an upper bound on growth for the whole family of groups {G ω }. Surprisingly, in the case of G their approach give the same upper bound as (9.2), so the question of improving it is quite intriguing (see problem 7).…”

“…The question of whether the upper bound obtained in [BA10,MP01a] is optimal (i.e. it coincides with the growth rate of the group G) is very intriguing and we formulate it as the next problem.…”

On growth of random groups of intermediate growth

“…The reason why oscillating groups are typical in the categorical sense is the existence of a countable dense subset in G 2 consisting of (virtually metabelian) groups of exponential growth and a dense subset of groups with the growth equivalent to the growth of the first Grigorchuk group G (012) ∞ , which is bounded by e n θ 0 due to a result of Bartholdi [Bar98] (see also [MP01]). Note also that this is the smallest upper bound known for any group of intermediate growth.…”

Critical constants for recurrence of random walks on $G$-spaces

“…-As it was already mentioned, if G = G 1 , G 2 or G(i), then the growth function of G satisfies v G (n) exp (n α ) for some α < 1 and any sufficiently large n ( [16]). It is known ( [5]) that in the case G = G 1 one can take α = log(2)/ log(2/X), where X is the positive solution of the equation X 3 + X 2 + X = 2 (see also [23] for certain generalizations of this type of estimates). One can check that for any > 0 and for any i large enough (depending on ), the growth function of the group G(i) satisfies v G(i) (n) exp (n α+ ) for any sufficiently large n. Let v sup G = lim sup log log v G,S (n)/ log n. (Clearly, this constant does not depend on the generating set S).…”