Growth of permutational extensions

Abstract

We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all k ∈ N a torsion group K k with growth function

Poisson–Furstenberg boundary and growth of groups

“…We have begun to study asymptotic properties of permutational wreath products in [7]. The asymptotic geometry of these groups turns out to be much richer than in the particular case of ordinary wreath products (namely, for which X " G).…”

“…It is easy to see that the word growth of A ≀ G is exponential whenever X " G is infinite and A is non-trivial. However, among permutational wreath products there are groups of intermediate growth, see [7].…”

“…The words w n in the statement of the lemma above were used in [7,Proposition 4.7] to estimate the growth of the permutational wreath product of the first Grigorchuk group.…”

“…Indeed, it is shown in [7] that the W above has intermediate growth if d " 1. In this paper we consider the groups W with d ě 2, and are in particular interested in the case d " 2.…”

Imbeddings into groups of intermediate growth

“…For example, only a finite number of primes appears as exponents in a Grigorchuk group [7]; see also [2, §3.6]. Extensions of Grigorchuk groups constructed by the authors in [3] admit a larger class of possible subgroups, but some restrictions appear nevertheless. In particular, Theorem A gives the first groups of subexponential growth containing Q.…”