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 vK k(n)~exp(n 1-(1-α)k), 2 3-3/α+2 2-2/α+2 1-1/α=2, and a torsion-free group H k with growth function vH k(n)~exp(log(n)n 1-(1-a)k). These are the first examples of groups of intermediate growth for which the asymptotics of their growth function is known. We construct a group of intermediate growth that contains the group of finitely supported permutations of a countable set as a subgroup. This gives the first example of a group of intermediate growth containing an infinite simple group as a subgroup. © 2011 The Author(s).