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 $$v_{K_k}(n)\sim\exp(n^{1-(1-\alpha)^k}),\qquad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2,$$ and a torsion-free group H k with growth function $$v_{H_k}(n)\sim\exp(\log(n)n^{1-(1-\alpha)^k}).$$ 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.