Growth of permutational extensions

TL;DR

This paper introduces new examples of groups with intermediate growth, including torsion and torsion-free groups, and explores their geometric properties, notably containing complex subgroups like infinite simple groups.

Contribution

It constructs the first known groups of intermediate growth with explicitly determined growth functions and includes the novel embedding of infinite simple groups.

Findings

Constructed torsion groups with explicit intermediate growth functions.

Developed torsion-free groups with intermediate growth characterized by logarithmic factors.

Embedded infinite simple groups within groups of intermediate growth.

Abstract

We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all natural number k a torsion group with growth function asymptotically $\exp(n^{1-(1-\alpha)^k}),\quad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2$, and a torsion-free group with growth function asymptotically $\exp(\log(n)n^{1-(1-\alpha)^k})$. These are the first examples of groups of intermediate growth for which the 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.