On Growth of Generalized Grigorchuk's Overgroups

TL;DR

This paper studies the growth properties of generalized overgroups of Grigorchuk's groups, showing that their growth type depends on the nature of the defining sequence, with polynomial growth for eventually constant sequences and intermediate growth otherwise.

Contribution

It introduces a family of generalized overgroups of Grigorchuk's groups and characterizes their growth types based on the properties of the defining sequences.

Findings

Overgroups with eventually constant sequences have polynomial growth.

Overgroups with non-constant sequences exhibit intermediate growth.

The growth behavior depends on the sequence's eventual constancy.

Abstract

Grigorchuk's Overgroup $\tilde{\mathcal{G}}$, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group $\mathcal{G}$ of intermediate growth constructed in 1980, but also has elements of infinite order. It's growth is substantially greater than the growth of $\mathcal{G}$. The group $\mathcal{G}$, corresponding to the sequence $(012)^\infty = 012012 ...$, is a member of the family $\{ G_\omega | \omega \in \Omega = \{ 0, 1, 2 \}^\mathbb{N} \}$ consisting of groups of intermediate growth when sequence $\omega$ is not virtually constant. Following this construction we define the family $\{ \tilde{G}_\omega, \omega \in \Omega \}$ of generalized overgroups. Then $\tilde{\mathcal{G}} = \tilde{G}_{(012)^\infty}$ and $G_\omega$ is a subgroup of $\tilde{G}_\omega$ for each $\omega \in \Omega$. We prove, if $\omega$ is eventually constant, then $\tilde{G}_\omega$ is of polynomial growth and if $\omega$ is not eventually constant, then $\tilde{G}_\omega$ is of intermediate growth.