Generalized Grigorchuk's Overgroups as points on $\mathcal{M}_k$

TL;DR

This paper generalizes Grigorchuk's overgroups within a family of groups parameterized by infinite sequences, analyzing their structure as points in the space of marked groups and exploring their growth and algebraic properties.

Contribution

It extends the classical Grigorchuk's overgroup construction to a broader family parameterized by sequences, and studies their closure in the space of marked groups.

Findings

The family of generalized overgroups forms a closed subset in the space of marked groups.

The groups are 8-generated and exhibit diverse growth behaviors.

The closure of the family reveals new algebraic and geometric properties.

Abstract

Following the construction from `Degrees of growth of finitely generated groups and the theory of invariant means' we generalize the Grigorchuk's overgroup $\tilde{\mathcal{G}}$, studied in `On parabolic subgroups and Hecke algebras of some fractal groups' to the family $\{ \tilde{G}_\omega, \omega \in \Omega = \{ 0, 1, 2 \}^\mathbb{N} \}$ of generalized Grigorchuk's overgroups. We consider these groups as 8-generated and describe the closure of this family in the space $\mathcal{M}_8$ of marked groups.