Full groups of bounded automaton groups

TL;DR

This paper demonstrates that all bounded automaton groups can be embedded into finitely generated, simple, amenable groups using topological full groups related to Schreier dynamical systems, revealing new structural insights.

Contribution

It introduces a novel embedding of bounded automaton groups into simple amenable groups via topological full groups, expanding understanding of their algebraic and dynamical properties.

Findings

Bounded automaton groups embed into finitely generated simple amenable groups.

The group generated by torsion elements in topological full groups has a simple commutator subgroup.

New connections between automaton groups, topological dynamics, and group simplicity.

Abstract

We show that every bounded automaton group can be embedded in a finitely generated, simple amenable group. The proof is based on the study of the topological full groups associated to the Schreier dynamical system of the mother groups. We also show that if $\mathcal{G}$ is a minimal \'etale groupoid with unit space the Cantor set, the group $[[\mathcal{G}]]_t$ generated by all torsion elements in the topological full group has simple commutator subgroup.