Palindromic subshifts and simple periodic groups of intermediate growth

TL;DR

This paper introduces a new class of simple groups of intermediate growth derived from dihedral group actions on Cantor sets, providing novel examples of such groups with specific properties.

Contribution

It presents a procedure to construct finitely generated periodic groups from dihedral group actions, including the first known simple groups of intermediate growth.

Findings

Groups of intermediate growth can be constructed from dihedral actions.

Schreier graphs' linear repetitiveness implies intermediate growth.

First examples of simple groups with intermediate growth are provided.

Abstract

We describe a new class of groups of Burnside type, giving a procedure transforming an arbitrary non-free minimal action of the dihedral group on a Cantor set into an orbit-equivalent action of an infinite finitely generated periodic group. We show that if the associated Schreier graphs are linearly repetitive, then the group is of intermediate growth. In particular, this gives first examples of simple groups of intermediate growth.