Two periodicity conditions for spinal groups

TL;DR

This paper establishes two dynamical system-based conditions under which constant spinal groups, subgroups of automorphisms of regular rooted trees, are periodic, thereby generalizing previous results and generating new examples of such groups.

Contribution

It introduces two new criteria based on dynamical systems for determining the periodicity of constant spinal groups, expanding the class of known periodic groups.

Findings

Provided two conditions for periodicity of constant spinal groups

Generalized results from Grigorchuk--Gupta--Sidki groups

Constructed new examples of finitely generated infinite periodic groups

Abstract

A constant spinal group is a subgroup of the automorphism group of a regular rooted tree, generated by a group of rooted automorphisms $A$ and a group of directed automorphisms $B$ whose action on a subtree is equal to the global action. We provide two conditions in terms of certain dynamical systems determined by $A$ and $B$ for constant spinal groups to be periodic, generalising previous results on Grigorchuk--Gupta--Sidki groups and other related constructions. This allows us to provide various new examples of finitely generated infinite periodic groups.