On topological full groups of Z^d-actions

TL;DR

This paper introduces new simple, finitely generated groups derived from free abelian group actions on Cantor sets, including examples related to interval exchanges and Penrose tilings, many of which are amenable.

Contribution

It provides novel constructions of topological full groups from Z^d-actions, expanding the class of known examples with specific geometric and dynamical properties.

Findings

Many groups in this class are amenable

Examples include groups of interval exchange transformations

A group associated with Penrose tilings is discussed

Abstract

We give new examples of simple finitely generated groups arising from actions of free abelian groups on the Cantor sets. As particular examples, we discuss groups of interval exchange transformations, and a group naturally associated with the Penrose tilings. Many groups in this class are amenable.