Eventually Self-Similar Groups acting on Fractals

TL;DR

This paper introduces ESS groups acting on fractal spaces constructed via hyperedge replacement systems, establishing their finiteness properties and providing new insights into their structure and relation to known groups.

Contribution

It develops a framework for analyzing groups acting on fractal topological spaces, extending previous work and establishing finiteness properties for new classes of groups.

Findings

Airplane and dendrite rearrangement groups have type F_infinity.

A group combining dendrite rearrangements and the Grigorchuk group is finitely generated.

Certain ESS groups of edge shifts have type F_infinity.

Abstract

Generalizing work by Belk and Forrest, we develop almost expanding hyperedge replacement systems that build fractal topological spaces as quotients of edge shifts under certain ``gluing'' equivalent relations. We define ESS groups, which are groups of homeomorphisms of these spaces that act as a finitary asynchronous transformations followed by self-similar ones, akin to the action of Scott-R\"{o}ver-Nekrashevych groups on the Cantor space. We provide sufficient conditions for finiteness properties of such groups, which allow us to show that the airplane and dendrite rearrangement groups have type $F_\infty$, that a group combining dendrite rearrangements and the Grigorchuk group is finitely generated, and that certain ESS groups of edge shifts have type $F_\infty$ (partially addressing a question of Deaconu), in addition to providing new proofs of previously known results about several Thompson-like groups.