Rearrangement groups of fractals

TL;DR

This paper introduces rearrangement groups for edge replacement systems, generalizing Thompson's groups, and demonstrates their actions on self-similar fractals, establishing properties like proper action on CAT(0) cubical complexes and type F infinity.

Contribution

It constructs a broad class of rearrangement groups for fractals, extending known groups and analyzing their geometric and algebraic properties.

Findings

Rearrangement groups act properly on CAT(0) cubical complexes.

Certain rearrangement groups are of type F infinity.

The framework generalizes Thompson's groups to fractal spaces.

Abstract

We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson's groups F, T, and V . Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite CAT(0) cubical complex, and we use this action to prove that certain rearrangement groups are of type F infinity.