Rearrangement Groups of Fractals: Structure and Conjugacy

Abstract

This dissertation is about rearrangement groups: a class of groups of homeomorphisms of fractal topological spaces. Introduced in 2019 by J. Belk and B. Forrest, this class generalizes the famous trio of Thompson groups $F$, $T$ and $V$ and includes some of their relatives and generalizations. After an introduction to this topic, this dissertation branches into different aspects of rearrangement groups. We first focus on a class of rearrangement groups of tree-like fractals known as Wa\.zewski dendrites. We find finite generating sets for them and their commutator subgroups, we prove that the commutator subgroups are simple (with one possible exception) and we show that these groups are countable dense subgroups of the groups of all homeomorphisms of dendrites. We then provide a sufficient condition to solve the conjugacy problem in rearrangement groups using strand diagram. This condition is not necessary, but understanding how to extend it is related to open problems in computer science. This method solves the conjugacy problem in essentially all known rearrangement groups. Next we study a group property known as invariable generation. With a dynamical approach we prove that transitive enough rearrangement groups are not invariably generated, which applies to most rearrangement group that has been considered so far. Then we study the gluing relation that defines the fractal topological spaces on which rearrangement groups act. We show that it is rational, i.e., there exists a finite-state automaton which reads a pair of elements if and only if they are related. We then show that finite and finitely generated abelian groups are rearrangement groups and that the stabilizers of finite sets of rational points of rearrangement groups are themselves rearrangement groups. Lastly, we prove that every rearrangement group embeds into Thompson's group $V$.