Rearrangement Groups of Fractals: Structure and Conjugacy

This dissertation is about rearrangement groups: a class of (countable) 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. Each rearrangement group and the space on which it acts is determined by an edge replacement system, which is a specific type of graph rewriting systems. The edge replacement system defines an edge shift and a gluing relation on it; the fractal space X is defined as the quotient under this relation, whenever it is an equivalence relation (for which there are natural sufficient conditions). Rearrangements are those homeomorphisms of X that descend from prefix-exchange homeomorphisms of the edge shifts that preserve the gluing relation. They are represented by isomorphisms between graphs that approximate the fractal. 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 Ważewski dendrites. We find finite generating sets for them and their commutator subgroups and we prove that the commutator subgroups are simple (with one possible exception). We also show that these groups are dense in the groups of all homeomorphisms of dendrites. We then extend our focus to the entire class of rearrangement groups and we provide a sufficient condition to solve the conjugacy problem (the decision problem of establishing whether two elements are conjugate) using the graphical tool of strand diagram. This condition is not necessary, but understanding how to extend it is related to open problems in computer science. However, this method solves the conjugacy problem in essentially all known rearrangement groups. Next we study invariable generation: the property of admitting a set of elements that generate the group even if each element is replaced by a conjugate. Generalizing results of Gelander, Golan and Juschenko about Thompson groups, 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 a rational relation, i.e., there exists a finite-state automaton which reads a pair of elements of the edge shift if and only if the two elements are related. In fact, we provide an algorithmic construction of such finite-state automata. Finally, we collect results that are based on constructions and modifications of edge replacement systems. We first prove that finite groups and finitely generated abelian groups are rearrangement groups. Then we show that the stabilizer of a finite set of rational points under the action of a rearrangement group is itself a rearrangement group. Lastly, we prove that every rearrangement group embeds into Thompson's group V.