Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated

TL;DR

This paper introduces conditions called vigorous and flawless for groups of homeomorphisms of the Cantor set, establishing when such groups are two-generated, simple, or both, and broadening the understanding of their algebraic structure.

Contribution

It provides new criteria for simplicity and two-generation in groups of Cantor set homeomorphisms, linking vigorous and flawless properties, and identifies many known groups as examples.

Findings

Simple vigorous groups are either two-generated by torsion elements or not finitely generated.

Vigorous groups are simple if and only if they are flawless.

The class of vigorous simple subgroups includes many well-known groups and is closed under natural constructions.

Abstract

Let $\mathfrak{C}$ be some Cantor space. We study groups of homeomorphisms of $\mathfrak{C}$ which are vigorous, or, which are flawless, where we introduce both of these terms here. We say a group $G\leq \operatorname{Homeo}(\mathfrak{C})$ is $vigorous$ if for any clopen set $A$ and proper clopen subsets $B$ and $C$ of $A$ there is $\gamma \in G$ in the pointwise-stabiliser of $\mathfrak{C}\backslash A$ with $B\gamma\subseteq C$. Being vigorous is similar in impact to some of the conditions proposed by Epstein in his proof that certain groups of homeomorphisms of spaces have simple commutator subgroups (and/or related conditions, as proposed in some of the work of Matui or of Ling). A non-trivial group $G\leq \operatorname{Homeo}(\mathfrak{C})$ is $flawless$ if for all $k$ and $w$ a non-trivial freely reduced product expression on $k$ variables (including inverse symbols), a particular subgroup $w(G)_\circ$ of the verbal subgroup $w(G)$ is the whole group. It is true, for instance, that flawless groups are both perfect and lawless. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) vigorous groups are simple if and only if they are flawless, and, 3) the class of vigorous simple subgroups of $\operatorname{Homeo}(\mathfrak{C})$ is fairly broad (it contains many well known groups such as the commutator subgroups of the Higman-Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, R\"{o}ver's group $V(\Gamma)$, and others of Nekrashevych's `simple groups of dynamical origin', and, the class is closed under various natural constructions).