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

We introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group G ≤ Homeo (𝕮) is vigorous if for any clopen set A and proper clopen subsets Band C of A there is γ ∈ G in the pointwise-stabiliser of 𝕮\A with Bγ ⊆ C. Anon-trivial group G ≤ Homeo (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)_◦ of the verbal subgroup w(G) is the whole group. We show: 1)simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of Homeo(𝕮) is fairly broad (the class is closed under various natural constructions and contains many well known groups such as the commutator subgroups of the Higman–Thompson groups Gn,r, the Brin-Thompson groups nV , Röver’s group V (Γ), and others of Nekrashevych’s ‘simple groups of dynamical origin’).