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Abstract
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 pointwisestabiliser of 𝕮\A with Bγ ⊆ C. Anontrivial group G ≤ Homeo (𝕮) is flawless if for all k and w a nontrivial 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 twogenerated 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 BrinThompson groups nV , Röver’s group V (Γ), and others of Nekrashevych’s ‘simple groups of dynamical origin’).
Original language  English 

Number of pages  34 
Journal  Journal of the Institute of Mathematics of Jussieu 
Volume  FirstView 
Early online date  20 Sept 2024 
DOIs  
Publication status  Epub ahead of print  20 Sept 2024 
Keywords
 Full groups
 Cantor space
 Verbal subgroups
 Simple groups
 Finite generation
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Dive into the research topics of 'Sufficient conditions for a group of homeomorphisms of the Cantor set to be twogenerated'. Together they form a unique fingerprint.Projects
 1 Finished

Bisynchronizing automata: Bisynchronizing automata, outer automorphism groups of HigmanThompson groups, and automorphisms of the shift
Bleak, C. P. (PI) & Cameron, P. J. (CoI)
1/05/18 → 30/04/21
Project: Standard