Constructing 2-generated subgroups of the group of homeomorphisms of Cantor space

Group theory;Cantor sets;Homeomorphisms

Hyde, James Thomas ; Bleak, Collin Patrick ; Bleak, Collin Patrick

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Group theory;    Cantor sets;    Homeomorphisms;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/11362/JamesHydePhDThesis.pdf?sequence=2&isAllowed=y

来源: DR-NTU

【 摘 要 】

We study finite generation, 2-generation and simplicity of subgroups of H[sub]c, thegroup of homeomorphisms of Cantor space.In Chapter 1 and Chapter 2 we run through foundational concepts and notation. In Chapter 3 we study vigorous subgroups of H[sub]c. A subgroup G of H[sub]cis vigorous if for any non-empty clopen set A with proper non-empty clopen subsets B and C there exists g ∈ G with supp(g) ⊑ A and Bg ⊆ C. It is a corollary of the main theorem of Chapter 3 that all finitely generated simple vigorous subgroups of H[sub]c are in fact 2-generated. We show the family of finitely generated, simple, vigorous subgroups of H[sub]cis closed under several natural constructions.In Chapter 4 we use a generalised notion of word and the tight completion construction from [13] to construct a family of subgroups of H[sub]cwhich generalise Thompson's group V . We give necessary conditions for these groups to be finitely generated and simple. Of these we show which are vigorous. Finally we give some examples.