2021/09/13 by Guelman, Nancy, Liousse, Isabelle, Arnoux, Pierre · 1 citation
#20E32 #20F12 #37E05 #57S30 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2109.05706
Let I=[0,1) and PC(I) [resp. PC+(I)] be the quotient group of the group of all piecewise continuous [resp. piecewise continuous and orientation preserving] bijections of I by its normal subgroup consisting in elements with finite support (i.e. that are trivial except at possibly finitely many points). Unpublished Theorems of Arnoux ([Arn81b]) state that PC+(I) and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix. Dealing with piecewise direct affine maps, we prove the simplicity of the group \mathcal A+(I) (see Definition 1.6). These results can be improved. Indeed, a group G is uniformly simple if there exists a positive integer N such that for any f,ϕ∈ G∖\Id\, the element ϕ can be written as a product of at most N conjugates of f or f-1. We provide conditions which guarantee that a subgroup G of PC(I) is uniformly simple. As Corollaries, we obtain that PC(I), PC+(I), PL+ (\mathbb S1), \mathcal A(I), \mathcal A+(I) and some Thompson like groups included the Thompson group T are uniformly simple.