Uniform simplicity for subgroups of piecewise continuous bijections of the unit interval

Abstract

AbstractLet and (resp. ) be the quotient group of the group of all piecewise continuous (resp. piecewise continuous and orientation preserving) bijections of by its normal subgroup consisting in elements with finite support (i.e., which are trivial except at possibly finitely many points). Arnoux's thesis states that and certain groups of interval exchanges are simple, and the proofs of these results are the purpose of the Appendix. We prove the simplicity of the group of locally orientation preserving, piecewise continuous, piecewise affine maps of the unit interval. These results can be improved. Indeed, a group is uniformly simple if there exists a positive integer such that for any , the element can be written as a product of at most conjugates of or . We provide conditions, which guarantee that a subgroup of is uniformly simple. As corollaries, we obtain that , , , , , and some Thompson‐like groups included that the Thompson group are uniformly simple.