Signature for piecewise continuous groups

Realizations of groups of piecewise continuous transformations of the circle

We study the near action of the group PC of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of PC is said to be realizable if it can be lifted to a group of permutations of the circle. We show that every finitely generated abelian subgroup of PC is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval …

分段连续变换群的可公度化作用

Commensurating actions for groups of piecewise continuous transformations

本文利用 Exel 的偏作用构造分段变换群的可公度化作用，借助横贯性质与刚性结果证明分段仿射、分段射影作用可共轭到保持几何结构的模型，并推出性质 T 及圆子群分类结论。

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

Let $I=[0,1)$ and $\mathcal{PC}(I)$ [resp. $\mathcal{PC}^+(I)$] 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 Unpublished Theorems of Arnoux ([Arn81b]) state that $\mathcal{PC}^+(I)$ and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix.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)…