Commensurating actions for groups of piecewise continuous transformations
Yves Cornulier
TL;DR
This work develops a unifying framework for commensurating actions via Exel's partial actions and applies it to groups that piecewise preserve affine or projective structures on 1-dimensional spaces. By encoding commensurating actions as cofinite partial actions and employing pseudogroups and modeled curves, the authors derive rigidity/conjugacy results for subgroups with Property FW or distorted cyclic subgroups, including a conjugacy to affine or projective actions on suitable models. Notably, they establish that PC_Aff$(oldsymbol{R}/oldsymbol{Z})$ has no infinite FW subgroups (and, similarly, PC_Proj$(oldsymbol{R}/oldsymbol{Z})$ has no infinite FW subgroups), and classify circle subgroups of PC_Proj$(oldsymbol{P}^1_{oldsymbol{R}})$ up to conjugacy, with refinements in the $C^1$ case. The work also provides regularization results that turn transfixed, piecewise actions into continuous $S$-preserving actions on appropriate cofinite models, yielding strong constraints on subgroups and their possible dynamics, including connections to Thompson-type groups and IET-like phenomena. Overall, the paper delivers a robust toolkit for translating geometric “piecewise” data into rigid algebraic conclusions with broad implications for 1-dimensional geometric group theory and dynamical systems on the circle.
Abstract
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geometric structure. We apply this to many groups with piecewise properties in dimension 1, notably piecewise of class C^k, piecewise affine, piecewise projective (possibly discontinuous). We derive various conjugacy results for subgroups with Property FW, or distorted cyclic subgroups, or more generally in the presence of rigidity properties for commensurating actions. For instance we obtain, under suitable assumptions, the conjugacy of a given piecewise affine action to an affine action on possibly another model. By the same method, we obtain a similar result in the projective case. An illustrating corollary is the fact that the group of piecewise projective self-transformations of the circle has no infinite subgroup with Kazhdan's Property T; this corollary is new even in the piecewise affine case. In addition, we use this to provide of the classification of circle subgroups of piecewise projective homeomorphisms of the projective line. The piecewise affine case is a classical result of Minakawa.