Commensurating actions for groups of piecewise continuous transformations

TL;DR

This paper develops a framework using partial actions to analyze groups acting piecewise on geometric structures, leading to new conjugacy results and classifications, including a novel proof that certain groups lack infinite Property T subgroups.

Contribution

It introduces a method to construct commensurating actions for groups with piecewise geometric actions, deriving conjugacy and classification results, and providing new insights into the structure of piecewise projective groups.

Findings

Conjugacy of piecewise affine actions to affine actions under certain conditions

Classification of circle subgroups in piecewise projective homeomorphisms

The group of piecewise projective circle transformations has no infinite Property T subgroup

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.