Commensurating actions for groups of piecewise continuous transformations
Yves Cornulier
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 piecewise groups in dimension 1, notably piecewise of class <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝒞</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:math> , piecewise affine, piecewise projective (possibly discontinuous). We derive various conjugacy results for subgroups with Property FW, or distorted cyclic subgroups. 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 the classification of circle subgroups of piecewise projective homeomorphisms of the projective line. The piecewise affine case is a classical result of Minakawa.