Mather invariant in groups of piecewise-linear homeomorphisms

Trends in Mathematics, Combinatorial and Geometric Group Theory, (2010), 251-260
http://dx.doi.org/10.1007/978-3-7643-9911-5_10 (preprint - http://arxiv.org/abs/0811.2370)

We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows us to easily recover centralizers and lends itself to generalization.