Mather invariants in groups of piecewise-linear homeomorphisms

TL;DR

This paper explores the relationship between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, providing a simplified framework that facilitates understanding centralizers and potential generalizations.

Contribution

It unifies two existing characterizations of conjugacy, offering a clearer perspective and enabling easier computation of centralizers in these groups.

Findings

Unified the two characterizations of conjugacy

Simplified the approach to compute centralizers

Enabled potential generalizations in the theory

Abstract

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.