The simultaneous conjugacy problem in groups of piecewise linear functions

TL;DR

This paper provides an elementary solution to the simultaneous conjugacy problem in Thompson's group F, using piecewise linear homeomorphisms, and extends techniques to larger groups of similar functions.

Contribution

It introduces a new elementary approach to solve the simultaneous conjugacy problem in Thompson's group F and generalizes these methods to larger groups of piecewise-linear homeomorphisms.

Findings

Solved the simultaneous conjugacy problem in Thompson's group F

Developed techniques to compute roots and centralizers

Extended methods to larger groups of piecewise-linear homeomorphisms

Abstract

Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.