Conjugacy and Dynamics in Thompson's Groups

TL;DR

This paper presents a unified approach to solving the conjugacy problem in Thompson's groups F, T, and V using strand diagrams, and explores the dynamics of elements in F through this new perspective.

Contribution

It introduces strand diagrams as a novel tool for analyzing conjugacy and dynamics in Thompson's groups, providing elegant proofs of existing results.

Findings

Unified solution to conjugacy problem for F, T, V

Strand diagrams relate to piecewise-linear functions

New insights into the dynamics of F elements

Abstract

We give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.