Combinatorial and metric properties of Thompson's group T

TL;DR

This paper explores the metric and combinatorial structures of Thompson's group T, introducing unique normal forms, analyzing word length, and comparing properties with Thompson's group F.

Contribution

It introduces a set of unique normal forms for T, relates properties to F, and discusses how to recognize torsion elements and estimate word length.

Findings

Number of carets estimates word length in T

F is undistorted in T

Methods to recognize torsion elements in T

Abstract

We discuss metric and combinatorial properties of Thompson's group T, including normal forms for elements and unique tree pair diagram representatives. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into natural pieces. We show that the number of carets in a reduced representative of an element of T estimates the word length, and that F is undistorted in T. We describe how to recognize torsion elements in T.