Combinatorial and metric properties of Thompson’s group 𝑇

Abstract

We discuss metric and combinatorial properties of Thompson’s group T T , including normal forms for elements and unique tree pair diagram representatives. We relate these properties to those of Thompson’s group F F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T 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 T estimates the word length, that F F is undistorted in T T , and we describe how to recognize torsion elements in T T .