Combinatorial and metric properties of Thompson's group t

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, that F is undistorted in T, and we describe how to recognize torsion elements in T. © 2008 American Mathematical Society Reverts to public domain 28 years from publication.