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 $T$.