Enumeration of mapping classes for the torus (Q5946234)

A choice of a set of generators for a group \(G\) defines the so-called word metric on \(G\). If \(a_n\) denotes the number of elements of \(G\) whose distance to the identity is exactly \(n\), then the formal power series \(\sum_{n=0}^{\infty} a_nx^n\) is called the growth function of \(G\) with respect to the chosen set of generators. In the paper under review, the growth function of \(SL(2,\mathbb{Z})\), the mapping class group of the \(2\)-torus, is computed. The growth functions of periodic, reducible and Anosov elements are also determined. This shows that almost all elements in \(SL(2,\mathbb{Z})\) are Anosov, i.e. if \(f_n\) and \(p_n\) denote the number of all and Anosov elements contained in the ball of radius \(n\) with center at the identity, then \(p_n /f_n \) approaches \(1\) as \(n\) grows. The proof uses the fact that \(PSL(2,\mathbb{Z})\) is isomorphic to free product of two cyclic groups of orders \(2\) and \(3\). Everything is first computed in \(PSL(2,\mathbb{Z})\) and then lifted to \(SL(2,\mathbb{Z})\).