Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms

Abstract

We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically, $\mathrm{SL}(n,\mathbb{Z})$ and $\mathrm{Sp}(2n,\mathbb{Z})$) to show that a random element in one of these lattices has irreducible characteristic polynomials (over $\mathbb{Z}$). The term random can be defined in at least two ways: first, in terms of height; second, in terms of word length in terms of a generating set. We show the result using both definitions.

We use these results to show that a random (in terms of word length) element of the mapping class group of a surface is pseudo-Anosov and that a random free group automorphism is irreducible with irreducible powers (or fully irreducible*)