  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 SL(n,Z) and Sp(2n, Z) to show
that a ``random'' element in one of these lattices has irreducible
characteristic polynomials (over the integers. The term ``random'' can be
defined in at least two ways (in terms of height and also in terms of word
length in terms of a generating set) -- we show the result using both
definitions.
  We use the above 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 a
random free group automorphism is irreducible with irreducible powers (or
strongly irreducible).
