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We prove sharp limit theorems on random walks on graphs with values in finitegroups. We then apply these results (together with some elementary algebraicgeometry, number theory, and representation theory) to finite quotients oflattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to showthat a ``random'' element in one of these lattices has irreduciblecharacteristic polynomials (over the integers. The term ``random'' can bedefined in at least two ways (in terms of height and also in terms of wordlength in terms of a generating set) -- we show the result using bothdefinitions.  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 arandom free group automorphism is irreducible with irreducible powers (orstrongly irreducible).

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