Mathematics – Number Theory
Scientific paper
2007-03-19
Mathematics
Number Theory
Revision (with different title and many logistical changes) of math.NT/0604489
Scientific paper
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).
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