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

By Igor Rivin. Duke mathematical journal, 2006

145 citations recorded by the source corpus.

We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebraic integers to be of the same degree, and give a simple proof (in the Appendix) that the distribution of cycle structures modulo a prime p for polynomials with a restricted coefficient is the same as that for general polynomials.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3edbe5c753dc8a6f8bba01bfa588717c9d867ec2

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In Geometry and Topology

1. Richard Aoun
2. Joseph Maher
3. Peter Sarnak

In Mathematical Physics

1. Richard Aoun
2. Joseph Maher
3. Peter Sarnak

Citations in the corpus

• Random subgroups of linear groups are free (2010, methodology): “ for all large n, Mn,1,...,Mn,⌊exp(Cn)⌋ generate a free group on ln = ⌊exp(Cn)⌋ generators remark 1.3. As explained above, our main result shares a common flavor with the works by Rivin and Kowalski - [Riv08] and [Kow08]-, in the sense that random elements in a finitely generated group are shown to ve”
• ON THE DISTRIBUTION OF GALOIS GROUPS (2010, unknown): “Since Gallagher’s paper [9] has had a few applications and extensions since its publication, for example to polynomials with some of the coefficients ai fixed ([5]), reciprocal polynomials ([6]), characteristic polynomials of unimodular matrices ([18]) or L-functions of algebraic curves ([3], [15]),”
• Random walks on the mapping class group (2006, background): “ element which is not pseudo-Anosov is conjugate to an element of relative length at most B, as shown in Lemma 5.5, and so this implies that wn is pseudo-Anosov with asymptotic probability one. Rivin [Riv08] and Kowalski [Kow08] have also shown that the nearest neighbour random walk on the mapping c”