Geometry, Topology and Dynamics Seminar
Ben Klaff
UIC
Rational Points on Modular Character Varieties of Hyperbolic Knots
Abstract: We develop aspects of Culler-Shalen theory for SL(2,k)-character varieties of finitely generated groups, where k denotes an algebraically closed field of positive characteristic. We show how topological and geometric properties of hyperbolic knots (and properties of their fundamental groups) are related to special number-theoretic and algebro-geometric properties of their modular SL(2,k) varieties. As an application, we show how the Lang-Weil estimates for the number of rational points on varieties over finite fields can be used to give upper bounds for the size of the smallest non-solvable finite quotient of the fundamental group of any "small" hyperbolic knot. This is joint work with Peter Shalen.
Monday April 25, 2011 at 3:00 PM in SEO 636