Speaker: Christopher Leininger
Title: Dynamics Versus Algebra Through Geometry.
Affiliation: University of Illinois at Urbana Champaign.
Date: Friday, March 2, 2007.
Place and Time: Room 101 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. I will start by describing the generic class of homeomorphisms of a surface S_g of genus g called the "pseudo-Anosov" homeomorphisms. These come with a basic measure of the dynamical complexity called the dilatation. If one considers all pseudo-Anosov homeomorphisms of a genus g surface, then there is a positive lower bound on the logarithm of dilatation. On the other hand, Penner has shown that if one allows g to tend to infinity, then the logarithm of the smallest possible dilatation of such a homeomorphism tends to zero on the order of 1/g. I will discuss joint work with Benson Farb and Dan Margalit that describes how one can use geometry to show that this type of behavior is prohibited if one imposes certain algebraic restrictions. As the simplest example, we prove that if a pseudo-Anosov homeomorphisms acts trivially on homology, then the logarithm of its dilatation is bounded below by .197 (independent of g).