Group theory, geometry and dynamics of surface homeomorphisms

Job Candidate Talk
Thursday, January 7, 2010 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 269
Dan Margalit – Tufts University – Dan.Margalit@tufts.edu
John Etnyre
Attached to every homeomorphism of a surface is a real number called its dilatation. For a generic (i.e. pseudo-Anosov) homeomorphism, the dilatation is an algebraic integer that records various properties of the map. For instance, it determines the entropy (dynamics), the growth rate of lengths of geodesics under iteration (geometry), the growth rate of intersection numbers under iteration (topology), and the length of the corresponding loop in moduli space (complex analysis). The set of possible dilatations is quite mysterious. In this talk I will explain the discovery, joint with Benson Farb and Chris Leininger, of two universality phenomena. The first can be described as "algebraic complexity implies dynamical complexity", and the second as "geometric complexity implies dynamical complexity".