A pressure metric for the Hitchin component
Time
Nov 7 2013 – 4:30pm – 5:30 pm
Location
CH 240
Speaker
Richard Canary (University of Michigan)
Abstract
If S is a closed surface, its Teichmuller space is the space of all (marked) hyperbolic structures on S. Hitchin showed that there is a component of the space of (conjugacy classes of) representatations of the fundamental group of a closed surface S into PSL(n,R) which is homeomorphic to an open ball. This component contains a copy of the Teichmuller space of S which we call the Fuchsian locus. We will begin by surveying basic facts about Teichmuller space and the Hitchin component. The pressure metric is a analytic Riemannian metric on the Hitchin component which is invariant under the action of the mapping class group and whose restriction to the Fuchsian locus is a multiple of the classical Weil-Petersson metric. We associate to each Hitchin representation a geodesic flow which is a Holder repameterization of the geodesic flow on S. We then use tools from the thermodynamic formalism to construct our metric. (This talk describes joint work with Bridgeman, Labourie and Sambarino.)