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Mathematics Colloquium
2014 - 2015

Tuesday, December 2, 2014
4:00 p.m.  //  151 Sloan
Richard Canary (University of Michigan) - The geometry of the Hitchin component

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 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.
 
In the first half of the talk we will introduced the Hitchin component and discuss various properties of the Hitchin component which lead one to think of it as a higher rank analogue of classical Teichmuller space. In the second half of the talk, we discuss an analytic Riemannian metric on the Hitchin component which is an analogue of the Weil-Petersson metric on Teichmuller space. In particular, it is mapping class group invariant and its restriction
to the Fuchsian locus is a constant multiple of the Weil-Petersson metric. One key tool is a metric Anosov flow associated to each Hitchin representation which is a Holder reparameterization of the geodesic flow on S whose periods encode the spectral radii of the images of the representation. (The second half of the talk describes joint work with Bridgeman, Labourie and Sambarino.)






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