Mathematics Colloquium

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.)