Geometry and Topology Seminar: Thurston’s Asymmetric Metric, Hölder Regularity, and Hitchin Components

Abstract: To study the Teichmüller space T(S) of a surface S from the point of view of hyperbolic geometry, Thurston introduced an asymmetric metric on T(S) in 1986. Our starting point is the observation that Thurston’s metric is encoded in the Hölder regularity of induced maps on the boundary of the hyperbolic plane.

Our main result is an extension of this circle of ideas to PSL(n,R) Hitchin components Hit(n,S) (n > 3). These are distinguished spaces of representations of pi_1(S) in PSL(n,R) that exhibit many unexpected similarities to Teichmüller space. In particular, we construct the first complete metric on Hit(n,S) (n > 3) that is defined by a geometric construction.

This lecture will be over Zoom. You can join the Zoom at the following link: https://wustl.zoom.us/j/95193027713?pwd=b3dkQmRrSHg1UGd0MzI5SzNNNUdydz09

Host: Charles Ouyang