We develop a theory of cusped Hitchin representations of geometrically finite Fuchsian groups into SL(d,R). When d=3, cusped Hitchin representations arise as holonomy maps of finite area real projective surfaces. More generally, they admit positive limit maps in the sense of Fock and Goncharov.

We develop general criteria for when one can obtain counting and equidistribution results for potentials on countable Markov shifts. We show that these general criteria are satisfied by roof functions associated to linear functionals giving "length functions" for cusped Hitchin representations.

The long term goal of this project is to develop a metric theory of the augmented Htichin component which generalizes the fact that augmented Teichmuller space is the metric completion of Teichmuller space with the Weil-Petersson metric. (This is joint work with Tengren Zhang and Andy Zimmer, and with Harry Bray, Nyima Kao and Giuseppe Martone). Speaker(s): Richard Canary (U Michigan)