# Richard Sharp

University of Warwick

### Representations of surface groups, zeta functions and counting

Ergodic Theory and Dynamical Systems Seminar
15th November 2018, 2:00 pm – 3:00 pm
Howard House, 2nd floor seminar room

Let $\Gamma$ be the fundamental group of a compact orientable surface $S$ with genus at least $2$. It is well known that a hyperbolic metric on $S$ is determined by a representations of $\Gamma$ into $\mathrm{PSL}(2,\mathbb R)$. Associated to such representations are the well-known zeta functions of Selberg and Ruelle. We will discuss analogous zeta functions for representations of $\Gamma$ into the higher rank groups $\mathrm{PSL}(d,\mathbb R)$, for $d \ge 3$. in particular we will discuss extensions of the zeta functions past their domain of convergence and the implication of this for counting weighted conjugacy classes. This is joint work with Mark Pollicott.