Richard Sharp

Representations of surface groups, zeta functions and counting

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.