Zeta functions in higher Teichmüller theory

In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in \(\textrm{PSL}(d, \mathbb R)\) $\text{PSL}(d,\mathbb{R})$ , for \(d \ge 3\) $d\ge 3$ . These are natural generalizations of the well known classical Selberg zeta function and L-function for Fuchsian groups, corresponding to the case \(d=2\) $d=2$ . We show that these complex functions have meromorphic extensions to the entire complex plane \(\mathbb C\) $\mathbb{C}$ .