Zeta functions in higher Teichmuller theory

TL;DR

This paper introduces zeta and L-functions for higher-dimensional surface group representations, generalizing classical functions from Fuchsian groups, and proves their meromorphic extension to the entire complex plane.

Contribution

It defines new zeta and L-functions for PSL(d,R) representations of surface groups and establishes their meromorphic continuation, extending classical theory to higher dimensions.

Findings

Zeta and L-functions are well-defined for higher-dimensional representations.

These functions have meromorphic extensions to the whole complex plane.

Generalizes classical Selberg functions to higher Teichmüller theory.

Abstract

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