Twisted Ruelle zeta function at zero for compact hyperbolic surfaces

TL;DR

This paper proves the meromorphic continuation of twisted Selberg and Ruelle zeta functions for compact hyperbolic surfaces and characterizes the order of their zero at zero, linking it to the representation dimension and surface genus.

Contribution

It establishes the meromorphic continuation of twisted zeta functions and determines the zero order at zero for these functions on compact hyperbolic surfaces.

Findings

Meromorphic continuation of twisted Selberg and Ruelle zeta functions.

Zero of the Ruelle zeta function at s=0 has order equal to the representation dimension times (2g-2).

Results connect spectral properties with geometric and algebraic data.

Abstract

Let $X$ be a compact, hyperbolic surface of genus $g\geq 2$. In this paper, we prove that the twisted Selberg and Ruelle zeta functions, associated with an arbitrary, finite-dimensional, complex representation $\chi$ of $\pi_1(X)$ admit a meromorphic continuation to $\mathbb{C}$. Moreover, we study the behaviour of the twisted Ruelle zeta function at $s=0$ and prove that at this point, it has a zero of order $\dim(\chi)(2g-2)$.