Twisted Ruelle zeta function at zero for compact hyperbolic surfaces

Abstract

Let $X$ be a compact, hyperbolic surface of genus $g\ge 2$. 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 $\dim (\chi )(2g-2)$.

v2: The functional equation for the twisted Ruelle zeta function is made more explicit

Cited by in corpus (2)

• Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus
• The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion