Twisted Ruelle zeta function at zero for compact hyperbolic surfaces

Abstract

Let X be an orientable compact connected hyperbolic surface of genus g. In this paper, we prove that the twisted Selberg and Ruelle zeta functions, associated with an arbitrary finite-dimensional complex representation χ of π1(X) admit a meromorphic continuation to C. Moreover, we study the behavior of the twisted Ruelle zeta function at s=0 and prove that at this point it has a zero of order dim⁡(χ)(2g−2).