Markov systems and transfer operators associated with cofinite Fuchsian groups

Abstract

In this paper we study a generalization of Mayer's result on the Selberg zeta function of $PSL(2, {\Bbb Z})$. Let $\Gamma$ be a cofinite Fuchsian group. We construct a Markov system ${\cal T}_{\Gamma}$ by modifying the Bowen–Series construction of a Markov map $T_{\Gamma}$ associated with $\Gamma$. The Markov system enables us to define transfer operators $L(s)$ for ${\cal T}_{\Gamma}$ so that they determine a meromorphic function taking values with nuclear operators on a nice function space. We show that the Selberg zeta function $Z(s)$ of $\Gamma$ has a determinant representation $Z(s)=\Det(I-L(s))F(s)$, where $\Det(I-L(s))$ is the Fredholm determinant of $L(s)$ and $F(s)$ is a meromorphic function depending only on a finite number of hyperbolic conjugacy classes of $\Gamma$. Combining such a representation and the investigation of the spectral properties of $L(s)$, we can also obtain some analytic information of $Z(s)$.