In this paper we study a generalization ofMayer's result on the Selberg zeta function of$PSL(2, {\Bbb Z})$. Let $\Gamma$ be a cofiniteFuchsian group. We construct a Markov system${\cal T}_{\Gamma}$ by modifying the Bowen–Seriesconstruction of a Markov map $T_{\Gamma}$ associatedwith $\Gamma$. The Markov system enables us todefine transfer operators$L(s)$ for ${\cal T}_{\Gamma}$ so that they determinea meromorphic function taking values with nuclear operatorson 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 theFredholm determinant of $L(s)$ and $F(s)$ is a meromorphicfunction depending only on a finite number ofhyperbolic conjugacy classes of $\Gamma$. Combining sucha representation and the investigation of the spectralproperties of $L(s)$, we can also obtain some analyticinformation of $Z(s)$.