A trace formula for Hecke operators on Fuchsian groups

In this paper we give a trace formula for Hecke operators acting on the cohomology of a Fuchsian group of finite covolume, with coefficients in a module $V$. The proof is based on constructing an operator whose trace on $V$ equals the Lefschetz number of the Hecke correspondence on cohomology, generalizing the operator introduced together with Don Zagier for the modular group.