Quantum fields in hyperbolic spacetimes with finite spatial volume

The 1-loop effective action for a massive self-interacting scalar field is investigated in four-dimensional ultrastatic spacetime R x H-3/Gamma, where H-3/Gamma is a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the zeta-function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that zeta(s) has a simple pole at s=0. The 1-loop effective action is analysed by means of proper-time regularizations and the 1-loop divergences are found explicitly. It is pointed out that, in this special case, zeta-function regularization also requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite-temperature effects are studied and a high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like spacetime is briefly discussed.