  The one-loop effective action for a massive self-interacting scalar field is
investigated in $4$-dimensional ultrastatic space-time $ R \times H^3/\Gamma$,
$H^3/\Gamma$ being 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 one-loop effective
action is analysed by means of proper-time regularisations and the one-loop
divergences are explicitly found. It is pointed out that, in this special case,
also $\zeta$-function regularisation requires a divergent counterterm, which
however is not necessary in the free massless conformal invariant coupling
case. Finite temperature effects are studied and the 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
space-time is briefly discussed.
