  We calculate the one-loop effective potential of a self-interacting scalar
field on the spacetime of the form $\reals^2\times H^2/\Gamma$. The Selberg
trace formula associated with a co-compact discrete group $\Gamma$ in
$PSL(2,\reals )$ (hyperbolic and elliptic elements only) is used. The closed
form for the one-loop unrenormalized and renormalized effective potentials is
given. The influence of non-trivial topology on curvature induced phase
transitions is also discussed.
