We calculate the one-loop effective potential of a self-interacting scalar field on the space-time of the form R2 X H-2/GAMMA. The Selberg trace formula associated with a co-compact discrete group GAMMA in PSL(2,R) (hyperbolic and elliptic elements only) is used. The dosed 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.