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Motivated by the study of quantum fields in a Friedman-Robertson-Walker (FRW)spacetime, the one-loop effective action for a scalar field defined in theultrastatic manifold $R\times H^3/\Gamma$, $H^3/\Gamma$ being the finitevolume, non-compact, hyperbolic spatial section, is investigated by ageneralisation of zeta-function regularisation. It is shown that additionaldivergences may appear at one-loop level. The one-loop renormalisability of themodel is discussed and making use of a generalisation of zeta-functionregularisation, the one-loop renormalisation group equations are derived.Comment: Latex, 16 pages, no figures; Latex mistakes corrected; accepted for  publication in Physical Review

Effective action for scalar fields and generalized zeta-function regularization