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The fundamental group of a compact hyperbolic space form has exponential growth, according to a well-known theorem of Milnor [Milnor, J. A note on curvature and fundamental group. J. Differential Geometry 2 1968 1--7. MR0232311]. Bieberbach groups are, on the other hand, polynomial: indeed, their translations subgroups have finite index, so by Gromov's theorem they have polynomial growth.