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](http://www.ams.org/mathscinet-getitem?mr=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.