We show that Bieberbach’s
rigidity theorem for flat manifolds still holds true for any continuous maps on
infra-nilmanifolds. Namely, every endomorphism of an almost crystallographic group
is semi-conjugate to an affine endomorphism. Applying this result to Fixed-point
theory, we obtain a criterion for the Lefschetz number and Nielsen number for a map
on infra-nilmanifolds to be equal.