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Every expanding map on a closed manifold is topologically conjugate to anexpanding map on an infra-nilmanifold, but not every infra-nilmanifold admitsan expanding map. In this article we give a complete algebraic characterizationof the infra-nilmanifolds admitting an expanding map. We show that, just as inthe case of Anosov diffeomorphisms, the existence of an expanding map dependsonly on the rational holonomy representation of the infra-nilmanifold. Asimilar characterization is also given for the infra-nilmanifolds with anon-trivial self-cover, which corresponds to determining whichalmost-Bieberbach groups are co-Hopfian. These results provide many newexamples of infra-nilmanifolds without non-trivial self-covers or expandingmaps.

Expanding maps and non-trivial self-covers on infra-nilmanifolds