  Every expanding map on a closed manifold is topologically conjugate to an
expanding map on an infra-nilmanifold, but not every infra-nilmanifold admits
an expanding map. In this article we give a complete algebraic characterization
of the infra-nilmanifolds admitting an expanding map. We show that, just as in
the case of Anosov diffeomorphisms, the existence of an expanding map depends
only on the rational holonomy representation of the infra-nilmanifold. A
similar characterization is also given for the infra-nilmanifolds with a
non-trivial self-cover, which corresponds to determining which
almost-Bieberbach groups are co-Hopfian. These results provide many new
examples of infra-nilmanifolds without non-trivial self-covers or expanding
maps.
