Expanding maps and non-trivial selfcovers on infra-nilmanifolds

Abstract

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 infranilmanifold. A similar characterization is also given for the infra-nilmanifolds with a nontrivial self-cover, which corresponds to determining which almost-Bieberbach groups are coHopfian. These results provide many new examples of infra-nilmanifolds without non-trivial self-covers or expanding maps.

Gradings on Lie algebras with applications to infra-nilmanifolds

“…Similarly as for expanding maps, we can combine Theorem 1.3 and Theorem 2.3 with the result [5,Theorem 5.4.] to find a complete algebraic characterization of the infra-nilmanifolds admitting a non-trivial self-cover, i.e.…”

“…In [5] is is showed that the existence of an expanding map depends only on the rational Mal'cev completion N Q and the representation H → Aut(N Q ). By combining Theorem 1.2 and Theorem 2.2, we have a complete algebraic description of the infra-nilmanifolds admitting an expanding map: Theorem 3.1.…”

“…In [5,Theorem 3.4] it is shown that this construction also works the other way around. Given an expanding automorphism ϕ we find a positive grading such that every automorphism ψ commuting with ϕ preserves the positive grading.…”

“…Given an expanding automorphism ϕ we find a positive grading such that every automorphism ψ commuting with ϕ preserves the positive grading. The result in [5] was only stated in the case where E = Q but in fact the proof works for any field E.…”

Pro-groups and generalizations of a theorem of Bing

“…For example, Belegradek considered in [8] when such a lattice must be co-Hopfian, and in particular when they are not. Non co-Hopfian subgroups of nilpotent Lie groups were also studied by Dekimpe, Lee and Potyagailo in [22,23,24], and by Cornulier in [20]. Here is a general version of Problem 8.1:…”