Groups, Geometry, and Dynamics

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Volume 11, Issue 1, 2017, pp. 105–120
DOI: 10.4171/GGD/390

Published online: 2017-04-20

Gradings on Lie algebras with applications to infra-nilmanifolds

Jonas Deré[1]

(1) KU Leuven Kulak, Kortrijk, Belgium

In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. The existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier and we give a different proof of this fact. Similarly, we prove that given a grading of one of these types and a finite group of automorphisms, there always exist a grading of the same type which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on the covering Lie group. Another application is the construction of a nilmanifold admitting an Anosov diffeomorphism but no non-trivial self-covers and in particular no expanding maps, which is the first known example of this type.

Keywords: Infra-nilmanifolds, nilpotent Lie algebras, expanding maps, Anosov diffeomorphisms

Deré Jonas: Gradings on Lie algebras with applications to infra-nilmanifolds. Groups Geom. Dyn. 11 (2017), 105-120. doi: 10.4171/GGD/390