Session 39. Contributed talks

Expanding maps and Anosov diffeomorphisms on infra-nilmanifolds

Jonas Deré, KU Leuven Kulak, Belgium
Infra-nilmanifolds play an important role in dynamical systems, especially when studying expanding maps or Anosov diffeomorphisms. Because of the algebraic nature of these manifolds, questions about self-maps can be translated into questions about endomorphisms of their fundamental group. In this way, it was shown by M. Gromov in [3] that every expanding map on a closed manifold is topologically conjugate to an affine infra-nilendomorphism. Up till now it is unknown if a similar statement also holds for Anosov diffeomorphisms, although some partial results point in that direction, e.g. for infra-nilmanifolds and for codimension one Anosov diffeomorphisms.

These results motivate the study of infra-nilmanifolds admitting an Anosov diffeomorphism or an expanding map. In this talk I will show how these questions are related to the existence of certain automorphisms of nilpotent Lie algebras. As a consequence, we can classify all infra-nilmanifolds modeled on a free nilpotent Lie group admitting an Anosov diffeomorphism, see [2]. Another consequence is that for nilmanifolds, the existence of an expanding map or a non-trivial self-covering does only depend on the commensurability class of the fundamental group of the manifold, answering a question stated in [1]. This allows us to construct examples of nilmanifolds admitting an Anosov diffeomorphism but no expanding map.

References
  1. I. Belegradek, On co-{H opfian nilpotent groups, The Bulletin of the London Mathematical Society 6, 2003, 805-811.
  2. K. Dekimpe, J. Deré, Existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups , preprint, 2013, arXiv:1304.6529.
  3. M. Gromov, Groups of polynomial growth and expanding maps , Institut des Hautes \'Etudes Scientifiques 53, 1981, 53-73.
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