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November 2021 Projective Anosov representations, convex cocompact actions, and rigidity
Andrew Zimmer
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J. Differential Geom. 119(3): 513-586 (November 2021). DOI: 10.4310/jdg/1635368438

Abstract

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain.

We then give three applications. First, we show that Anosov representations into general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). Next, we prove a rigidity result involving the Hilbert entropy of a projective Anosov representation. Finally, we prove a rigidity result which shows that the image of the boundary map associated to a projective Anosov representation is rarely a $C^2$ submanifold of projective space. This final rigidity result also applies to Hitchin representations.

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Andrew Zimmer. "Projective Anosov representations, convex cocompact actions, and rigidity." J. Differential Geom. 119 (3) 513 - 586, November 2021. https://doi.org/10.4310/jdg/1635368438

Information

Received: 13 May 2017; Accepted: 27 January 2020; Published: November 2021
First available in Project Euclid: 1 November 2021

Digital Object Identifier: 10.4310/jdg/1635368438

Subjects:
Primary: 20F65 , 20F67 , 20H10 , 22E40 , 37D40 , 53C24 , 57S30

Rights: Copyright © 2021 Lehigh University

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Vol.119 • No. 3 • November 2021
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