Convex projective structures on nonhyperbolic three-manifolds

Y Benoist proved that if a closed three-manifold $M$ admits an indecomposable convex real projective structure, then $M$ is topologically the union along tori and Klein bottles of finitely many submanifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist’s theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.

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Keywords

real projective structures, three-manifolds, moduli spaces, representations of groups, divisible convex sets

Mathematical Subject Classification 2010

Primary: 57M50

Secondary: 20H10, 53A20, 57M60, 57S30

References

Publication

Received: 26 September 2016

Revised: 21 August 2017

Accepted: 15 October 2017

Published: 16 March 2018

Proposed: Ian Agol

Seconded: Bruce Kleiner, Dmitri Burago

Authors

Samuel A Ballas
Department of Mathematics Florida State University Tallahassee, FL United States
https://www.math.fsu.edu/~ballas/
Jeffrey Danciger
Department of Mathematics The University of Texas Austin, TX United States
https://www.ma.utexas.edu/users/jdanciger/
Gye-Seon Lee
Mathematisches Institut Ruprecht-Karls-Universität Heidelberg Heidelberg Germany
https://www.mathi.uni-heidelberg.de/~lee/

Volume 22, issue 3 (2018)

Samuel A Ballas, Jeffrey Danciger and Gye-Seon Lee

Abstract