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Convex projective structures on nonhyperbolic three-manifolds

Samuel A Ballas, Jeffrey Danciger and Gye-Seon Lee

Geometry & Topology 22 (2018) 1593–1646

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.

real projective structures, three-manifolds, moduli spaces, representations of groups, divisible convex sets
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 20H10, 53A20, 57M60, 57S30
Received: 26 September 2016
Revised: 21 August 2017
Accepted: 15 October 2017
Published: 16 March 2018
Proposed: Ian Agol
Seconded: Bruce Kleiner, Dmitri Burago
Samuel A Ballas
Department of Mathematics
Florida State University
Tallahassee, FL
United States
Jeffrey Danciger
Department of Mathematics
The University of Texas
Austin, TX
United States
Gye-Seon Lee
Mathematisches Institut
Ruprecht-Karls-Universität Heidelberg