Geometry & Topology

Convex projective structures on nonhyperbolic three-manifolds

Samuel A Ballas, Jeffrey Danciger, and Gye-Seon Lee

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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.

Article information

Geom. Topol., Volume 22, Number 3 (2018), 1593-1646.

Received: 26 September 2016
Revised: 21 August 2017
Accepted: 15 October 2017
First available in Project Euclid: 31 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 53A20: Projective differential geometry 57M60: Group actions in low dimensions 57S30: Discontinuous groups of transformations

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


Ballas, Samuel A; Danciger, Jeffrey; Lee, Gye-Seon. Convex projective structures on nonhyperbolic three-manifolds. Geom. Topol. 22 (2018), no. 3, 1593--1646. doi:10.2140/gt.2018.22.1593.

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