Convex projective structures on nonhyperbolic three-manifolds

Ballas, Samuel¹ ; Danciger, Jeffrey² ; Lee, Gye-Seon³

¹ Department of Mathematics, Florida State University, Tallahassee, FL, United States
² Department of Mathematics, The University of Texas, Austin, TX, United States
³ Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

Geometry & topology, Tome 22 (2018) no. 3, pp. 1593-1646.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.1593

Classification : 57M50, 20H10, 53A20, 57M60, 57S30
Keywords: real projective structures, three-manifolds, moduli spaces, representations of groups, divisible convex sets

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