Properly Convex Bending of Hyperbolic Manifolds

Samuel A. Ballas (1) , Ludovic Marquis (2)

1 FSU Mathematics - Department of Mathematics [Tallahassee, FL]
2 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geqslant 3$ there are examples finite volume, but non-compact, properly convex $d$-manifolds. Furthermore, we show that the examples can be chosen to be either strictly convex or non-strictly convex.

Mots clés: Hilbert geometry; Hyperbolic manifold

Domaines: Topologie géométrique [math.GT]; Théorie des groupes [math.GR]; Géométrie métrique [math.MG]

Samuel A. Ballas, Ludovic Marquis. Properly Convex Bending of Hyperbolic Manifolds. Groups, Geometry, and Dynamics, 2020, 14 (2), pp. 653-688. ⟨hal-01363922v2⟩