On Lorentz spacetimes of constant curvature

We describe in parallel the Lorentzian homogeneous spaces $G=\mathrm{PSL}(2,\mathbb{R})$ and $\mathfrak{g}=\mathfrak{psl}(2,\mathbb{R})$, and review some recent results relating the geometry of their quotients by discrete groups.Comment: 16 pages, 2 figures. Appeared in the Proceedings of the December 2012 conference "Groups, Geometry, Dynamics" (Almora, India). Editors: C.S. Aravinda, W.M. Goldman, K. Gongopadhyay, A. Lubotzky, Mahan Mj, A. Weave

Lengthening deformations of singular hyperbolic tori

Let $S$ be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of $S$ lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when $S$ becomes Euclidean, i.e. very small.Comment: 17 pages, 5 figures. To appear in: Annales de la facult\'e des sciences de Toulouse (volume dedicated to Michel Boileau's sixtieth birthday

Angled decompositions of arborescent link complements

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements.Comment: 42 pages, 23 figures. Slightly expanded exposition and reference

Convex cocompact actions in real projective geometry

We study a notion of convex cocompactness for (not necessarily irreducible) discrete subgroups of the projective general linear group acting on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good properties of classical convex cocompact subgroups in rank-one Lie groups. Extending our earlier work arXiv:1701.09136 from the context of projective orthogonal groups, we show that for word hyperbolic groups preserving a properly convex open set in projective space, the above general notion of convex cocompactness is equivalent to a stronger convex cocompactness condition studied by Crampon-Marquis, and also to the condition that the natural inclusion be a projective Anosov representation. We investigate examples.Comment: 77 pages, 6 figures. Added appendix. Removed section on Anosov right-angled reflection groups, which will appear as a separate pape