François Guéritaud

On canonical triangulations of once-punctured torus bundles and two-bridge link complements

We prove the hyperbolization theorem for punctured-torus bun- dles and two-bridge links by decomposing them into ideal tetrahedra which are then given hyperbolic structures.

Anosov representations and proper actions

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

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 The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements. In the 1990s, Andrew Casson introduced a powerful technique for constructing and studying cusped hyperbolic 3-manifolds. His idea was to subdivide a manifold M into angled ideal tetrahedra: that is, tetrahedra whose vertices are removed and whose edges carry prescribed dihedral angles. When the dihedral angles of the tetrahedra add up to 2! around each edge of M, the triangulation is called an angled triangulation. Casson proved that every orientable cusped 3-manifold that admits an angled triangulation must also admit a hyperbolic metric, and outlined a possible way to find the hyperbolic metric by studying the volumes of angled tetrahedra — an idea also developed by Rivin [17]. The power of Casson’s approach lies in the fact that the defining equations of an angled triangulation are both linear and local, making angled triangulations relatively easy to find and deform (much easier than to study an actual hyperbolic triangulation, as in [14, 20] or in some aspects of Thurston’s seminal approach [22]). The aim of this paper is to extend this approach to larger and more complicated building blocks. These blocks can be ideal polyhedra instead of tetrahedra, but they may also have nontrivial topology. In general, an angled block will be a 3-manifold whose boundary is subdivided into faces looking locally like the faces of an ideal polyhedron (in a sense to be defined). The edges between adjacent faces carry prescribed dihedral angles. In Section 2, we will describe the precise combinatorial conditions that the dihedral angles must satisfy. These conditions will imply the following generalization of a result by Lackenby [11, Corollary 4.6].

Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j, in the sense that the hyperbolic translation lengths for j are uniformly larger than for rho; conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation rho, whose Euler class can be prescribed. This has applications to compact anti-de Sitter 3-manifolds.

Explicit angle structures for veering triangulations

Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3‐manifolds. We prove, by a constructive argument, that every veering triangulation admits positive angle structures, recovering a result of Hodgson, Rubinstein, Segerman, and Tillmann. Our construction leads to explicit lower bounds on the smallest angle in this positive angle structure, and to information about angled holonomy of the boundary tori. 57M50, 57R05

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups

Veering triangulations and Cannon–Thurston maps

\mathrm {PO}(p,q)

Formal Markoff maps are positive

PO(p,q) by considering their action on the associated pseudo-Riemannian hyperbolic space