Cones Embedded in Hyperbolic Manifolds

Cones Embedded in Hyperbolic Manifolds

We show that the existence of a maximal embedded tube in a hyperbolic n-manifold implies the existence of a certain conical region. One application is to establish a lower bound on the volume of the region outside the tube, thereby improving estimates on volume and estimates on lengths of geodesics in small volume hyperbolic 3-manifolds. We also provide new bounds on the injectivity radius and ...

Intersections in hyperbolic manifolds

We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension ≥ 3, only finitely many hyperbolic n–manifolds are total spaces of orientable vector bundles over M . AMS Classification numbers Primary: 30F40, 53C23, 57R20 Secondary: 22E40, 32H20, 51M10

Horocycles in hyperbolic 3-manifolds

Hyperideal polyhedra in hyperbolic manifolds

Let (M, ∂M) be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric (but is not a solid torus). We consider the hyperbolic metrics on M such that ∂M looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon [BB02] on hyperideal polyhedra, and those of Rousset [Rou02...

Arithmetic of Hyperbolic Manifolds

By a hyperbolic 3-manifold we mean a complete orientable hyperbolic 3-manifold of finite volume, that is a quotient H/Γ with Γ ⊂ PSL2C a discrete subgroup of finite covolume (here briefly “a Kleinian group”). Among hyperbolic 3-manifolds, the arithmetic ones form an interesting, and in many ways more tractable, subclass. The tractability comes from the availability of arithmetic tools and invar...