Related papers: Injectivity Radius Bounds in Hyperbolic Convex Cor…

For any hyperbolic 3-manifold $M$ with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of…

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations…

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the…

We prove that the continuous cohomology of $\text{Isom}^+(\mathbb{H}^n)$ can be measurably realized on the boundary of hyperbolic space. This implies in particular that for $\text{Isom}^+(\mathbb{H}^n)$ the comparison map from continuous…

We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and…

We show there is an upper bound on the diameter of a closed, hyperbolic 3-manifold in terms of the length of any presentation of its fundamental group.

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend…

We show that the diameter of the skinning map of an acylindrical hyperbolic 3-manifold M is bounded on thick Teichmueller geodesic rays by a constant depending only on the thickness of the ray and the topological type of the boundary of M.

We consider quasifuchsian manifolds with "particles", i.e., cone singularities of fixed angle less than $\pi$ going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity…

We classify convex disks with a fixed characteristic foliation and Legendrian boundary, up to contact isotopy relative to the boundary, in every closed overtwisted contact 3-manifold. This classification covers cases where the neighborhood…

In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$ of the first type, the systole of most of the principal congruence coverings $M_{I}$ satisfy $$sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$$ where $c$ is…

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence $(M_{i}%, p_{i}) $ of pointed…

Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N,C) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set C of simple closed curves. We prove that the map…

We show that the base space of a homotopy cofibration is locally hyperbolic under various conditions. In particular, if these manifolds admit a rationally elliptic closure, then almost all punctured manifolds and almost all manifolds with…

The continuity of the injectivity radius of a compact manifold under $C^2$ perturbation of the Riemannian metric was originally proved by P. Ehrlich (Composito Math., 1974), and later the proof was simplified by T. Sakai (Math. J. Okayama…

We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each g at least 2, to identify a constant r_{g-1,2} with the property that the set of closed genus-g hyperbolic surfaces with…

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific…

Making use of the dual Bonahon-Schl\"afli formula, we prove that the dual volume of the convex core of a quasi-Fuchsian manifold $M$ is bounded by an explicit constant, depending only on the topology of $M$, times the Weil-Petersson…

We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.