Hyperbolic Manifold Of Finite Volume Research Articles

The volume of hyperbolic alternating link complements

If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v3(t(D) − 2)/2 and v3(10t(D) − 10), where v3 is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite-volume hyperbolic 3-manifolds, in the geometric topology. 2000 Mathematics Subject Classification 57M25, 57N10.

Efficient fundamental cycles of cusped hyperbolic manifolds

Let M be a manifold (with boundary) of dimension > 3, such that its interior admits a hyperbolic metric of finite volume. We discuss the possible limits arising from sequences of relative fundamental cycles approximating the simplicial volume ∥M, ∂M∥, using ergodic theory of unipotent actions. As applications, we extend results of Jungreis and Calegari from closed hyperbolic to finite-volume hyperbolic manifolds: a) Strict subadditivity of simplicial volume with respect to isometric glueing along geodesic surfaces, and b) nontriviality of the foliated Gromov norm for most foliations with two-sided branching.

The co-rank conjecture for 3–manifold groups

In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3-manifold group (also known as the cut number) is bounded below by one-third the first Betti number.

The Gauss–Bonnet Formula for Hyperbolic Manifolds of Finite Volume

Let M denote an even-dimensional noncompact hyperbolic manifold of finite volume. We show that such manifolds are candidates for minimal volume. Generalizing H. Hopf's ideas around the Curvatura integra for compact Clifford–Klein space forms, we present an elementary combinatorial-metrical proof of the Gauss–Bonnet formula for M. In contrast to former results of G. Harder and M. Gromov, our approach doesn't make use of the arithmetical and differential geometrical machinery.

The Dirac operator on hyperbolic manifolds of finite volume

The Dirac operator on hyperbolic manifolds of finite volume

Volumes of cusped hyperbolic manifolds

For n-dimensional hyperbolic manifolds of finite volume with m ⩾ 1 cusps a new lower volume bound is presented which is sharp for n = 2,3. The estimate depends upon m and the ideal regular simplex volume. The proof makes essential use of a density argument for ball packings in Euclidean and hyperbolic spaces and explicit formulae for the simplicial density function. Examples, consequences for the Gromov invariant, and-for n even-the maximal number of cusps are discussed.

Feuilletages géodésiques des variétés localement symétriques

We prove here that a hyperbolic manifold of finite volume (not necessarily complete!) does not possess a locally Lipschitz foliation with totally geodesic and complete leaves.

Spectral Convergence for Degenerating Sequences of Three Dimensional Hyperbolic Manifolds

For degenerating sequences of three dimensional hyperbolic manifolds of finite volume, we prove convergence of their eigenfunctions, heat kernel and spectral measure.

Geodesic excursions into cusps in finite-volume hyperbolic manifolds.

Geodesic excursions into cusps in finite-volume hyperbolic manifolds.