An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let be a hyperbolic 3-manifold obtained by identifying the faces of convex ideal polyhedra . If the faces of are glued to , then can be decomposed into ideal tetrahedra by subdividing the 's.

     

Minimum volume hyperbolic 3-manifolds

We enumerate the small-volume manifolds that can be obtainedby Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai,Meyerhoff, and the author. In so doing we complete the proofthat the Weeks manifold is the compact hyperbolic 3-manifoldof minimum volume, as well as enumerating the ten smallest one-cuspedhyperbolic 3-manifolds. Received October 21, 2008.     

Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let ${\mathcal{T}}$ be an ideal triangulation of M. The deformation variety ${\mathfrak{D}(\mathcal{T})}$ , a subset of which parameterises (incomplete) hyperbolic structures obtained on M using ${\mathcal{T}}$ , is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.     

Positively oriented ideal triangulations on hyperbolic three-manifolds

Let M³ be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetrahedra. We show that the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components of M³. Moreover, we show that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of hyperbolic Dehn surgery space of once-punctured torus bundles are given.     

Length multiplicities of hyperbolic 3-manifolds

LetM = ℍ³/Γ be a hyperbolic 3-manifold, where Γ is a non-elementary Kleinian group. It is shown that the length spectrum ofM is of unbounded multiplicity.     

Isometry transformations of hyperbolic 3-manifolds

The isometry group of a compact hyperbolic manifold is known to be finite. We show that every finite group is realized as the full isometry group of some compact hyperbolic 3-manifold.     

Cusp transitivity in hyperbolic 3-manifolds

Geometriae Dedicata - In this paper, we study multiply transitive actions of the group of isometries of a cusped finite-volume hyperbolic 3-manifold on the set of its cusps. In particular, we prove...     

Commensurability of 1-cusped hyperbolic 3-manifolds

We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres where every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.

     

On certain classes of hyperbolic 3-manifolds

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links. Received: 27 November 1998 / Accepted: 12 May 1999     

Vol3 and other exceptional hyperbolic 3-manifolds

D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.

     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

The known examples of transitive partially hyperbolic diffeomorphisms on 3-manifolds belong to 3 basic classes: perturbations of skew products over an Anosov map of T², perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T³. In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.