A Note on Surfaces Bounding Hyperbolic 3-Manifolds

We consider the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold (as some or as the only boundary component). We discuss some explicit examples of hyperbolic surfaces, in particular the surface associated to the small stellated dodecahedron (one of the four Kepler-Poinsot polyhedra) which is the boundary of a hyperbolic icosahedral 3-manifold.     

The Cannon–Thurston map for punctured-surface groups

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.     

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.     

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.     

Hyperbolic 3-Manifolds With Nonintersecting Closed Geodesics

A hyperbolic 3-manifold is said to have the spd-property if all its closed geodesics are simple and pairwise disjoint. For a 3-manifold which supports a geometrically finite hyperbolic structure we show the following dichotomy: either the generic hyperbolic structure has the spd-property or no hyperbolic structure has the spd-property. Both cases are shown to occur. In particular, we prove that the generic hyperbolic structure on the interior of a handlebody (or a surface cross an interval) of negative Euler characteristic has the spd-property. Simplicity and disjointness are consequences of a variational result for hyperbolic surfaces. Namely, the intersection angle between closed geodesics varies nontrivially under deformation of a hyperbolic surface.     

Maskit combinations of Poincaré-Einstein metrics

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3-manifold which is a finite connected sum of quotients of S³ and S²×S¹ bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.     

All flat three-manifolds appear as cusps of hyperbolic four-manifolds

There are ten diffeomorphism classes of compact, flat 3-manifolds. It has been conjectured that each of these occurs as the boundary of a 4-manifold whose interior admits a complete, hyperbolic structure of finite volume. This paper provides evidence in support of the conjecture. In particular, each diffeomorphism class of compact, flat 3-manifolds is shown to appear as one of the cusps of a complete, finite-volume, hyperbolic 4-manifold. This is done with a construction that uses special coverings of ³ by 3-balls. A further consequence of the construction is a finer result about the geometric structures which can be induced on cusps of complete, finite-volume, hyperbolic 4-manifolds. Using Mostow's Rigidity Theorem, one can show that not every flat structure occurs in this way. However, the fact that the flat structures induced on cusps of such 4-manifolds are dense in their respective moduli spaces follows from the construction.     

From the Boundary of the Convex Core to the Conformal Boundary

If N is a hyperbolic 3-manifold with finitely generated fundamental group, then the nearest point retraction is a proper homotopy equivalence from the conformal boundary of N to the boundary of the convex core of N. We show that the nearest point retraction is Lipschitz and has a Lipschitz homotopy inverse and that one may bound the Lipschitz constants in terms of the length of the shortest compressible curve on the conformal boundary.     

Detecting Incompressibility of Boundary in 3-Manifolds

A construction is presented which can be utilized to prove incompressibility of boundary in a 3-manifold W. One constructs a new 3-manifold DW by doubling W along a subsurface in its boundary. If DW is hyperbolic, and if W has compressible boundary, then DW must have a longitude of 'length' less than 4. This can be applied to show that an arc that is a candidate for an unknotting tunnel in a 3-manifold cannot be an unknotting tunnel. It can also be used to show that a 'tubed surface' is incompressible. For knot and link complements in S ³, and an unknotting tunnel, DW is almost always hyperbolic. Empirically, this construction appears to provide a surprisingly effective procedure for demonstrating that specific arcs are not unknotting tunnels.     

Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori. S. Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00024). M. Teragaito was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19540089.     

Totally knotted Seifert surfaces with accidental peripherals

We show that if there exists an essential accidental surface in the knot exterior, then a closed accidental surface also exists. As its corollary, we know boundary slopes of accidental essential surfaces are integral or meridional. It is shown that an accidental incompressible Seifert surface in knot exteriors in is totally knotted. Examples of satellite knots with arbitrarily high genus Seifert surfaces with accidental peripherals are given, and a Haken 3-manifold which contains a hyperbolic knot with an accidental incompressible Seifert surface of genus one is also given.

     

Plissage des variétés hyperboliques de dimension 3

We give a characterization of the measured geodesic laminations which can occur as the bending measured lamination of some geometrically finite metric on a 3-manifold. When the 3-manifold has incompressible boundary, such a characterization has already been given by F. Bonahon and J.-P. Otal. Here we deal with the general case.      

On Minimizing Problems with a Volume Constraint in Hyperbolic 3-Manifolds

This paper investigates the existence of an area (or Dirichlet integral) minimizing parametric surface in a hyperbolic 3-manifold subject to a volume constraint. The existence of a minimizing surface is proved, assuming some conditions on the prescribed free homotopy class. This result implies a non-existence result of minimizing surfaces of prescribed mean curvature. A criterion for the existence of surfaces of prescribed mean curvature, which turns out to be optimal in view of the non-existence result, is also obtained.     

Telescopic actions

A group action H on X is called ??telescopic?? if for any finitely presented group G, there exists a subgroup H?? in H such that G is isomorphic to the fundamental group of X/H??. We construct examples of telescopic actions on some CAT[?C1] spaces, in particular on 3 and 4-dimensional hyperbolic spaces. As applications we give new proofs of the following statements: (1) Aitchison??s theorem: Every finitely presented group G can appear as the fundamental group of M/J, where M is a compact 3-manifold and J is an involution which has only isolated fixed points; (2) Taubes?? theorem: Every finitely presented group G can appear as the fundamental group of a compact complex 3-manifold.     

A quadratic bound on the number of boundary slopes of essential surfaces with bounded genus

Let M be an orientable 3-manifold with ∂M a single torus. We show that the number of boundary slopes of immersed essential surfaces with genus at most g is bounded by a quadratic function of g. In the hyperbolic case, this was proved earlier by Hass et al.     

Hausdorff Dimension and Limits of Kleinian Groups

In this paper we prove that if M is a compact, hyperbolizable 3-manifold, which is not a handlebody, then the Hausdorff dimension of the limit set is continuous in the strong topology on the space of marked hyperbolic 3-manifolds homotopy equivalent to M. We similarly observe that for any compact hyperbolizable 3-manifold M (including a handlebody), the bottom of the spectrum of the Laplacian gives a continuous function in the strong topology on the space of topologically tame hyperbolic 3-manifolds homotopy equivalent to M. Submitted: January 1998.     

The fixed subgroups of homeomorphisms of Seifert manifolds

Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold B M,and fπ : π1(M) →π1(M) be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the rank of the fixed subgroup of fπ, namely, rank Fix(fπ) 2rankπ1(M),which is an analogue of inequalities on surface groups and hyperbolic 3-manifold groups.     

Thick Surfaces in Hyperbolic 3-Manifolds

We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian.     

On(e)-reducible 3-manifolds Which Admit Complete Surface Systems

In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M , then there exists a complete surface system S for M , which is equivalent to S′, such that D is disjoint from S . We also obtain some properties of such 3-manifolds which can be embedded in S3.     

On the complexity and volume of hyperbolic 3-manifolds

We compare the volume of a hyperbolic 3-manifold M of finite volume and a complexity of its fundamental group.