Hyperbolic Complements Research Articles

Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links

We present explicit geometric decompositions of the hyperbolic complements of alternating k-uniform tiling links, which are alternating links whose projection graphs are k-uniform tilings of S2, E2, or H2. A consequence of this decomposition is that the volumes of spherical alternating k-uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral realization of the k-uniform tiling on H2. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces Sg×I with genus g≥2 and totally geodesic boundaries. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for all hyperbolic links in Sg×I, ranging over all g≥2, is a dense subset of the interval [0,2voct], where voct≈3.66386 is the volume of the ideal regular octahedron.

Hyperbolicity of links in thickened surfaces

Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in S3. We prove a similar result for links in closed thickened surfaces S×I. We define a link to be fully alternating if it has an alternating projection from S×I to S where the interior of every complementary region is an open disk. We show that a prime, fully alternating link in S×I is hyperbolic. Similar to Menasco, we also give an easy way to determine primeness in S×I. A fully alternating link is prime in S×I if and only if it is “obviously prime”. Furthermore, we extend our result to show that a prime link with fully alternating projection to an essential surface embedded in an orientable, hyperbolic 3-manifold has a hyperbolic complement.

A Cantor set with hyperbolic complement

We construct a Cantor set in S 3 \mathbb {S}^3 whose complement admits a complete hyperbolic metric.

Angled decompositions of arborescent link complements

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements.

Rational homotopy type of subspace arrangements with a geometric lattice

Let A = { x 1 , … , x n } \mathcal {A} = \{x_1, \dotsc , x_n\} be a subspace arrangement with a geometric lattice such that codim ⁡ ( x ) ≥ 2 \operatorname {codim}(x) \geq 2 for every x ∈ A x \in \mathcal {A} . Using rational homotopy theory, we prove that the complement M ( A ) M(\mathcal {A}) is rationally elliptic if and only if the sum x 1 ⊥ + … + x n ⊥ x_1^\perp + \dotso + x_n^\perp is a direct sum. The homotopy type of M ( A ) M(\mathcal {A}) is also given: it is a product of odd-dimensional spheres. Finally, some other equivalent conditions are given, such as Poincaré duality. Those results give a complete description of arrangements (with a geometric lattice and with the codimension condition on the subspaces) such that M ( A ) M(\mathcal {A}) is rationally elliptic, and show that most arrangements have a hyperbolic complement.

Quasi-Fuchsian Seifert surfaces

Let $K \subset S^3$ be a non fibered knot with hyperbolic complement. Given a Seifert surface of minimal genus for $K$ , we prove that it corresponds to a quasi-Fuchsian group, by showing it has no accidental parabolics. In particular, this shows that any lift of the Seifert surface to the universal cover is a quasi-disk and its limit set is a quasi-circle in the sphere at infinity.

Hyperbolic surfaces in ${\mathbb P}^3(\mathbb C)$

We show a class of perturbations $X$ of the Fermat hypersurface such that any holomorphic curve from $\mathbb C$ into $X$ is degenerate. Applying this result, we give explicit examples of hyperbolic surfaces in $\mathbb P^3(\mathbb C)$ of arbitrary degree $d\ge 22$, and of curves of arbitrary degree $d\ge 19$ in $\mathbb P^2(\mathbb C)$ with hyperbolic complements.

An embedding of ? in ?2 with hyperbolic complement

An embedding of ? in ?2 with hyperbolic complement

On cyclic branched coverings of hyperbolic links

Two of the main methods for the construction of closed orientable 3-manifolds, and in particular of hyperbolic 3-manifolds, are surgery on links and branched coverings of links. If the link is hyperbolic, i.e., has hyperbolic complement, by results of Thurston most of the resulting 3-manifolds are hyperbolic. In the present paper, for a fixed integer n ⩾ 2, we consider hyperbolic 3-manifolds M n, k which are cyclic n-fold branched coverings of a hyperbolic link with two components. Our main theorem relates the classification up to isometry or homeomorphism of these manifolds to the symmetry group of the link and allows a complete classification of these manifolds in various cases; as an example, we consider cyclic branched coverings of the Whitehead link. The classification resembles the classification of lens spaces which are the cyclic branched coverings of the Hopf link (which is not hyperbolic, however); it generalizes to links with more than two components. For the proof, which consists in a mixture of geometric and algebraic arguments, we extend methods used in [5] in a special case.