Hyperbolic Polyhedra Research Articles

On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles

In 1832, Jakob Steiner (Systematische Entwicklung der Abhangigkeit geometrischer Gestalten von einander, Reimer (Berlin)) asked for a characterization of those planar graphs which are combinatorially equivalent to polyhedra inscribed in the sphere. This question was answered in the 1990s by Igor Rivin (Ann. of Math. 143(1996), 51–70), as a byproduct of his classification of ideal polyhedra in hyperbolic 3-space. Rivin also proposed a more direct approach to these results in Rivin (Ann. of Math. 139 (1994), 553–580). In this paper, we prove a combinatorial result (Theorem 6) which enables one to complete the program of Rivin (Ann. Math. 139(1994), 553–580).

The Laplacian and Dirac operators on critical planar graphs

On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs.

ON ARCHIMEDEAN LINK COMPLEMENTS

We study a subclass of alternating links for which the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.

Gluing copies of a 3-dimensional polyhedron to obtain a closed nonpositively curved pseudomanifold

Let S be a smooth 3-dimensional nonpositively curved Riemannian manifold with corners, whose boundary consists of a finite number of geodesically convex nonpositively curved faces (for example, a Euclidean or hyperbolic polyhedron). We show that it is always possible to glue together finitely many copies of S so as to get a nonpositively curved pseudomanifold

A Schläfli-type formula for convex cores of hyperbolic 3-manifolds

In 3-dimensional hyperbolic geometry, the classical Schlafli formula expresses the variation of the volume of a hyperbolic polyhedron in terms of the length of its edges and of the variation of its dihedral angles. We prove a similar formula for the variation of the volume of the convex core of a geometrically finite hyperbolic 3--manifold M, as we vary the hyperbolic metric of M. In this case, the pleating locus of the boundary of the convex core is not constant any more, but we showed in an earlier paper that the variation of the bending of the boundary of the convex core is described by a geodesic lamination with a certain transverse distribution. We prove that the variation of the volume of the convex core is then equal to 1/2 the length of this transverse distribution.

Dihedral Branches Coverings of Hyperbolic Orbifolds

We consider the class of hyperbolic 3-orbifolds whose underlying topological space is the 3-sphere S3 and whose singular set is a trivalent graph with singular index 2 along each edge (an important special case occurs when the trivalent graph is the 1-skeleton of a hyperbolic polyhedron). Our main result is a classification of the D-branched coverings of these orbifolds (where D2 is the dihedral group of order 4) under some general conditions on their isometry groups or the lengths of their geodesics.

A note on hyperbolic 3-manifolds of the same volume

We describe a method for constructing an arbitrary number of closed hyperbolic 3-manifolds of the same volume. In fact we prove that many hyperbolic 3-manifolds of finite volume have an arbitrary number of non-homeomorphic finite convering spaces of the same degree and hence the same volume. This applies, for example, to all hyperbolic 3-manifolds whose universal covering group is a subgroup of finite index in a Coxeter group generated by the reflections in the faces of a hyperbolic Coxeter polyhedron. It also applies to all hyperbolic 3-manifolds of finite volume with at least one cusp.

Smallest universe of negative curvature.

The smallest known three-dimensional closed manifold of curvature k\ensuremath{\equiv}-1 was discovered a few years ago by Weeks. This kind of manifold is constructed from a hyperbolic polyhedron with faces pairwise identified. Here it is used as the comoving spatial section of a Friedmann cosmological model, in the spirit of Ellis and Schreiber's idea of small universes. Its nontrivial global topology has the effect of producing multiple images of single cosmic sources, and this is the basis of an attempt to solve a famous controversy about the redshifts of quasars.

On the volume and surface area of hyperbolic polyhedra

A truncated tetrahedron is a building block of hyperbolic 3-manifolds with totally geodesic boundary. We study the relation between the volume of a truncated tetrahedron and the area of its faces which form the boundary of manifolds.

Whitehead groups of certain hyperbolic manifolds

An aspherical manifold is a connected manifold whose universal cover is contractible. It has been conjectured that the Whitehead groups Whj (π1 M) (including the projective class group, the original Whitehead group of π1M, and the higher Whitehead groups of [9]) vanish for any compact aspherical manifold M. The present paper considers this conjecture for twelve hyperbolic 3-manifolds constructed from regular hyperbolic polyhedra. Hyperbolic manifolds are of special interest in this regard since so much is known about their topology and geometry and very little is known about the algebraic K-theory of hyperbolic manifolds whose fundamental groups are not generalized free products.

The Hausdorff dimension of the limit set of a geometrically finite Kleinian group

We consider in this paper groups of MObius transformations of l~=RtJ {zc}. Such a group is called Kleinian if it acts discontinuously somewhere in 1~. The action of G extends to the (n+ D-dimensional hyperbolic space H-I=R• zc) and G is geometrically finite if there is a hyperbolic fundamental polyhedron with a finite number of faces for the action of G in H ' ' t (for a more precise definition see [15. 1 B]). We prove in this paper that the Hausdor[f dimension dim~L(G) of the limit set L(G) of a geometrically finite Kleinian group G of 1~ is less than n (Theorem D). Our proof of this theorem is based on the following obser~'ation, Assume that G is of compact t3'pe, i.e. if IzI-I=H-IuI~ ''. then ((-I-~\L(G))/G is compact. Then there is an integer such that if we divide any n-cube Q of R into q equal subcubes, then at least one of these subcubes does not touch L(G). Let L4~Q) be the family of these subcubes which touch L(G). Then the n-measures of Q' E,Le(Q) do not add up to the n-measure of Q and we get

The orbit space of a Kleinian group: Riley’s modest example

In the previous paper, Robert Riley [4] and his computer file Poincare found a fundamental domain for the action of a discrete group G of isometries of hyperbolic space HI generated by three parabolics. In this paper, we show that the orbit space H3/G is homeomorphic to a complement S3 k*, where k* is k union a point and where k is the (3, 3, 3) pretzel knot. Furthermore, H 3/G is equipped with an infinite volume hyperbolic orbifold structure. This should not be confused with the complete finite volume hyperbolic structure on S3 k. A neighborhood of k in H 3/G is not the quotient of a horoball by a group of parabolic isometries with common fixed point. It is instead the quotient of a neighbourhood of the domain of discontinuity for G. In addition, G has elliptic elements of order three which give rise to three singular axes in the hyperbolic structure. These three axes meet at a point at infinity in H 3/G. This accounts for the additional missing point in 3k*. A neighbourhood of this point is the quotient of a horoball by the Euclidean (3,3,3) triangle group. For a treatment of hyperbolic and orbifold structures, see Thurston [5]. The computer output form Riley's program consists of a picture of the fundamental domain 6D, and a data output giving the face pairings and face pairing transformations. For this paper, only the picture of 6D appearing in Section 3 of the preceding paper is used, as the information that it contains is sufficient to determine the face pairings uniquely. As a result, our labellings are different from those appearing in the data output, but the face pairings are the same. To show the homeomorphism H03/G = 3k*, we will glue up by identifying paired faces. The geometric structure will then arise as a direct consequence of the gluing. The domain 6X is an infinite volume hyperbolic polyhedron. In the upper half space model G&3 it lies between two EH-planes parallel to the imaginary axis of the boundary complex plane iro. The E-closure of 6D, GD, contains three subsets of i0. Two of these are compact, and are labelled Y and Z. The third region, labelled X, has connected closure in ro* and intersects any neighborhood of the point { oo Figure 1 shows a slightly altered version of the original computer drawing with added labellings of faces and edges. Some of the edges of 6D are EH-lines and, as such, they do not appear on the computer drawing. This poses no difficulty, as these edges will be subsumed by the first gluing step and will play no further part in the discussion.