On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus $$g>1$$ is $$24(g-1)$$, and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order $$|G| = 24(g-1)$$ is obtained, for some G-invariant Heegaard splitting of genus $$g>1$$. If M is reducible then it is obtained by doubling an action of maximal possible order $$24(g-1)$$ on a handlebody of genus g. If M is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.     

Small volume closed hyperbolic 4-manifolds

Further examples of non-orientable compact hyperbolic 4-manifoldsof volume 32²/3 arising from torsion-free subgroups of the [5,3, 3, 3] Coxeter group are given. These are the smallest knownclosed hyperbolic 4-manifolds and arise by consideration ofmaps from the [5, 3, 3, 3] Coxeter group onto the simple simplecticgroup S₄(4).     

Relative hyperbolicity and Artin groups

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a two-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for hyperbolicity of the Deligne complex which applies to a large class of these groups for which the underlying Coxeter group is hyperbolic. The key tool in the proof is an extension of the Milnor-Svarc Lemma which states that if a group G admits a discontinuous, co-compact action by isometries on a Gromov hyperbolic metric space, then G is weakly hyperbolic relative to the isotropy subgroups of the action.      

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any $k\ge 2$, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.     

A note on the Chern-Simons invariant of hyperbolic 3-manifolds

In this note we study how the Chern-Simons invariant behaves when two hyperbolic 3-manifolds are glued together along incompressible
thrice-punctured spheres. Such an operation produces many hyperbolic 3-manifolds with different numbers of cusps sharing the same volume and the same Chern-Simons invariant. The results in this note, combined with those of Meyerhoff and Ruberman, give an algorithm for determining the unknown constant in Neumann's simplicial formula for the Chern-Simons invariant of hyperbolic 3-manifolds.

     

Hyperbolic Coxeter N-Polytopes with n+2 Facets

In this paper, we classify all the hyperbolic noncompact Coxeter polytopes of finite volume whose combinatorial type is either that of a pyramid over a product of two simplices or a product of two simplices of dimension greater than one. Combined with the results of Kaplinskaja (1974) and Esselmann (1996), this completes the classification of hyperbolic Coxeter N-polytopes of finite volume with n+2 facets.     

Rigidity of Geometrically Finite Hyperbolic Cone-Manifolds

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three-dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically finite hyperbolic 3-manifolds.     

Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron

It is known that the volume function for hyperbolic manifolds of dimension 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4π²/3. This is “half” the set of possible values for volumes, which is the integral multiples of 4π²/3 due to the Gauss-Bonnet formula Vol(M) = 4π²/3 · χ(M).     

Coxeter groups and hyperbolic manifolds

The theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6-dimensions, are given, each of very small volume, and in one case of smallest possible volume.The author is grateful to Patrick Dorey for a number of helpful conversations.Revised version: 22 December 2003     

On the Volume Formula for Hyperbolic Tetrahedra

We present a volume formula for arbitrary hyperbolic tetrahedra and give applications for Coxeter tetrahedra. Received November 6, 1997, and in revised form April 29, 1998.     

Dissecting orthoschemes into orthoschemes

We exhibit a dissection, with one degree of freedom, of an arbitrary orthoscheme in Euclidean, spherical or hyperbolic d-space into d+1 orthoschemes (Section 2); this can be interpreted as a set of relations in the scissors congruence group or, weaker, as a set of functional equations for the volume. Besides special cases where the dissection is into mutually congruent parts, we obtain, in the spherical case and for a special value of the parameter, scissors congruence formulae similar to Schläfli's period formulae for the spherical orthoscheme volume (see Section 5). In Section 6 we use the dissection to explain the structure of the volume formula for asymptotic hyperbolic 3-orthoschemes due to Lobachevsky. Finally, in Section 7, by exploiting symmetries, we show that two systems of special volume relations of Schläfli (in spherical d-space) and Coxeter (for all three geometries in dimension 3) hold even on the level of dissection. In particular, it seems that all the presently known exact values for the volume of special spherical 3-simplexes hold, independently of Schläfli's differential formula, as consequences of scissors congruence relations.     

On medial links and hyperbolic 3-manifolds with large isometry groups

Starting from the regular Platonic solids we construct links, generalizing the Borromean rings, with few components but large finite symmetry groups. We consider the 3-manifolds obtained by equivariant surgeries on these links, most of them hyperbolic, and the quotient orbifolds obtained from these group actions, among them various of the smallest known hyperbolic 3-orbifolds. Also, various of the manifolds obtained by equivariant surgery on these links are maximally symmetric hyperbolic 3-manifolds.     

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.     

Hyperbolic 3-manifolds with geodesic boundary: Enumeration and volume calculation

We describe a natural strategy to enumerate compact hyperbolic 3-manifolds with geodesic boundary in increasing order of complexity. We show that the same strategy can be applied in order to analyze simultaneously compact manifolds and finite-volume manifolds with toric cusps. In contrast, we show that if one allows annular cusps, the number of manifolds grows very rapidly and our strategy cannot be employed to obtain a complete list. We also carefully describe how to compute the volume of our manifolds, discussing formulas for the volume of a tetrahedron with generic dihedral angles in hyperbolic space.     

On a class of hyperbolic 3-manifolds and groups with one defining relation

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, arbitrarily many of the same volume. The fundamental groups of these 3-manifolds are groups with one defining relation. Our main result is a classification of these manifolds up to homeomorphism, resp. isometry.     

On quotients of Coxeter groups under the weak order

We consider quotients of finitely generated Coxeter groups under the weak order. Björner and Wachs proved that every such quotient is a meet semi-lattice, and in the finite case is a lattice [Björner and Wachs, Trans. Amer. Math. Soc. 308 (1988) 1–37]. Our result is that the quotient of an affine Weyl group by the corresponding finite Weyl group is a lattice, and that up to isomorphism, these are the only quotients of infinite Coxeter groups that are lattices. In this paper, we restrict our attention to the non-affine case; the affine case appears in [Waugh, Order 16 (1999) 77–87]. We reduce to the hyperbolic case by an argument using induced subgraphs of Coxeter graphs. Within each quotient, we produce a set of elements with no common upper bound, generated by a Maple program. The number of cases is reduced because the sets satisfy the following conjecture: if a set of elements does not have an upper bound in a particular Coxeter group, then it does not have an upper bound in any Coxeter group whose graph can be obtained from the graph of the original group by increasing edge weights.     

Spherical coxeter groups and hyperelliptic 3-manifolds

Reflection groups of Coxeter polyhedra in three-dimensional Thurston geometries are examined. For a wide class of Coxeter groups, the existence of subgroups of finite index that uniformize hyperelliptic 3-manifolds is established. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 173–177, August, 1999.     

Determining Knotsw and Links by Cyclic Branched Coverings

It is well known that different knots or links in the 3-sphere can have homeomorphic n-fold cyclic branched coverings. We consider the following problem: for which values of nis a knot of link determined by itsn-fold cyclic branched covering? We consider the class of hyperbolic resp.2π/n-hyperbolic links. The isometry or symmetry groups of such links are finite, and their n-fold branched coverings are hyperbolic 3-manifolds. Our main result states that if ndoes not divide the order of the finite symmetry group of such a link, then the link is determined by its n-fold branched covering. In a sense, the result is best possible; the key argument of its proof is algebraic using some basic result about finite p-groups. The main result applies, for example, to the cyclic branched coverings of the 2-bridge links; in particular, it gives a classification of the maximally symmetricD₆-manifolds which are exactly the 3-fold branched coverings of the 2-bridge links.     

Lower bounds on volumes of hyperbolic Haken 3-manifolds

We prove a volume inequality for 3-manifolds having metrics ``bent' along a surface and satisfying certain curvature conditions. The result makes use of Perelman's work on the Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume of orientable hyperbolic 3-manifolds. An appendix compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with experimental data.