Cyclically symmetric hyperbolic spatial graphs in 3-manifolds

The aims of this paper is to prove that every closed connected orientable 3-manifold with an orientation-preserving periodic diffeomorphism contains infinitely many, setwise invariant, spatial graphs whose exteriors are hyperbolic 3-manifolds.     

Margulis numbers for Haken manifolds

For every closed orientable hyperbolic Haken 3-manifold and, more generally, for any orientable hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If H ₁(M;ℚ) ≠ 0, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.     

Complexity of Geometric Three-manifolds

We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev's complexity implied by our computations are sharp for thousands of manifolds, and we conjecture they are for infinitely many, including all Seifert manifolds. Our computations and estimates apply to all the Dehn fillings of M 6 ₁ ³ (the complement of the three-component chain-link, conjectured to be the smallest triply cusped hyperbolic manifold), whence to infinitely many among the smallest closed hyperbolic manifolds. Our computations are based on the machinery of the decomposition into ‘bricks’ of irreducible manifolds, developed in a previous paper. As an application of our results we completely describe the geometry of all 3-manifolds of complexity up to 9.     

Boundary slopes of knots

The results in this paper show that simple connectivity of a 3-manifold is reflected in the behavior of essential surfaces in exteriors of knots in the manifold. A corollary of the main theorem is that any non-trivial knot, with irreducible complement, in a homotopy 3-sphere must have two boundary slopes that differ by at least 2. This statement is false for knots in a homology 3-sphere. The main theorem itself applies more generally to knots in closed orientable 3-manifolds with cyclic fundamental group. Received: January 20, 1998.     

Hyperbolic spatial graphs in 3-manifolds

For any closed connected orientable 3-manifold M, we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S³ via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M.     

An algorithm for finding planar surfaces in three-manifolds

By a slope in the boundary ∂M of a 3-manifold, we mean the isotopy class α of a finite set of disjoint simple closed curves in ∂M that are nontrivial and pairwise nonparallel. In this paper, we construct an algorithm to decide whether or not a given orientable 3-manifold M contains an essential planar surface whose boundary has a given slope α. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 197–202, 2005.     

On the doubled tetrus

The “tetrus” is a member of a family of hyperbolic 3-manifolds with totally geodesic boundary, described by Paoluzzi–Zimmerman, which also contains W.P. Thurston’s “tripus.” Each member of this family has a branched cover to B ³ over a certain tangle T. This map on the tripus has degree three, and on the tetrus degree four. We describe a cover of the double of the tetrus, itself a double across a closed surface, which fibers over the circle.     

On foliated circle bundles over closed orientable 3-manifolds

We show that there exists a family of smooth orientable circle bundles over closed orientable 3-manifolds each of which has a codimension-one foliation transverse to the fibres of class C ⁰ but has none of class C ³ . There arises a necessary condition induced from the Milnor-Wood inequality for the existence of a foliation transverse to the fibres of an orientable circle bundle over a closed orientable 3-manifold. We show that with some exceptions this necessary condition is also sufficient for the existence of a smooth transverse foliation if the base space is a closed Seifert fibred manifold. Received: May 13, 1996     

Spectral geometry,link complements and surgery diagrams

We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require ‘complicated’ surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds.     

Length and Stable Length

This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive ϵ there is a positive δ depending only on n and on ϵ such that an element of π₁(M) with stable commutator length less than δ is represented by a geodesic with length less than ϵ. Moreover, for any such M, the first accumulation point for stable commutator length on conjugacy classes is at least 1/12. Conversely, “most” short geodesics in hyperbolic 3-manifolds have arbitrarily small stable commutator length. Thus stable commutator length is typically good at detecting the thick-thin decomposition of M, and 1/12 can be thought of as a kind of homological Margulis constant. Received: June 2006 Revision: May 2007 Accepted: June 2007     

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.     

On the Generation of Oriented Matroids

We provide a multiple purpose algorithm for generating oriented matroids. An application disproves a conjecture of Grünbaum that every closed triangulated orientable 2-manifold can be embedded geometrically in R ³ , i.e., with flat triangles and without self-intersections. We can show in particular that there exists an infinite class of orientable triangulated closed 2-manifolds for each genus g \geq 6 that cannot be embedded geometrically in Euclidean 3-space. Our algorithm is interesting in its own right as a tool for many investigations in which oriented matroids play a key role. Received January 7, 1999, and in final form July 16, 1999.     

Cartesian product stabilization of 3-manifolds

The Cartesian product of a closed, orientable prime geometric 3-manifold and a closed orientable surface is unique except for the case of the Cartesian product of a special class of Seifert manifolds and a torus. The same type of uniqueness holds for stabilization of 3-manifolds by an n-dimensional torus. Cartesian squares of Seifert fibered 3-manifolds are completely classified.     

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.

     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

Cusp size bounds from singular surfaces in hyperbolic 3-manifolds

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, -curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, -curve length and maximal cusp volume for hyperbolic knots in depending on crossing number. Particular improved bounds are obtained for alternating knots.

     

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.     

Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in ℝ³. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. Correcting a previous error in the literature, we construct the first infinite family of such nonrealizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g., for face lattices of polytopes.     

Construction and recognition of hyperbolic 3-manifolds with geodesic boundary

We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical decomposition, to an algorithm that computes the Kojima decomposition.

Our theory has been exploited to classify all the orientable finite-volume hyperbolic -manifolds with non-empty compact geodesic boundary admitting an ideal triangulation with at most four tetrahedra. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boundary in which the boundary is non-compact. We include this case using a suitable adjustment of the notion of ideal triangulation, and we show how this case arises within the theory of knots and links.