Any closed, oriented, hyperbolic three-manifold with nontrivial second homology has many quasigeodesic flows, where quasigeodesic means that flow lines are uniformly efficient in measuring distance in relative homotopy classes. The flows are pseudo-Anosov flows which are almost transverse to finite depth foliations in the manifold. The main tool is the use of a sutured manifold hierarchy which has good geometric properties.
We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over gamma is a hyperbolic metric space. As an application, for complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically inf…
In this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover. We prove that such flows are quasigeodesic if and only if there is an upper bound, depending only on the flow, to the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow. The main ingredient is a proof that under t…
We show that if M is a hyperbolic 3-manifold which admits a quasigeodesic flow, then pi_1(M) acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic 3-manifold without a quasigeodesic flow, answering a long-standing question of Thurston.
Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a π_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has a natural compactification as a closed disc that inherits a π_1 action. The embedding of P in H^3 extends continuously to the compactification and the restriction to the boundary is a surjective π_1-equivariant map from S^1 to S^2_\infty. This generalizes the result of Cannon and Thurston for fibered hyperbolic 3-manifolds.
We construct one dimensional foliations which are subfoliations of two dimensional foliations in 3-manifolds. The subfoliation is by quasigeodesics in each two dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves. This article answers the question whether the funnel property is a consequence of the leafwise quasigeodesic property or not.
If M is a hyperbolic 3-manifold with a quasigeodesic flow then we show that π_1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is Möbius-like but not conjugate into PSL(2, R). We conjecture that the latter possibility cannot occur.
We introduce the notion of controlled Floyd separation between geodesic rays starting at the identity in a finitely generated group G. Two such geodesic rays are said to be Floyd separated with respect to quasigeodesics if the (Floyd) length of c-quasigeodesics (for fixed but arbitrary c) joining points on the geodesic rays is asymptotically bounded away from zero. This is always satisfied by Morse geodesics. The main purpose of this paper is to furnish an example of a finitely generated group $G$ such that 1) all finitely presented subgroups of G are hyperbolic, 2) G has an uncountable family…
We consider two manifestations of non-positive curvature: acylindrical actions on hyperbolic spaces and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for studying many important families of groups, including mapping class groups, right-angled Coxeter and Artin groups, most 3-manifold groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces, so it is natural to search for a "best" one. The set of all cobounded acylindric…
Florida State University（佛罗里达州立大学）；CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche（法国国家科学研究中心，儒勒·让数学家研究所-巴黎左岸）；Facultad de Ciencias, Universidad de la República（乌拉圭共和国大学理学院）
We construct a locally hyperbolic 3-manifold $M$ such that $π_ 1(M)$ has no divisible subgroups. We then show that $M$ is not homotopy equivalent to any complete hyperbolic manifold.
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
We construct a locally hyperbolic 3-manifold $M_\infty$ such that $π_ 1(M_\infty)$ has no divisible subgroup. We then show that $M_\infty$ is not homeomorphic to any complete hyperbolic manifold. This answers a question of Agol [DHM06,Mar07].
This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also show how this result can be used to construct a complete list of all one-cusped hyperbolic three-manifolds with volume <= 2.848 and all closed hyperbolic three-manifolds with volume <= 0.943. In particular, the Weeks manifold is the unique smallest volume closed orientable hyperboli…
In this paper, we describe all the hyperbolic 24-cell 4-manifolds with exactly one cusp. There are four of these manifolds up to isometry. These manifolds are the first examples of one-cusped hyperbolic 4-manifolds of minimum volume.
A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3-manifolds which do not satisfy this condition.
In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery on geometrically finite hypebolic cone-manifolds.
We give a bounded runtime solution to the homeomorphism problem for closed hyperbolic 3-manifolds. This is an algorithm which, given two triangulations of hyperbolic 3-manifolds by at most $t$ tetrahedra, decides if they represent the same hyperbolic 3-manifold with runtime bounded by \[ 2^{2^{t^{O(t)}}}.\] We do this by first finding a hyperbolic structure on each manifold given as a geometric triangulation and then comparing the two as geometric manifolds.
We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures on closed 3-manifolds.
We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings; to show that the figure-8 knot complement and its sister are the unique hyperbolic 3-manifolds with minimal volume maximal cusps; and to extend results on determining low volume closed and cusped hyperbolic 3-manifolds.
We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.
This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This technique is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold. We prove the following result: \it\noindent Let $N$ be a closed hyperbolic 3-manifold. Then \begin{enumerate} \item[(1)] If $f\colon M \to N$ is a homotopy equivalence where $M$ is a closed irreducible 3-manifold, then $f$ is homotopic to a homeomorphism. \item[(2)] If $f,g\colon M\to N$…
We show the manifolds at infinity of the complex hyperbolic triangle groups $Δ_{3,4,4;\infty}$ and $Δ_{3,4,6;\infty}$,are one-cusped hyperbolic 3-manifolds $m038$ and $s090$ in the Snappy Census respectively.That is,these two manifolds admit spherical CR uniformizations. Moreover, these two hyperbolic 3-manifolds above can be obtained by Dehn surgeries on the first cusp of the two-cusped hyperbolic 3-manifold $m295$ in the Snappy Census with slopes $2$ and $4$ respectively. In general,the main result in this paper allow us to conjecture that the manifold at infinity of the complex hyperbolic t…
We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifolds and the Fibonacci 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.
We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic 4-manifolds contain L-spaces as hypersurfaces.
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.
We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold.
By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.
We discuss two families of closed orientable three-dimensional manifolds which arise as cyclic generalizations of two hyperbolic icosahedral manifolds listed by Everitt. Everitt's manifolds are cyclic coverings of the lens space $L_{3,1}$ branched over some 2-component links. We present results on covering properties, fundamental groups, and hyperbolic volumes of the manifolds belonging to these families.
We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.
This paper is the first in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Here we introduce Mom technology and enumerate the hyperbolic Mom-n manifolds for n <= 4.
We provide, for hyperbolic and flat 3-manifolds, obstructions to bounding hyperbolic 4-manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.
We prove that there are infinitely many pairwise non-commensurable hyperbolic $n$-manifolds that have the same ambient group and trace ring, for any $n \geq 3$. The manifolds can be chosen compact if $n \geq 4$.
We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total spaces of orientable vector bundles over M.
We show there is an upper bound on the diameter of a closed, hyperbolic 3-manifold in terms of the length of any presentation of its fundamental group.
We prove that the deformation space $AH(M)$ of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold $M$ with incompressible boundary is locally connected at quasiconformally rigid points.
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we try to establish an upper bound of the length of $n^{th}$ shortest closed geodesic of $M\setminus L$ in terms of $V$.
A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated to one of the families. It is conjectured that there are exactly 6 exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea's census of closed hyperbolic 3-manifolds, associated to 5 other families. Along w…
We construct here two new examples of non-orientable, non-compact, hyperbolic 4-manifolds. The first has minimal volume $v_m = 4π^2/3$ and two cusps. This example has the lowest number of cusps among known minimal volume hyperbolic 4-manifolds. The second has volume $2\cdot v_m$ and one cusp. It has lowest volume among known one-cusped hyperbolic 4-manifolds.
This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov--Pyatetski-Shapiro and Agol--Belolipetsky--Thomson non-arithmetic manifolds embed geodesically. Moreover, we show that the number of commensurability classes of hyperbolic manifolds with a representative of volume $\leq v$ that bounds geometrically is at least $v^{Cv}$, for $v$ large enough.
In this paper, we develop general techniques for computing the G-index of a closed, spin, hyperbolic 2- or 4-manifold, and apply these techniques to compute the G-index of the fully symmetric spin structure of the Davis hyperbolic 4-manifold.
We enumerate the small-volume manifolds that can be obtained by Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai, Meyerhoff, and the author. In so doing we complete the proof that the Weeks manifold is the minimum-volume compact hyperbolic 3-manifold, as well as enumerating the 10 smallest one-cusped hyperbolic 3-manifolds.
L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so-called canonical decomposition of hyperbolic manifolds.
In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold has many pairs of non-isometric finite covers with identical spectra. Forgetting multiplicities, we can also construct pairs where the volume ratio is unbounded.
