Hyperbolic three-manifolds with trivial finite type invariants

Simple closed geodesics in cusped hyperbolic 3-manifolds

We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.

Hyperbolic Dehn surgery on geometrically infinite 3-manifolds

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.

Genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume

We describe a class of genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume.

The Homeomorphism Problem For Hyperbolic Manifolds I

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.

双曲三维流形的几何理想三角剖分

Geometric ideal triangulations of hyperbolic 3-manifolds

该研究证明了每个有限体积尖点双曲三维流形均存在有限几何理想三角剖分，从而解决了瑟斯顿理论中的长期猜想。

Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-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.

Hyperbolic 3-manifolds of low cusp volume

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.

Pants immersed in 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.

Homotopy Hyperbolic 3-Manifolds are Hyperbolic

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 $f,g\colon M\to N$…

Systoles of hyperbolic 4-manifolds

We prove that for any \e>0, there exists a closed hyperbolic 4-manifold with a closed geodesic of length < \e.

Three-manifolds at infinity of complex hyperbolic orbifolds

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…

Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds

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.

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.

$S^4$中二维环面的双曲链环补中的全测地双曲3-流形

Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in $S^4$

该研究针对四维球面中二维环面的双曲链环补空间，证明其内部不存在闭的嵌入全测地双曲3-流形，为高维双曲流形的几何结构研究提供了新结论。

Kirby diagrams and the Ratcliffe-Tschantz hyperbolic 4-manifolds

We show how to construct a Kirby diagram for a large class of finite volume hyperbolic 4-manifolds constructed by J. Ratcliffe and S. Tschantz.

双曲4-流形上的相交配对

Intersection pairing on hyperbolic 4-manifolds

该研究针对双曲4-流形的同调类相交配对问题，证明了可通过同调类的Thurston范数对其进行估计的结论。

Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants

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.

Curvature bounds for surfaces in hyperbolic 3-manifolds

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.

Convex plumbings in closed hyperbolic 4-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.

Right-angled Coxeter polytopes, hyperbolic 6-manifolds, and a problem of Siegel

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.

Cyclic generalizations of two hyperbolic icosahedral manifolds

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.

Thick surfaces in hyperbolic 3-manifolds

We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.

Mom technology and volumes of hyperbolic 3-manifolds

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.

On the geometric boundaries of hyperbolic 4-manifolds

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.

Non-commensurable hyperbolic manifolds with the same trace ring

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 $n \geq 4$.

关于双曲曲面到双曲3-流形的等变嵌入

On equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds

本文研究了闭双曲曲面到闭双曲3-流形的全测地嵌入及其等变版本，特别探讨了可定向曲面上的大反向作用及不可定向曲面上的大作用。

Intersections in hyperbolic manifolds

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.

A diameter bound for closed, hyperbolic 3-manifolds

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.

Local topology in deformation spaces of hyperbolic 3-manifolds II

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.

Geodesic Bounds of Hyperbolic Link Complements in Hyperbolic $3$-Manifold

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 $V$.

Exceptional regions and associated exceptional hyperbolic 3-manifolds

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…

Hyperbolic cone-manifolds with large cone-angles

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.

Some hyperbolic 4-manifolds with low volume and number of cusps

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.

Embedding non-arithmetic hyperbolic 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.

Counting Essential Surfaces in a Closed Hyperbolic 3-Manifold

Let M be a closed hyperbolic 3-manifold. We show that the number of genus g surface subgroups of the fundamental group of M grows like g^{2g}.

The G-index of a closed, spin, hyperbolic manifold of dimension 2 or 4

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.

球面流形中的双曲小结

Hyperbolic small knots in spherical manifolds

针对Lopez关于闭不可约非Haken 3-流形包含小结的猜想，本文构造了除棱镜流形外大多数闭可定向球面3-流形中双曲小结的显式例子。

Minimum-volume hyperbolic 3-manifolds

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.

A classification of hyperbolic manifolds related to the three-dimensional torus

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.

The genus spectrum of a hyperbolic 3-manifold

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.

双曲3-流形群中有界秩的子群

Subgroups of bounded rank in hyperbolic 3-manifold groups

本文证明双曲3-流形群中有界秩子群的有限性定理，并由此推出闭双曲3-流形的有界秩覆盖塔必为与1-轨道形上纤维化相关的有限覆盖塔。

Self-bumping of deformation spaces of hyperbolic 3-manifolds

Let $N$ be a hyperbolic 3-manifold and $B$ a component of the interior of $AH(π_1(N))$, the space of marked hyperbolic 3-manifolds homotopy equivalent to $N$. We will give topological conditions on $N$ sufficient to give $ρ\in \bar{B}$ such that for every small neighborhood $V$ of $ρ$, $V \cap B$ is disconnected. This implies that $\bar{B}$ is not manifold with boundary.

Closed Geodesic Length Bounds of Hyperbolic Link Complements in Hyperbolic $3$-Manifolds

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 establish an upper bound of the length of an $n^{th}$ shortest closed geodesic as a logarithmic function of $V$ in $M\setminus L$. Our works complement the work of Lakeland and Leininger \cite{Christopher} on the upper bound of systole length.

Commensurators of cusped hyperbolic manifolds

This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all non-arithmetic hyperbolic once-punctured torus bundles over the circle. For hyperbolic 3-manifolds, the algorithm has been implemented using Goodman's computer program Snap. We use this to determine the commensurability classes of all cusped hyp…

某些 cusps 转移的双曲 4 维流形

Some cusp-transitive hyperbolic 4-manifolds

本文研究了4种闭合可定向平坦3维流形作为双曲4维流形的cusp截面，其对称群在cusp集合上转移。

Regenerating hyperbolic cone 3-manifolds from dimension 2

We prove that a closed 3-orbifold that fibers over a hyperbolic polygonal 2-orbifold admits a family of hyperbolic cone structures that are viewed as regeneration of the polygon, provided that the perimeter is minimal.

Angle structures and hyperbolic $3$-manifolds with totally geodesic boundary

This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic $3$-manifold with totally geodesic boundary has an ideal triangulation that admits angle structures.

The smallest hyperbolic 6-manifolds

By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.

The curve complex and covers via hyperbolic 3-manifolds

Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via hyperbolic 3-manifolds.

Injectivity Radius and Fundamental Group of Hyperbolic 3-Manifolds

We show that there is an upper bound on the injectivity radius of a hyperbolic 3-manifold in terms of the the number of generators of its fundamental group.

有界秩的算术双曲流形

Arithmetic hyperbolic manifolds with bounded rank

该研究证明有界秩的闭算术双曲n-流形具有有界伴随迹场次数与下界有界内射半径，进而得到闭算术双曲3-流形的可公度类有限、海加德亏格与秩的关系等推论。

Lower bounds on volumes of hyperbolic Haken 3-manifolds

In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a hyperbolic manifold M contains an acylindrical surface S, then Vol(M)>= -2 V_3 chi(S). We also show that if beta_1(M)>= 2, then Vol(M)>= 4/5 V_3.

具有有限体积的非紧致双曲四维流形不存在自旋结构

A cusped hyperbolic 4-manifold without spin structures

研究构建了不存在自旋结构的非紧致双曲四维流形，探讨其几何与拓扑特性

Subsurface distances for hyperbolic 3-manifolds fibering over the circle

For a hyperbolic fibered 3-manifold M, we prove results that uniformly relate the structure of surface projections as one varies the fibrations of M. This extends our previous work from the fully-punctured to the general case.