All flat manifolds are cusps of hyperbolic orbifolds

On volumes of complex hyperbolic orbifolds

We construct an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.

Non-isometric hyperbolic 3-orbifolds with the same topological type and volume

We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.

具有翻转端的双曲3-轨形上的射影形变

Projective deformations of hyperbolic 3-orbifolds with turnover ends

本文研究具有翻转端的双曲3-轨形的射影形变，证明其唯一形式为射影尖点开口，保持适当凸性，并完整刻画了翻转$S^2(3,3,3)$在SL(4,R)中的特征簇。

The Dehn filling space of a certain hyperbolic 3-orbifold

We construct the first example of a ``one-cusped'' hyperbolic 3-orbifold for which we see the true shape of the space of hyperbolic Dehn fillings.

The non-arithmetic cusped hyperbolic 3-orbifold of minimal volume

We show that the 1-cusped quotient of the hyperbolic space $\mathbb{H}^3$ by the tetrahedral Coxeter group $Γ_*=[5,3,6]$ has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice $Γ_*$ is incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in the work of C. Adams. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp.

Volumes of Picard modular surfaces

We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that simultaneously cover both minimal orbifolds.

On the volume and the Chern-Simons invariant for the hyperbolic alternating knot orbifolds

We extend the Neumann's methods and give the explicit formulae for the volume and the Chern-Simons invariant for hyperbolic alternating knot orbifolds.

Immersed Turnovers In Hyperbolic 3-Orbifolds

We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic boundary, called the "turnover core,'' whose volume is bounded from above by a function depending only on the area of the given turnover. Furthermore, we show that, for a given type of turnover, there are only finitely many possibilities for the turnover core. As a corollary, if the volume of a complete orientable hyperbolic 3-orbifold is at least 2πand if the fund…

The maximum injectivity radius of hyperbolic orbifolds

For two-dimensional orientable hyperbolic orbifolds, we show that the radius of a maximal embedded disk is greater or equal to an explicit constant ρ_T, with equality if and only if the orbifold is a sphere with three cone points of order 2, 3 and 7.

Infinitesimal rigidity of a compact hyperbolic 4-orbifold with totally geodesic boundary

Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.

Projective structures on a hyperbolic 3-orbifold

We compute and analyse the moduli space of those real projective structures on a hyperbolic 3-orbifold that are modelled on a single ideal tetrahedron in projective space. Parameterisations are given in terms of classical invariants, traces, and geometric invariants, cross ratios.

Simple closed geodesics in cusped hyperbolic 3-manifolds

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

On volumes of quaternionic hyperbolic n-orbifolds

By use of H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group, we construct an explicit lower bound for the volume of a quaternionic hyperbolic orbifold that depends only on dimension.

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.

Explicit formulae for Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

Hyperbolic 24-cell 4-manifolds with one cusp

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.

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.

Guts and volume for hyperbolic $3$-orbifolds with underlying space $S^3$

For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain a lower bound on its volume in the case that it contains an essential $2$-suborbifold with underlying space the $2$-sphere with four cone points. Our techniques involve computing the guts of the orbifold split along the $2$-suborbifold via a careful analysis of its topology. We also characterize the orbifolds of this type that have empty guts.

Kleinian orbifolds uniformized by RP groups with an elliptic and a hyperbolic generator

We consider non-elementary Kleinian groups Γ, without invariant plane, generated by an elliptic and a hyperbolic element with their axes lying in one plane. We find presentations and a complete list of orbifolds uniformized by such Γ.

On volumes of hyperbolic orbifolds

In this paper we derive an explicit lower bound on the volume of a hyperbolic $n$-orbifold for dimensions greater than or equal to four. Our main tool is H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a discrete subgroup of a semisimple Lie group.

Densities of Hyperbolic Cusp Invariants

We find that cusp densities of hyperbolic knots in the 3-sphere are dense in [0,0.6826...] and those of links are dense in [0,0.853...]. We define a new invariant associated with cusp volume, the cusp crossing density, as the ratio between the cusp volume and the crossing number of a link, and show that cusp crossing density for links is bounded above by 3.1263.... Moreover, there is a sequence of links with cusp crossing density approaching 3. The least upper bound for cusp crossing density remains an open question. For two-component hyperbolic links, cusp crossing density is shown to be dens…

Minimum volume cusped hyperbolic three-manifolds

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…

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。

Small hyperbolic polyhedra

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic turnover, we classify the triangle subgroups of the fundamental groups of orientable 3-dimensional hyperbolic tetrahedral orbifolds in the case when all of the vertices of the tetrahedra are non-finite, and we provide a conjectural classification of all the triangle subgroups of the…

双曲4流形的尖点与可公度类

Cusps and Commensurability Classes of Hyperbolic 4-Manifolds

本文为算术双曲4流形的可公度类何时包含给定尖点类型的代表元提供判据，并首次给出无穷多个不含某些尖点类型的可公度类例子。

Cusp transitivity in hyperbolic 3-manifolds

In this paper, we study multiply transitive actions of the group of isometries of a cusped finite-volume hyperbolic 3-manifold on the set of its cusps. In particular, we prove a conjecture of Vogeler that there is a largest $k$ for which such $k$-transitive actions exist, and that for each $k \geq 3$, there is an upper bound on the possible number of cusps.

Explicit formulae for Chern-Simons invariants of the hyperbolic orbifolds of the knot with Conway's notation $C(2n, 3)$

We calculate the Chern-Simons invariants of the hyperbolic orbifolds of the knot with Conway's notation $C(2n, 3)$ using the Schläfli formula for the generalized Chern-Simons function on the family of $C(2n,3)$ cone-manifold structures. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee's methods. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic $C(2n,3)$ orbifolds.

可定向紧边双曲3流形的完整10四面体普查

The complete $10$-tetrahedra census of orientable cusped hyperbolic $3$-manifolds

本研究通过普查10个四面体的双曲3流形，发现新的流形及例外Dehn填充，并揭示了新的最简单双曲结外皮。

The minimal volume orientable hyperbolic 3-manifold with 4 cusps

We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also need to estimate the volume of a hyperbolic 3-manifold with totally geodesic boundary which contains an essential surface with non-separating boundary.

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

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…

Achiral 1-cusped hyperbolic 3-manifolds not coming from amphicheiral null-homologous knot complements

It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.

Cusp transitivity in hyperbolic 3-manifolds

Let $M$ be a cusped finite-volume hyperbolic three-manifold with isometry group $G$. Then $G$ induces a $k$-transitive action by permutation on the cusps of $M$ for some integer $k\ge 0$. Generically $G$ is trivial and $k=0$, but $k>0$ does occur in special cases. We show examples with $k=1,2,4$. An interesting question concerns the possible number of cusps for a fixed $k$. Our main result provides an answer for $k=2$ by constructing a family of manifolds having no upper bound on the number of cusps.

构造等谱但不等距的多尖点双曲流形

Constructing multi-cusped hyperbolic manifolds that are isospectral and not isometric

本文将Garoufalidis与Reid的1-尖点双曲3-流形研究拓展至多尖点场景，借助Sunada方法和强逼近定理构造出任意多尖点的等谱不等距双曲3-流形，且这类流形具有相同的Eisenstein级数、无限离散谱与复长度谱。

A generalization of the Dijkgraaf-Witten invariants for cusped 3-manifolds

We introduce a generalization of the Dijkgraaf-Witten invariants for cusped or compact oriented 3-manifolds. We show that the generalized DW invariants distinguish some pairs of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same Turaev-Viro invariants. We also present an example of a pair of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same homology groups, whereas with distinct generalized DW invariants.

On quotient orbifolds of hyperbolic 3-manifolds of genus two

We analyze the orbifolds that can be obtained as quotients of hyperbolic 3-manifolds admitting a Heegaard splitting of genus two by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the hyperbolic 2-fold branched coverings of 3-bridge links. If the 3-bridge link is a knot, we prove that the underlying topological space of the quotient orbifold is either the 3-sphere or a lens space and we describe the combinatorial setting of the singular set for each possible isometry group. In the case of 3-bridge links with two or three components, the situation …

算术双曲流形的尖点类型

Cusp types of arithmetic hyperbolic manifolds

本文建立了平坦流形作为尖点横截面出现在算术双曲流形共形类中的充要条件，将问题简化为和乐群的有理表示问题，并证明了和乐表示对拟算术流形构成阻碍。

Guts 与具有 3 个尖点的最小体积可定向双曲 3-流形

Guts and The Minimal Volume Orientable Hyperbolic 3-Manifold with 3 Cusps

本文借助缝合流形与 pared 流形中的 guts，证明满足所有二阶同调类均为 libroid 的三尖点可定向双曲 3-流形体积不低于 6 倍 Catalan 常数。

Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study the deformation space of real projective structures on a compact Coxeter $n$-orbifold $Q$ admitting a hyperbolic structure. Let $e_+(Q)$ be the number of ridges of order $\geq 3$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_+(Q) …

具有孤立奇点的复双曲2-轨道形

Complex hyperbolic 2-orbifolds with isolated singularities

本文对每个素数 p 构造了具有孤立奇点的紧致复双曲 2-轨道形，并指出 p=2 情形的构造与复双曲格在挠元上的共轭可分性密切相关。

A new family of minimal ideal triangulations of cusped hyperbolic 3-manifolds

Previous work of the authors with Bus Jaco determined a lower bound on the complexity of cusped hyperbolic 3-manifolds and showed that it is attained by the monodromy ideal triangulations of once-punctured torus bundles. This paper exhibits an infinite family of minimal ideal triangulations of Dehn fillings on the link $8^3_9$ that also attain this lower bound on complexity.

Volume and Homology for Hyperbolic $3$-Orbifolds, I

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is irreducible, and $|{\mathfrak M}|$ is irreducible for every two-sheeted (orbifold) covering $\widetilde{\mathfrak M}$ of ${\mathfrak M}$, and if ${\rm vol} {\mathfrak M}\le1.72$, then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 15$. Furthermore, if ${\rm vol} {\mathfrak M}\le1.22$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 11$, and if ${\rm vol} {\mathfrak M}\le0.…

一个带尖点的小型双曲4-流形

A small cusped hyperbolic 4-manifold

本研究通过拼接Kerckhoff和Storm的多面体制成了已知最小的、与理想24-胞腔等不可公度的带尖点算术可定向双曲4-流形，其体积为最小体积的两倍。

The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold

Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to the circle, according as the center of the fundamental group of M is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings of…

Combinatorial Ricci flow on cusped 3-manifolds

Combinatorial Ricci flow on a cusped $3$-manifold is an analogue of Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact $3$-manifolds with boundary for finding complete hyperbolic metrics on cusped $3$-manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped $3$-manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow…

算术双曲流形的尖点横截面现象

Cusp cross-section phenomena for arithmetic hyperbolic manifolds

本文研究算术双曲流形中平坦流形作为尖点横截面的唯一性（UCC性质），在维度≥32构造具有该性质的例子，并证明此类可公度类数量无界，同时给出维度≥24中不能共现的流形对。

Arithmetic cusp shapes are dense

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat $n$-manifold $M$, we show that the set of similarity classes of flat metrics on $M$ which occur as a cusp cross-section of a hyperbolic $(n+1)$-orbifold is dense in the space of similarity classes of flat metrics on $M$. The set used for density is precisely the set of those classes which arise in arithmetic orbifolds.

The panted cobordism group of cusped hyperbolic 3-manifolds

For any oriented cusped hyperbolic $3$-manifold $M$, we study its $(R,ε)$-panted cobordism group, which is the abelian group generated by $(R,ε)$-good curves in $M$ modulo the oriented boundaries of $(R,ε)$-good pants. In particular, we prove that for sufficiently small $ε>0$ and sufficiently large $R>0$, some modified version of the $(R,ε)$-panted cobordism group of $M$ is isomorphic to $H_1(\text{SO}(M);\mathbb{Z})$.

The three smallest compact arithmetic hyperbolic 5-orbifolds

We determine the three hyperbolic 5-orbifolds of smallest volume among compact arithmetic orbifolds, and we identify their fundamental groups with hyperbolic Coxeter groups. This gives two different ways to compute the volume of these orbifolds.

1-efficient triangulations and the index of a cusped hyperbolic 3-manifold

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index structure if and only if T is 1-efficient and (b) if M is hyperbolic, it has a canonical set of 1-efficient ideal triangulations related by 2-3 and 0-2 moves which preserve the 3D index. We illustrate our results with several examples.

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.

Commensurability of 1-cusped hyperbolic 3-manifolds

We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres with the property that every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.

Currents on cusped hyperbolic surfaces and denseness property

The space $\mathrm{GC} (Σ)$ of geodesic currents on a hyperbolic surface $Σ$ can be considered as a completion of the set of weighted closed geodesics on $Σ$ when $Σ$ is compact, since the set of rational geodesic currents on $Σ$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(Σ)$. We prove that even when $Σ$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(Σ)$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $Σ$ but also to weighted geodesics connecting two cusps. In addition, we …

带尖点边界的双曲流形的Bridgeman-Kahn恒等式

The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary

本研究将Bridgeman-Kahn恒等式推广到带全测地边界的有限体积可定向双曲$n$-流形，针对非紧边界情形补充了边界尖点内蕴不变量相关项，完善了该恒等式的适用范围。