Paul Norbury
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.
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Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs of (non-)orientable surfaces are 9-hyperbolic.
This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that the manifold is geometrically finite or that it has finitely generated fundamental group.
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。
We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most πcan be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local rigidity by Hodgson and Kerckhoff implies the global rigidity for compact orientable hyperbolic 3-cone-manifolds under the same angle assumption.
The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $n \geq 2$, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic $n$-manifolds. An analogous result holds for non-compact orientable hyperbolic arithmetic $n$-manifolds of finite volume that are geometric boundaries, for $n \geq 2$.
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.
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。
The work of Jorgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We show that there is an infinite sequence of closed orientable hyperbolic 3-manifolds, obtained by Dehn filling on the figure eight knot complement, that are uniquely determined by their volumes. This gives a sequence of distinct volumes x_i converging to the volume of the figure eight knot complement with N(x_i) = 1 for each i. We also give an infinite sequence of 1-cusped hyperbolic 3-manifolds, obtained by Dehn filling one cusp of the (-2,3,8)-pretzel li…
Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume $\frac{34π^2}{3}\cdot 16$ by using the small cover theory. In particular, we classify all of the orientable four-dimensional small covers over the right-angled 120-cell up to homeomorphism; these are all with even intersection forms.
We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.
We develop a way of seeing a complete orientable hyperbolic $4$-manifold $\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset \mathbb{H}^4$ that has a facet colouring. We also develop a way of finding totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing the result of mutations along $\mathcal{N}$. As an application of our method, we construct an example of a complete orientable hyperbolic $4$-manifold $\mathcal{X}$ with a single non-toric cusp and a complete orientable hyperbolic $4$-manifold $\mathcal{Y}$ with a single toric cusp. Both $\mathc…
Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.
In this paper, we show that any open orientable surface S can be properly embedded in H^3 as a minimizing H-surface for any 0<=H<1. We obtained this result by proving a version of the bridge principle at infinity for H-surfaces. We also show that any open orientable surface S can be nonproperly embedded in H^3 as a minimal surface, too.
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 provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen theorem on the parametrization of the Teichmüller space of the surface, the Dehn-Thurston theorem on the parametrization of measured foliations in the surface, and the Hatcher-Thurston theorem, which gives a complete minimal set of moves between pair of pants decompositions of th…
Puncture loops on a non-orientable surface
本文研究非可定向曲面上由两条单侧闭合测地线交点光滑化所得环绕过穿刺点的现象，证明其自由同伦类至多对两个奇数 $m$ 值发生绕穿刺，且若存在两个这样的 $m$，它们必为连续奇数。
This article explores closed geodesics on hyperbolic surfaces. We show that, for sufficiently large $k$, the shortest closed geodesics with at least $k$ self-intersections, taken among all hyperbolic surfaces, all lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.
Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.
In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3-manifolds that have the same set of surfaces.
Totally Geodesic Spanning Surfaces of Knots and Links in 3-Manifolds
本文在多种3-流形中构造了具有全测地张成曲面的无限纽结与链环族，并指出其与虚拟纽结的联系。
It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zeta function of the canonical component of the ${\rm PSL}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.
Geodesic simplices of pseudo-hyperbolic space
本文通过上同调解释伪双曲空间测地单纯形，并给出了其体积有限的充要条件，推导出二维双曲空间中理想测地多面体具有有限体积。
Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length of closed geodesics.
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…
The deformation space of non-orientable hyperbolic 3-manifolds
本文计算了具有理想三角剖分的有限体积不可定向双曲3-流形的形变空间及其表示簇，发现当端不可定向时存在无法由三角剖分形变实现的表示形变，并在 Gieseking 流形上验证了结论。
This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of minimal length. The main result is that, if the surface has at least one cusp, their self-intersection numbers are exactly $k$ for large enough $k$.
The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius only depending on the topology of the surface (and the size of the first embedded disk), which is disjoint from any simple closed geodesic.
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.
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。
We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. Diff. Geom. 38 (1993) 545-558], [Experimental Mathematics 10(3) (2001) 419-436]. In this paper we show that all cusped orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. Our proof is constructive, and the infinite family of geodesic knots prod…
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。
In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as Lanneau and Thiffeault's conjectural minima for orientable mapping classes, when g = 2,4 (mod 6). Our examples also show that the minimum dilatation for orientable mapping classes is strictly greater than the minimum dilatation for non-orientable ones when g = 4,6,8.
Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic $3$-Manifolds
本文用Thurston范数界定双曲3-流形上L²调和1-形式的L²-范数，推广Brock-Dunfield不等式，并研究其锐性，统一结果并提出猜想。
Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to the word metric). Also suppose that if n=3, then the boundaries of M and of M' are compact. We show that M is commensurable with M'. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as above and G is a finitely generated …
The polyhedral decomposition of cusped hyperbolic $n$-manifolds with totally geodesic boundary
该研究针对带全测地边界的体积有限非紧完备双曲$n$-流形，证明其存在由理想多面体或带单个截断面的部分截断多面体构成的多面体分解，用两种方法验证结论并证明分解数量有限，完善了相关流形分解理论。
Shortest k-Geodesics on Hyperbolic Surfaces
本文研究双曲面上闭测地线的长度与自交数的关系，改进了现有估计，将最大自交数的渐进行界从512降至128。
$L^2$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds
本文研究了凸余紧双曲3-流形钻孔短测地线后边界复射影结构的变化，给出了不依赖于边界单射半径的$L^2$-界，证明其变化受普适常数乘以测地线长度平方根的约束。
On geodesic embeddings of hyperbolic surfaces into hyperbolic 3-manifolds
本文研究闭可定向双曲曲面全测地嵌入闭可定向双曲3-流形的问题，证明所有有限不可约群作用下的准柏拉图曲面均存在等变测地嵌入，并讨论更一般有限群作用的情形。
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
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…
We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.
We enumerate all spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to non-manifold points, we remove small open neighbourhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary. There are 298 combinatorially inequivalent gluing patterns, and we show that they define 191 distinct manifolds, of which 132 are hyperbolic and 59 are not. All the 132 hyperbolic manifolds were already considered in different contexts by other authors, and we provide here their known `…
We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M < 3.44, then H_1(M;Z/2Z) has dimension at most 7.
In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give criteria for when the totally geodesic subspaces of such an orbifold determine its commensurability class. We give a parametrization of the commensurability classes of finite volume quaternionic hyperbolic orbifolds in terms of arithmeti…
On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class
本文构造了允许共定向紧致叶面但无消失欧拉类的3维流形，揭示了其与L空间猜想及左可序基本群的关系。
The lengths of geodesics on hyperbolic surfaces satisfy intriguing equations, known as identities, relating these lengths to geometric quantities of the surface. This paper is about a large family of identities that relate lengths of closed geodesics and orthogeodesics to boundary lengths or number of cusps. These include, as particular cases, identities due to Basmajian, to McShane and to Mirzakhani and Tan-Wong-Zhang. In stark contrast to previous identities, the identities presented here include the lengths taken among all closed geodesics.
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。
The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A. Epstein to arbitrary surfaces, compact or not, with or without boundary and orientable or not. We give a new method of deducing the theorem about homotopic homeomorphisms from the results about homotopic curves via the hyperbolic geometry of surfaces. This works on all but 13 surfaces where ad hoc proofs are needed.
We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics
The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable manifolds with toric boundary and we give two-sided bounds for the graph complexity of tetrahedral manifolds. As a consequ…
Let M be an orientable hyperbolic surface without boundary and let $γ$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $γ$ in H2 is shorter than $γ$.
In the present paper, we show that the minimal length of closed geodesics on finite-type hyperbolic surfaces with self-intersection number $k$ has order $2\log k$ as $k$ gets large.
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 $V$.
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.
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.
Shortest filling geodesics on hyperbolic surfaces
该研究确定了模空间中亏格$g$双曲曲面上填充（多）测地线的最短长度，证明其由补集为直角正$(8g-4)$边形的测地线实现，并构造了相应单条测地线。
We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first exampl…
In all dimensions $n \ge 5$, we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text{spin}^c$ structure, and in fact we show that there are infinitely many commensurability classes of such manifolds. These manifolds all have non-vanishing third Stiefel--Whitney class $w_3$ and are all arithmetic of simplest type. More generally, we show that for each $k \ge 1$ and $n \ge 4k+1$, there exist infinitely many commensurability classes of closed orientable hyperbolic $n$-manifolds $M$ with $w_{4k-1}(M) \ne 0$.
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.
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…
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