David Gabai
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.
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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.
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.
Toward the profinite rigidity of hyperbolic 3-manifolds
本文综述了可定向有限体积双曲三维流形投射刚性问题的研究进展，梳理了将问题归结到双曲情形的已有结果，并总结了支持肯定答案的证据、现有思路及所用方法。
We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to isometry by its curvature and a decorated ideal hyperbolic polyhedral metric is determined up to isometry and change of decorations by its curvature. The main tool used in the proof is the Fenchel dual of the volume function.
We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already described in the papers arXiv:2009.04997 [math.GT] and arXiv:2105.14795 [math.GT]. The 5-dimensional example is diffeomorphic to $N \times \mathbb{R}$ for some aspherical 4-manifold $N$ which does not admit any hyperbolic structure. To this purpose we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by …
We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are un…
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 note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theory this is also true in dimension $n \geq 3$. Our current result extends a previous result that flat tori in dimension $n \geq 2$ that are focally equivalent are isometric modulo rescaling.
We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditio…
Conformal tetrahedra and global rigidity of sphere packings in constant curvature background geometry
本文通过共形四面体的对偶刻画和扩展变分原理，证明了常曲率空间中四面体及3-流形上球堆积的全局刚性，并推广至高维共形单形。
Local $C^0$-semi-rigidity of meandering-hyperbolic actions
研究蜿蜒双曲作用的局部 $C^0$-半刚性问题，通过证明其在该拓扑下的局部半刚性，将稳定性理论从利普希茨 $C^0$ 拓扑扩展到完整 $C^0$ 拓扑，进而从单一原理恢复相关作用及格的 $C^0$-局部半刚性。
We consider a coarse version of the marked length spectrum rigidity: given a group with two left invariant metrics, if the marked length spectrum (the translation length function) under the two metrics are the same, then the two metrics are uniformly close. We prove the rigidity theorem for relatively hyperbolic groups. This generalizes a result of Fujiwara.
We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we show that, up to isometry, there are only finitely many hyperbolic structures on a surface that share a given ortho spectrum.
We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, $SU(np,p), Sp(2n+2,\mathbb R), SO^*(2n+2), SO(2n,2)$. This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv, Pozzetti}.
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbolic turnovers. To that end we first generalize our local rigidity result to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the geometric techniques developed by Boileau, Leeb and Porti to deform the cone-manifold structure to a complete non-sing…
This paper concerns with a rigidity of core geodesics in hyperbolic Dehn fillings. For instance, for an $n$-cusped hyperbolic $3$-manifold $M$ having non-symmetric cusp shapes, we show any Dehn filling of $M$ with sufficiently large coefficient is uniquely determined by the product of the holonomies of its core geodesics. We also explore various implications of the main results. An appendix by I. Agol provides an alternative geometric proof of one of the corollaries of our main arguments.
This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictly between r-1 and r+1. A corollary is that any small knot which has only integer boundary slopes has Property P. The proof uses connections between the topology of the complement of the knot, M, and the PSL(2, C) character variety of M that were discovered by Culler and Shalen. T…
We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by $-6$. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic $3$-space.
We give the first examples of closed fibered hyperbolic 3-manifolds whose fundamental groups are distinguished from every other finitely generated, residually finite group by their finite quotients. One of the examples is also the first example of a non-orientable profinitely rigid hyperbolic 3-manifold.
Orthospectrum and simple orthospectrum rigidity: finiteness and genericity
本文研究带测地边界的紧致双曲曲面的正交谱与简单正交谱，证明共享同一正交谱或简单正交谱的双曲曲面数量有限，一般曲面可由这两种谱唯一确定，还修正了论文中的一处证明错误。
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the existence of three different types of circles in the hyperbolic plane. The second part will be concerned with the problem of maximizing the area of polygons with fixed sidelengths. By the metric character of these problems it is necessary to dispense to a certain extent with classical hyperbolic trigonometry because angles must be replaced by length quantities…
Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities
该研究证明双曲曲面简单闭曲线长度恒等式在泰希米勒空间的刚性，构造瘦集证明一般双曲曲面的无标记简单长度谱可作为等距类的模，推广了紧曲面情形的相关结果。
In this survey paper, we outline the proofs of the rigidity results for simple, thick, hyperbolic P-manifolds found in our three earlier papers math.GR/0506518, math.GT/0410476, and math.GR/0409586. We discuss how the arguments change in the two, three, and higher dimensional settings. This paper was written for the 22nd Annual Workshop in Geometric Topology (Colorado Springs, June 2005).
Systole, inradius and rigidity of cusped hyperbolic 3-manifolds
建立了有限体积双曲三维流形中 systole 和内半径与体积之间的最优不等式，并确定了极值流形，包括八之字结补及其姐妹流形、Weeks-Matveev-Fomenko 流形等。
We show local rigidity of hyperbolic triangle groups generated by reflections in pairs of $n$-dimensional subspaces of $R^{2n}$ obtained by composition of the geometric representation in $PGL(2, R)$ with the diagonal embeddings into $PGL(2n, R)$ and $PSp^\pm(2n, R)$.
A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.
Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a space $\mathcal{T}$ of complete hyperbolic metrics on $N$ with cone singularities along the edges of the tetrahedra. We prove that $\mathcal{T}$ is homeomorphic to a Euclidean space and we compute its dimension. By means of examples, we examine if the elements of $% \mathcal{T}$ are uniquely determined by the angles around the edges of $N.$
Finite Index Rigidity of Relatively Hyperbolic Groups
本研究证明无挠且外围子群非相对双曲的相对双曲群的同构有限指数子群指数相同，推广至保持外围结构同构的情形，通过单形数量线性增长性质论证，推广了此前双曲群的相关结果。
Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space
研究双曲三维空间中非紧凸集的刚性，证明若两个非紧闭凸子集的理想边界为圆型闭集且连通分支可数，则边界间的内蕴等距可延拓为整体等距，推广了Luo-Luo-Rao的刚性定理，并建立了Weyl问题的唯一性结果。
Rigidity, volume and angle structures of 1-3 type hyperbolic polyhedral 3-manifolds
本文研究1-3型双曲多面体3-流形的刚性与角结构体积优化，证明带装饰的1-3型双曲多面体度量由曲率唯一确定，验证了Casson-Rivin体积优化方案对该类流形的适用性，还得到强1-高效三角剖分结果。
Profinite rigidity and hyperbolic four-punctured sphere bundles over the circle
本文证明了圆周上的双曲四穿孔球丛可由其基本群的有限商群在所有3-流形群中被唯一区分，并通过改进Liu的结果证明了纤维拓扑类型可由profinite完备性检测。
We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichmüller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichmüller space with the Teichmüller space of complete hyperbolic surfaces with punctures, by sho…
Poisson boundaries of building lattices and rigidity with hyperbolic-like targets
本文证明正则厚欧氏建筑及其格的泊松边界为无穷远房室空间并带调和测度类，进而推广了从建筑格到负曲率群的态射与上闭链的刚性结果。
We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set gra…
A Rigidity Theorem for Convex Sets in Hyperbolic 3-Space
本文研究了双曲三维空间中闭合凸集的刚性问题，证明了在极限点集具有零长度时，其边界度量可确定该集，且该条件最优，类比复分析中的Painlevé定理。
Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds
本文通过研究带凹口的双曲3流形的Dehn填充，证明了某些流形在所有紧致定向3流形中是有限扩张刚性的，并改进了双曲结或双曲型卫星结的有限扩张刚性判据。
Given the fundamental group $Γ$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $ρ:Γ\rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $ρ$ is maximal, then $ρ$ must be conjugated to the holonomy of the hyperbolic structure of $M$. This paper generalizes this rigidity result by showing that if a sequence of representations of $Γ$ into $\text{Isom}(\mathbb{H}^3)$ satisfies $\lim_{n \to \infty} \text{Vol}…
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.
For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of $H_i$ is a codimension one topological sphere. 2) limit set of $H_i$ is an even dimensional topological sphere. 3) $H_i$ is a codimension one duality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could be any freely indecomposable subgroup of $G_i$. 4) $H_i$ is an odd-dimensional Poincare Duality group $PD(2k+1)$. This generalizes (2). We…
The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo, who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo's conjecture was solved by Bobenko-Pinkall-Springborn, who also introduced the vertex scaling for piecewise hyperbolic metrics and proved its global rigidity. Bobenko-Pinkall-Spingborn's proof is based on their observation of the connection of vertex scaling and the geometry of polyhedra in $3$-dimensional hyperbolic space and the concavity of the volume of ideal and hyper-ideal tetrahedra. I…
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem…
Profinite rigidity and geometric convergence
本文证明 profinitely rigid 有限体积双曲流形在几何拓扑下闭合并借助 bubble-drilling 构造推广到一大类带尖点双曲流形，涵盖 Whitehead link 与 Borromean ring 补空间。
For an $n$-dimensional real hyperbolic manifold $M$, we calculate the Zariski tangent space of a character variety $χ(π_1(M),SL(n+1,\mathbb R)), n>2$ at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using real projective structures.
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).
We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficien…
For 3-dimensional hyperbolic cone structures with cone angles $θ$, local rigidity is known for $0 \leq θ\leq 2π$, but global rigidity is known only for $0 \leq θ\leq π$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $π$ do not degenerate in deformations decreasing cone angles to zero. In this paper, we give an example of a degeneration of hyperbolic cone structures with decreasing cone angles less than $2π$. These cone structures are constructed on a certain alternating link in the thickened torus by gluing four copies …
The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Weiß when the cone angles are less than π. We pr…
We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove t…
We prove that simple, thick hyperbolic P-manifolds of dimension >2 exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension >2. The key tool in the proofs of these rigidity results is a strong form of the Jordan separation theorem, for maps from S^n to S^{n+1} which are not necessarily injective.
A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.
A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients.
For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson--Courtois--Gallot's entropy rigidity result to Hilbert geometries.
Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space
本文研究了双曲三维空间中无限圆图案和凸多面体的特征化问题，证明了当0≤Θ<π时无限圆图案的刚性和统一化定理，并解决了正则圆图案的分类问题。
Failure of quasi-isometric rigidity for infinite-ended groups
本研究证明了满足特定条件的无限端群不具备拟等距刚性，结合已有工作得出剩余有限多端双曲群拟等距刚性的充要条件，证明方法改编自Whyte的相关论证。
Horoboundary and rigidity of filling geodesic currents
该研究通过构建双曲面上测地流的非对称度量，揭示了不同亏格表面的测地流空间在度量上的刚性差异。
Projective Rigidity of Once-Punctured Torus Bundles via the Twisted Alexander Polynomial
本文通过扭曲Alexander多项式验证双曲单穿孔环面丛的射影刚性，并将其与字符簇切空间上的诱导作用及谱序列中的群论作用联系起来。
We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the product of the hyperbolic plane with the real line are large-scale conformally rigid. This implies new characterizations of groups that can act properly, cocompactly by isometries on those manifolds.
Graphically discrete groups and rigidity
本文引入群的图离散性概念，证明顶点群为图离散群的群图具强刚性，包括自由积的作用刚性与特定双曲群图的拟等距刚性，还给出例子并指出其非可公度不变性。
The Manhattan curve, ergodic theory of topological flows and rigidity
本文为一般非初等双曲群引入曼哈顿曲线，利用拓扑流遍历理论和Patterson-Sullivan测度证明其可微性与刚性，并给出三角形群等显式计算。
We prove a topological rigidity result for simple, thick, hyperbolic P-manifolds of dimension 2: isomorphism of the fundamental groups implies homeomorphism of the P-manifolds. An immediate application is a diagram rigidity theorem for certain amalgamations of free groups: the direct limits of two such diagrams are isomorphic if and only if there is an isomorphism between the respective diagrams.
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