Yair N. Minsky
These revised lecture notes are an expository account of part of the proof of Thurston's Ending Lamination Conjecture for Kleinian surface groups, which states that such groups are uniquely determined by invariants that describe the asymptotic structure of the ends of their quotient manifolds.
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This paper studies the combinatorial Yamabe flow on hyperbolic bordered surfaces. We show that the flow exists for all time and converges exponentially fast to conformal factor which produces a hyperbolic surface whose lengths of boundary components are equal to prescribed positive numbers. This provides an algorithm to such problems.
This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long time behavior of the flow and the geometric meaning is investigated.
For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than $2π$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.
Geometrically and topologically random surfaces in a closed hyperbolic three manifold
本文研究闭双曲三维流形中随机极小曲面的极限测度，发现拓扑极限测度在全测地子曲面存在时完全疤痕，而几何极限测度则不然。
Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow
本文通过引入无限理想圆模式的组合Ricci流，在特定组合条件下证明了其刻画，从而肯定地解决了Rivin关于双曲三维空间中无限理想多面体的长期公开问题。
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.
Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces
研究无限三角剖分双曲曲面上组合Yamabe流，在特定条件下建立其光滑解短时存在性与唯一性，引入扩展流并证明其解全局存在与唯一性，为该流提供适定性理论。
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.
Combinatorial Calabi flows with ideal circle patterns
本文将理想圆型的组合Ricci流推广到双曲与欧几里得背景下的组合Calabi流，证明解长期存在并指数收敛到相应几何度量。
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume is roughly equal to 29.311.
Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry
该研究针对双曲背景几何中退化圆填充能否实现给定全测地曲率数据的问题，刻画了充要条件并证明唯一性，还借鉴已有方法引入组合Ricci流求解目标退化圆填充曲面。
In this paper we provide the first examples of arithmetic hyperbolic 3-manifolds that are rational homology spheres and bound geometrically either compact or cusped hyperbolic 4-manifolds.
Combinatorial Ricci Flow and Thurston's Triangulation Conjecture
本文提出组合Ricci流作为处理Thurston三角剖分猜想的一般方法，证明流收敛当且仅当三角剖分是几何的，并建立了一般双曲多面体3-流形的刚性结果。
We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.
This paper investigates the combinatorial $α$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $α$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. To handle…
We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.
Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry
本文研究双曲背景几何下带边界曲面上具锥奇点的广义圆填充，利用变分原理证明其存在性与刚性，并首次引入组合p次Calabi流以求解预设全测地曲率与离散高斯曲率的圆填充。
In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness for the extended Ricci flow for decorated hyperbolic polyhedral metrics. We prove that the extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature. If it is the case, the flow converges exponentially fast. These results apply for cusped hyperbolic …
We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo-Luo's rigidity of the discrete conformal structures and Guo's rigidity of vertex scaling on surface with boundary. As a result, new convexities of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow-Luo's combinatorial Ricci flow and Luo's c…
We present a statistical approach for the discovery of relationships between mathematical entities that is based on linear regression and deep learning with fully connected artificial neural networks. The strategy is applied to computational knot data and empirical connections between combinatorial and hyperbolic knot invariants are revealed.
We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequence of this result is that closed arithmetic hyperbolic $4$-manifolds have nonLERF fundamental groups. Along with the author's previous work, we get that, for any arithmetic hyperbolic manifold with dimension at least $4$, with possible exceptions in $7$-dimensional manifolds de…
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…
We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e. the (extended) solid angle at each vertex is equal to 4π. In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (e…
In this paper we deepen the analysis of certain classes M_{g,k} of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of M_{g,k} is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusps. We study small deformations of the complete hyperbolic structure of manifolds in M_{g,k} via a close analysis of their geodesic triangulations. We prove that several elements in M_{g,k} admit non-homeomorphic hyperbolic Dehn fillings sharing the same volume, homology, cusp vol…
In this note, we show that there exist cusped hyperbolic $3$-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic $4$-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic hyperbolic manifolds.
Combinatorial Ricci flows on infinite disk triangulations
本文首次在无限圆盘三角剖分上研究欧氏与双曲背景几何中的组合里奇流，证明其存在唯一性与收敛性，并应用于无限锥角圆填充度量的存在性。
A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and $120$-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k\geqslant 1$, we build an orientable compact closed $3$-manifold tessellated by $16k$ right-a…
We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume bounds follow from techniques related to the proof of Thurston's Orbifold Theorem, Schläfli's formula, and previous results of the author giving volume bounds for right-angled hyperbolic polyhedra.
We introduce a combinatorial curvature flow for PL metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow are established. The most important ones is that the evolution of the PL curvature satisfies a combinatorial heat equation. It implies that the total curvature decreases along the flow. The local convergence of the flow to the hyperbolic metric is also established if the triangu…
A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups
本研究提出基于双曲单纯复形作用的层级双曲群组合判据，证明映射类群模Dehn扭大幂次的商群为层级双曲群，构造非初等双曲商群并将相关有限商问题归约为特定双曲群的剩余有限性问题。
We prove a partial generalization of Bonahon's tameness result to surfaces inside irreducible 3-manifolds with hyperbolic fundamental group. Bonahon's result states that geometrically infinite ends of freely indecomposable hyperbolic 3-manifolds are simply degenerate. It is easy to see that a geometrically infinite end gives rise to a sequence of curves on the corresponding surface whose geodesic representatives are not contained in any compact set. The main step in his proof is showing that one may assume that these curves are simple on the surface. In this paper, we generalize the main step …
Bending parameterization of geometrically finite hyperbolic manifolds
本文证明非 Fuchs 型几何有限双曲结构由弯曲叶状结构唯一确定，弯曲映射为同胚，推广了凸余紧情形，并建立了边界纤维可收缩性判据。
Hyperbolic spaces with geometric and geometrically finite quasi-actions are symmetric
研究证明，若一个度量空间与有限生成群及具有上空的有限生成群空间准等距，则该空间与一维对称空间或实直线准等距。
The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3-manifold with no cusps and incompressible ends.
A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces
本文提出通过组合方法构建新的分层超几何空间，并揭示了分层超几何空间与格理论之间的联系。
We suggest a method of computing volume for a simple polytope $P$ in three-dimensional hyperbolic space $\mathbb{H}^3$. This method combines the combinatorial reduction of $P$ as a trivalent graph $Γ$ (the $1$-skeleton of $P$) by $I-H$, or Whitehead, moves (together with shrinking of triangular faces) aligned with its geometric splitting into generalised tetrahedra. With each decomposition (under some conditions) we associate a potential function $Φ$ such that the volume of $P$ can be expressed through a critical values of $Φ$. The results of our numeric experiments with this method suggest th…
We introduce a new technique for finding CAT(-1) surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is geometrically and topologically tame.
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.
In this paper we study $\times_0$-products of Lannér diagrams. We prove that every $\times_0$-product of at least four Lannér diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order $2$. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to $12$.
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner p…
We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This the first example of geometrically bounding hyperbolic knot complement and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume.
Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that such a manifold admits a piecewise Euclidean metric of non-positive curvature and the universal cover contains no isometrically embedded flat planes. The proof involves a mixture of computer computation and techniques from small cancellation theory.
Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact $3$-Manifolds with Boundary
本文研究了具有边界的紧致3流形上的组合里奇流，证明在边度数至少为9的情况下，流能指数收敛到双曲度量，并解决了相关猜想。
We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups.
Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, the number of Laplacian eigenvalues of the congruence cover Gamma(q)\ H^n smaller than lambda_0 is at most of order [Gamma:Gamma(q)]^c for some c=c(λ_0)>0.
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.
Ending currents of hyperbolic group actions
该研究建立双曲群作用的末端电流理论，推广Thurston的末端层，明确不同推广形式的差异，探讨其刻画几何有限作用的条件，给出相关例子与等价条件。
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Löbell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes …
Notions of Anosov representation of relatively hyperbolic groups
本文证明了发散的、扩展几何有限表示可视为受限Anosov表示，且此类表示在小变形下稳定，举例说明通过Galois覆盖诱导的表示具有非同胚边界扩展。
Geometrically bounding 3-manifold, volume and Betti number
本文针对几何有界闭双曲3-流形稀少的问题，利用小覆盖理论与已有结果，构造出满足指定第一贝蒂数与体积条件、可作为全测地双曲4-流形边界的闭双曲3-流形。
Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of (j,rho)-equivariant Lipschitz maps from the real hyperbolic space H^n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is "maximally stretched" by all such maps when the minimal constant is at least 1. As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to …
Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets
本文对具有圆形谢尔宾斯基极限集的几何有限双曲三维流形，分类了标架丛上水平圆流的轨道闭包，证明水平圆闭包为恰当浸入子流形，推广了凸余紧情形下的已知结果。
Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic metrics on 3-manifolds of bounded type and large heights, and prove existence of a bilipschitz diffeomorphism to a combinatorial model described explicitly in terms of the list of irreducible manifolds, the topology of the identification, and the combinatorics of the gluing maps.
We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric decomposition of M. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo [Luo05] for pseudo 3-manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.
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.
Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the number of points in w_0Gamma of norm at most T, provided that the Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the Gamma skinning size of w_0 are finite. The main ergodic ingredient in our approac…
We study unions of fundamental domains of a Fuchsian group, especially those with hyperbolic plane metric realizing the metric of the corresponding hyperbolic surface. We call these unions the \textit{geodesic covers} of the Fuchsian group or the hyperbolic surface. The paper contributes to showing that finiteness of geodesic covers is basically another characterization of geometrically finiteness. The resolution of geometrically finite case is based on Shimizu's lemma.
Finding Fibonacci in the Hyperbolic Plane
本文通过双曲平面的组合逼近考察圆盘周长、面积的指数增长及线性等周不等式，并发现这些计算与 Fibonacci 数存在意外联系。
We prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of …
Topological Characterizations of Geometrically Infinite Convergence Group Actions and Orbit Uniform Metrics
本文针对一般收敛群作用，引入轨道一致度量概念，刻画几何无限性，推广了Gromov双曲空间上作用的对应结果。
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