Yair N. Minsky
We give the first part of a proof of Thurston's Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a ``Lipschitz model'' for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick part, and to give a-priori geometric bounds.
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We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.
Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in math.GT/0302208, and a subsequent paper will establish the Ending Lamination Conjecture in general.
In this paper we provide the complete classification of convex cocompact Kleinian group of Hausdorff dimensions less than $1.$ In particular, we prove that every convex cocompact Kleinian group of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. The result implies that the converse of Burside's conjecture \cite{Burside} is true: All non-classical Schottky groups must have Hausdorff dimension $\ge1$. The prove of the theorem relies on the result of Hou \cite{Hou}.
We give a full proof to Agol's announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.
We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly determined by the translation lengths of images of elements of $π_1(M)$ represented by simple curves on the boundary of $M$. As a second application, we show the group of diffeomorphisms of quasifuchsian space which preserve the renormalized intersection number is generated by the (…
In this paper we provide the complete classification of Kleinian groups of Hausdorff dimensions less than $1.$ In particular, we prove that every purely loxodromic Kleinian groups of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. As an application, the result of \cite{H} then implies that, every closed Riemann surface is uniformizable by a classical Schottky group. The prove relie on the result of Hou \cite{Hou}, and space of rectifiable $\G$-invariant closed curves.
We give an alternative proof to Agol's classification of parabolic generating pairs of non-free Kleinian groups generated by two parabolic transformations. As an application, we give a complete characterisation of epimorphims between $2$-bridge knot groups and a complete characterisation of degree one maps between the exteriors of hyperbolic $2$-bridge links.
We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.
Quasisymmetric universality, quasi-isometric classification and topological rigidity of Kleinian groups
该研究针对克莱因群极限集，通过刻画几何有限极限集的拟对称普适性，实现有限生成克莱因群的拟等距分类及几乎拓扑刚性几何有限克莱因群的分类
For geometrically finite Kleinian surface groups, Bonahon and Otal proved the existence part, and partly the uniqueness part of the bending lamination conjecture. In this paper, we generalise the existence part to general Kleinian surface groups including geometrically infinite ones. We furthermore prove the compactness of the set of Kleinian surface groups realising an arbitrarily fixed data of bending laminations and ending laminations. Our proof is independent of that of Bonahon and Otal.
Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.
We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the bending data of its convex core.
We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the algebraic limit wraps within the geometric limit under the natural locally isometric covering map.
For any closed surface $S$ of genus $g \geq 2$, we show that the deformation space of marked hyperbolic 3-manifolds homotopy equivalent to $S$, $AH(S \times I)$, is not locally connected. This proves a conjecture of Bromberg who recently proved that the space of Kleinian punctured torus groups is not locally connected. Playing an essential role in our proof is a new version of the filling theorem that is based on the theory of cone-manifold deformations developed by Hodgson, Kerckhoff, and Bromberg.
Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3-manifolds homotopy equivalent to S. This 3-dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has bizarre local topology, possessing many points that are not closed. There is, however, a natural embedding of M(S) into AI(S) and a compactification of AI(S) such that this embedding extends to an embedding of the Deligne-Mumford compactification of M(S) into the compactification of AI(S).
We prove an effective version of a theorem relating curve complex distance to electric distance in hyperbolic 3-manifolds, up to errors that are polynomial in the complexity of the underlying surface. We use this to give an effective proof of a result regarding maps between curve complexes of surfaces induced by finite covers. As applications, we effectively relate the electric circumference of a fibered manifold to the curve complex translation length of its monodromy, and we give quantitative bounds on virtual specialness for cube complexes dual to curves on surfaces.
We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have `bounded geometry' (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invari…
One of the basic problems in studying topological structures of deformation spaces for Kleinian groups is to find a criterion to distinguish convergent sequences from divergent sequences. In this paper, we shall give a sufficient condition for sequences of Kleinian groups isomorphic to surface groups to diverge in the deformation spaces.
Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups
本文研究余紧Kleinian群中曲面子群的几何控制，证明对任意亏格2拟Fuchs群和余紧Kleinian群，存在曲面子群与有限指标子群拟共形共轭。
We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations. Applications include an improvement to the bounded geometry versions of Thurston's ending lamination conje…
Birational Kleinian groups
该文聚焦双有理克莱因群的研究，明确其为复射影簇上满足自由、纯不连续且余紧作用的双有理变换群，并完成了二维维度下该类群的分类工作。
Classical Kleinian groups are discrete subgroups of isometries of H n. The well-known theory of Kleinian groups starts with the definition of their associated limit set in the boundary of H n , and includes the geometric properties of the quotient hyperbolic space. This approach, naively applied, fails in the Lorentzian analogue anti-de Sitter space: discrete subgroups do not act properly discontinuously, and in many cases the set of accumulation points of orbits at the conformal boundary at infinity depends on the orbit. In this survey, we point out a way to extend this classical theory by in…
Bending parameterization of one-sided degenerate Kleinian surface groups
本文针对出现在Bers切片边界中的单侧退化非拟富克斯流形，证明其由退化端的端结构（含抛物轨迹及补集分支的终结叶状结构/共形结构）与另一端的弯曲叶状结构唯一确定，拓展了拟富克斯流形的相关唯一性结果。
Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y \subseteq \mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $\mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In t…
Troels Jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in 'most' cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty. We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian grou…
We prove the existence of Cannon-Thurston maps for simply and doubly degenerate surface Kleinian groups. As a consequence we prove that connected limit sets of finitely generated Kleinian groups are locally connected.
We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon-Thurston maps for degenerate free groups without parabolics correspond to end-points of leaves of an ending lamination in the Masur domain, whenever a point has more than …
In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups - discrete nonelementary groups of isometries of hyperbolic $3$-space. In particular we make a detailed study of two-generator groups and discuss the classification of the arithmetic generalised triangle groups (and their near relatives). This work is mainly based around my collaborations over the last two decades with Fred Gehring and Colin Maclachlan, both of whom passed away in 2012. There are many others involved as well. Over the last few decades the theory of Kleinian groups has flour…
Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree $n_k$ of these fields in dimension 3: $n_k$ does not exceed 9. Combined with previous results of Maclachlan and Nikulin, this leads to a new bound $n_k \le 25$ which is valid for all dimensions. We also obtain upper bounds for the discriminants of these fields and give some heuristic results which may be useful for the clas…
In earlier work we introduced geometrically natural probability measures on the group of all Möbius transformations in order to study "random" groups of Möbius transformations, random surfaces, and in particular random two-generator groups, that is groups where the generators are selected randomly, with a view to estimating the likely-hood that such groups are discrete and then to make calculations of the expectation of their associated parameters, geometry and topology. In this paper we continue that study and identify the precise probability that a Fuchsian group generated by two parabolic M…
Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Be…
In this paper, we classify completely hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to $π_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we construct bi-Lipschitz model manifolds for such hyperbolic 3-manifolds, which have a structure called brick decomposition and are embedded in $S \times (0,1)$. In the second theorem, we show that conversely, any such model manifold admitting a brick decomposition with reasonable conditions is bi-Lipschitz homeomorphic to a hyperbolic manifold corresponding to some ge…
Geometrical finiteness for automorphism groups via cone conjecture
本文通过锥猜想研究K3、恩里ques、Coble表面及不可约辛流形的自同构群在超球面上的等距作用的几何有限性，证明此类群非正曲率特性，并揭示K3表面的Kleinian晶格与(-2)曲线关系。
Combining Rational maps and Kleinian groups via orbit equivalence
本研究提出复多项式与Kleinian曲面群全纯配对的轨道等价框架，定义高阶Bowen-Series映射，明确配对Teichmüller空间分支，证明圆周Fuchsian群无轨道等价刚性，并分类可配对的Kleinian Bers边界群。
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.
The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map. This is an exposition of a special case of the main result of arXiv:math/0607509.
We prove a continuity property for ending invariants of convergent sequences of Kleinian surface groups. We also analyze the bounded curve sets of such groups and show that their projections to non-annular subsurfaces lie a bounded Hausdorff distance from geodesics joining the projections of the ending invariants.
We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.
It is well known that the collection of uniformizations of a closed Riemann surface $S$ is partially ordered; the lowest ones are the Schottky unformizations, that is, tuples $(Ω,Γ,P:Ω\to S)$, where $Γ$ is a Schottky group with region of discontinuity $Ω$ and $P:Ω\to S$ is a regular holomorphic cover map with $Γ$ as its deck group. Let $τ:S \to S$ be a conformal (respectively, anticonformal) automorphism of $S$ of finite order $n$, and let $(Ω,Γ,P:Ω\to S)$ be a Schottky uniformization of $S$. Assume that $τ$ lifts with respect to the previous Schottky uniformization, that is, there exists a Mö…
We survey the analogy between Kleinian groups and subgroups of the mapping class group of a surface.
Peripheral subgroups of Kleinian groups
研究Kleinian群外围结构的准等距变形空间，通过折叠组合学构造稳定的基本域区域。
A virtual Schottky group is a Kleinian group $K$ containing a Schottky group $G$ as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of their Schottky uniformizations. In this paper we provides a geometrical structural decomposition of $K$ in the particular case when $K/G$ is an abelian group.
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian punctured surface groups without accidental parabolics. In this note we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. This extends earlier work done for closed surface groups.
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This completes the proof of the conjectural picture of Cannon-Thurston maps for surface groups.
We strengthen the analogy between convex co-compact Kleinian groups and convex co-compact subgroups of the mapping class group of a surface (in the sense of B. Farb and L. Mosher).
We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich implies that this space is homeomorphic to the Gromov boundary of the complex of curves. It follows that the boundary of the complex of curves is connected in these cases, answering the conjecture of P. Storm. Other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate Kleinian groups.
We give a complete topological classification of minimal surfaces in Euclidian three-space.
We present a local classification of smooth projective surfaces in 3-space via projective transformations in accordance with singularity types of central projections up to codimension 4. We also discuss relations between our classification of Monge forms and bifurcations of parabolic curves and flecnodal curves.
We are interested in the local extrinsic geometry of smooth surfaces in 4-space, and classify jets of Monge forms by projective transformations according to $\mathcal{A}^3$-types of their central projections.
Manifold classification from the descriptive viewpoint
本文从描述集合论角度研究流形和李群离散子群的分类问题，揭示了流形同胚问题的Borel复杂性及双曲流形等距关系的分类特性。
Let (ρ_\la)_{\la\in \La} be a holomorphic family of representations of a surface group π_1(S) into PSL(2,C), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bifurcation current on the parameter space \La, that is a (1,1) positive closed current attached to the bifurcations of the family. It is defined as the $dd^c$ of the Lyapunov exponent of the representation with respect to the Brownian motion on the Riemann surface S, endowed with its Poincare metric. We show that this bifurcat…
Given a symmetry $τ$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $Γ$ uniformizing $S$, such that $K/Γ$ induces $\langle τ\rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known. A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configurati…
This article was born as a generalisation of the analysis made by Series, where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. We use the Top Terms' Relationship proved by the author and Series to determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface S as the bending measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit embedding M of a surface S is the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end i…
Infinite-type Schottky groups and group actions on infinite-type surfaces
本文研究了无穷Schottky群的构造及其在无穷类型Riemann面中的作用，提出退折定理并推广了有限秩Schottky群的情况。
Cubulating Surface-by-free Groups
本文研究了闭曲面基本群与自由群构成的正合序列，提供了双曲群可立方化的充分条件并构造实例，主要结果为非拟凸合并子群下虚拟特殊双曲群的组合定理。
Description of Origamis by Schottky groups
本文研究用虚拟 Schottky 群描述 origami 对，定义了 origami-Schottky 群，并借助 Klein-Maskit 组合定理给出了这类群的几何结构图景。
We prove that the foliated Euler caracteristic classifies amenable measured foliations up to those defined by ergodic actions of the euclidian plane.
The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with $b_2>0$ has $b_2$ curves. By the results of Kato, Nakamura and Dloussky/Oeljeklaus/Toma, this conjecture (if true) would solve this classification problem completely. We explain a new approach (based on techniques from Donaldson theory) to prove existence of curves on class VII su…
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