Kleinian groups with discrete length spectrum

Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions

Using the rings of Lipschitz and Hurwitz integers $\mathbb{H}(\mathbb{Z})$ and $\mathbb{H}ur(\mathbb{Z})$ in the quaternion division algebra $\mathbb{H}$, we define several Kleinian discrete subgroups of $PSL(2,\mathbb{H})$

克莱因观点在高秩世界中的应用

Kleinian viewpoints on higher rank worlds

本文探讨了克莱因群理论在高秩李群中的应用，旨在引导研究者关注离散子群的性质与相关数学结构。

A characterization of quaternionic Kleinian groups in dimension 2 with complex trace fields

Let $G$ be a non-elementary discrete subgroup of $\mathrm{Sp}(2,1)$. We show that if the sum of diagonal entries of each element of $G$ is a complex number, then $G$ is conjugate to a subgroup of $\mathrm{U}(2,1)$.

Primitive stable representations of free Kleinian groups

In this paper, we give a complete criterion for a discrete faithful representation $ρ:F_n \ra \pslc$ to be primitive stable. This will answer Minsky's conjectures about geometric conditions on $\H^3/ρ(F_n)$ regarding the primitive stability of $ρ$.

Random Kleinian Groups, II : Two parabolic generators

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…

具有大临界指数的复双曲Kleinian群

Complex hyperbolic Kleinian groups of large critical exponents

该研究证明二维和三维复双曲空间中存在临界指数可任意接近但严格小于最大值的离散等距群，表明复双曲Kleinian群的临界指数取值无间隙。

克莱因群路径的组合与层析

Combinations and chromatography of paths of Kleinian groups

研究\(n\)维双曲空间等距群无挠离散子群集合\(\mathcal{D}_n\)上查布蒂拓扑的连续路径，证明组合定理构造奇异路径，引入‘层析’技术证明分解定理刻画\(\mathcal{D}_3\)中凸紧群路径。

Commensurators of non-free finitely generated Kleinian groups

Suppose G is a non-free finitely generated Kleinian group without parabolics which is not a lattice and let C(G) denote the commensurator in PSL(2,C). We prove that if the limit set of G is not a round circle, then C(G) is discrete. Furthermore, G has finite index in C(G) unless G is a fiber group in which case C(G) is a lattice.

The Geometry and Arithmetic of Kleinian Groups

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…

纯loxodromic自由Kleinian群中长度为$1$的等距与迹不等式

Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities

本文推广了PSL₂(C)子群的离散性判据，证明纯loxodromic自由Kleinian群中满足特定位移条件时存在迹不等式，常数由KKT理论确定。

自由 Klein 群的对称分解与双曲位移

Symmetric Decompositions of Free Kleinian Groups and Hyperbolic Displacements

研究自由 Klein 群中由对称等距子集产生的双曲位移，证明二生成元群在长度至多 k 时的统一位移下界，并对纯双曲有限生成自由群提出进一步下界猜想。

Polynomial Trace Identities in $SL(2,{\bf C})$, Quaternion Algebras, and Two-generator Kleinian Groups

We study certain polynomial trace identities in the group $SL(2,\IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in §4 and then in §5 we give a new proof for Gehring and Martin's polynomial trace identities for good words, and extend that result to a larger class which is also closed under a semigroup operation inducing polynomial composition. This new approach is through the use of quaternion algebras over indefinites and an associated group of units. We obtain structure theorems for these quaternion algebras which appe…

Lorentzian Kleinian Groups

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…

Bounded geometry for Kleinian groups

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…

Covers of surfaces, Kleinian groups, and the curve complex

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.

Strong convergence of Kleinian groups: the cracked eggshell

In this paper we give a complete description of the set of discrete faithful representations SH(M) uniformizing a compact, orientable, hyperbolizable 3-manifold M with incompressible boundary, equipped with the strong topology, with the description given in term of the end invariants of the quotient manifolds. As part of this description, we introduce coordinates on SH(M) that extend the usual Ahlfors-Bers coordinates. We use these coordinates to show the local connectivity of SH(M) and study the action of the modular group of M on SH(M).

在紧致的弗希茨群和克莱因群上进行随机游走

Random walks on cocompact Fuchsian and Kleinian groups

该研究证明了在紧致的弗希茨和克莱因群上，某些随机游走的击中测度在无穷远处是奇异的。

The topology of deformation spaces of Kleinian groups

Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π_1(M)) denote the space of (conjugacy classes of) discrete faithful representations of π_1(M) into PSL 2 (C). The components of the interior MP(π_1(M)) of AH(π_1(M)) (as a subset of the appropriate representation variety) are enumerated by the space A(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(π_1(M)) and hence a conjectural topological enumeration of …

克莱因群强收敛的一致谱隙与正交测地线计数

Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

本文研究克莱因群强收敛下几何有限双曲流形的小特征值收敛性，利用极限流形的谱隙与测地流指数混合性质，建立了凸集间正交测地线数量的渐近一致计数公式，并应用于 Margulis 管、测地环及本原闭测地线。

The continuous part of the axial distance spectrum for Kleinian groups

Elements $f$ of finite order in the isometry group of hyperbolic three-space $\IH^3$ have a hyperbolic line as a fixed point set, this line is the axis of $f$. The possible hyperbolic distances between axes of elements of order $p$ and $q$, not both two, among {\em all} discrete subgroups $Γ$ of $Isom^+(\IH^3)$ has an initial discrete spectrum \[ 0 =δ_0< δ_1 < δ_2 < \ldots <δ_\infty,\] each value taken with finite multiplicity, and above $δ_\infty$ this spectrum of possible distances is continuous. The value $δ_\infty$ is the smallest number with the property that for each $λ<1$ there are only…

Flows, Fixed Points and Rigidity for Kleinian Groups

We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is not conformal then this group contains a non-trivial one parameter subgroup. This leads to rigidity results; for example, Mostow rigidity is an immediate consequence. We are also able to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group $G$, by a $G$-invariant pattern we mean a $G$-invariant collection of closed pro…

Length multiplicities of hyperbolic 3-manifolds

Let $M = H^3/Γ$ be a hyperbolic 3-manifold, where $Γ$ is a non-elementary Kleinian group. It is shown that the length spectrum of $M$ is of unbounded multiplicity.

关于具有三个抛物元素的二阶克莱因群

On rank two Kleinian groups with three parabolics

研究二阶自由群到\(\mathsf{PSL}(2,\mathbb{C})\)的离散表示，使三条闭曲线映为抛物元素，证明无限余体积群是特定极大尖点群，还发现了非来自特定链环希盖尔分裂表示的有限余体积群。

Kleinian群由抛物线对生成的几何学

Geometry for Kleinian Groups Generated by a Parabolic Pair

本文提出了一种基于三角剖分的计算框架，用于研究由抛物线对生成的Kleinian群的双曲几何及2桥链接补集的结构。

小临界指数的离散子群

Discrete subgroups of small critical exponent

本文证明了小临界指数的有限生成高维Kleinian群必为凸余紧，并推广了临界指数小于1的完备双曲负曲率流形的几何性质。

二生成元无挠 Klein 群与双曲位移

Two-Generator Free Kleinian Groups and Hyperbolic Displacements

本文针对无挠离散 Klein 群改进 $\log 3$ 位移定理，证明在相同条件下可从 $ξ$、$η$、$ξη$ 中找到等距，使任一点位移至少为 $(1/2)\log(5+3\sqrt{2})$。

Dynamics for discrete subgroups of $\operatorname{SL}_2(\mathbb C)$

Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups that "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title of Theory of discrete subgroups of $\operatorname{SL}_2(\mathbb C)$." In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works.

On Discreteness of Commensurators

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurators. This generalizes a Theorem of Greenberg for Kleinian groups. We then prove that for all finitely generated, Zariski dense, infinite covolume discrete subgroups of $Isom ({\mathbb{H}}^3)$, commensurat…

The outside of the Teichmuller space of punctured tori in Maskit's embedding

We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each rational ray there is a sequence of parameters in $\overline\Bbb H\setminus\Cal M$ accumulating at the boundary point of $\Cal M$ on the ray. These are the unique parameters on the extended $p/q$ ray for which the special word $W_{p/q}$ is a primitive elliptic Möbius transformat…

On the displacement of generators of free Fuchsian groups

We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the $\log (2k-1)$ Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of closed curves, which improves a result of Buser's.

通过自连接研究Kleinian群的刚性

Rigidity of Kleinian groups via self-joinings

本文通过引入自连接高阶动力学的新视角，证明了在极限集上具有共形边界映射的非Fuchsian Kleinian群的新刚性定理，并回答了McMullen关于零体积四面体顶点映射刚性的问题。

临界指数小于1的可及CAT(-1)群

Accessible CAT$(-1)$ groups of critical exponent less than one

该研究针对可及CAT(-1)群，证明临界指数小于1时群的结构性质，给出相关推论，覆盖更大类群并补充JSJ相关结论。

Kleinian群的外围子群

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 note on discrete Holonomy through directed edges, with no lengths

This note demonstrates how both the concept of distance and the concept of holonomy can be constructed from a suitable network with directed edges (and no lengths). The number of different edge types depends on the signature of the metric and the dimension of the holonomy group. If the holonomy group is of dimension one and the metric is positive-definite, a single type of directed edges is needed.

Borel and volume classes for dense representations of discrete groups

We show that the bounded Borel class of any dense representation $ρ: G\to \PSL_n\bC$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the $3$-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $ρ: G\to \PSL_2\bC$ is uniformly separated in semi-norm from any other representation $ρ': G\to \G$ for which there is a subgroup $H\le G$ on which $ρ$ is still dense but $ρ'$ is discrete or indiscrete but stabilizes a point…

通过自联结的克莱因群刚性：测度论准则

Rigidity of Kleinian groups via self-joinings: measure theoretic criterion

本文通过自联结子群的遍历理论与共形测度理论，证明了扎里斯基稠密凸紧克莱因群表示的测度论刚性二分法，并将结果推广至发散型子群。

Effective counting in sphere packings

Given a Zariski-dense, discrete group, $Γ$, of isometries acting on $(n + 1)$-dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain $Γ$-orbits. In particular, this allows us to obtain a best-known effective error rate for the Apollonian and (more generally) Kleinian sphere packing counting problems, that is, counting the number of spheres in such with radius bounded by a growing parameter. Our method extends the method of Kontorovich [Kon09], which was itself an extension of the orbit counting method of Lax-Phillips [LP82], in…

Duality structures and discrete conformal variations of piecewise constant curvature surfaces

A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Additional structure is defined either by giving a geometric structure to the Poincare dual of the triangulation or by assigning a discrete metric, a way of assigning length to oriented edges. This notion leads to a notion of discrete conformal structure, generalizing the discrete …

无限类型双曲面的等谱性

Isospectrality and isometry groups for infinite-type hyperbolic surfaces with discrete length spectrum

本文证明了通过Sunada方法得到的无限类型双曲面的等谱族在特定条件下具有有限性，并展示了有限群可以作为这些表面的全等距群。

Bounded Outdegree and Extremal Length on Discrete Riemann Surfaces

Let $T$ be a triangulation of a Riemann surface. We show that the 1-skeleton of $T$ may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed from $T$ by attaching new edges and vertices and subdividing its faces. Such refinements provide a mechanism of convergence of the discrete triangulation to the classical surface. We will prove a bound on the distortion of the discrete extremal lengths of path families on $T$ under the refinement process. Our bound will depend only on the refinement and not …

Discrete surfaces with length and area and minimal fillings of the circle

We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian surfaces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrical…