Ilya Kapovich, Richard Weidmann
We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.
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We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has finite index in the reflection group of a right-angled ideal polyhedra in $\mathbb{H}^3$ then it has a co-final tower of finite sheeted covers with positive rank gradient. The manifolds we provide are also known to have co-final towers of covers with zero rank gradient.
We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.
We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a round sphere.
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。
In this paper, we construct Kleinian groups $Γ<\mathrm{Isom}(\mathbb{H}^{2n})$ from the direct product of $n$ copies of the rank 2 free group $F_2$ via strict hyperbolization. We give a description of the limit set and its topological dimension. Such construction can be generalized to other right-angled Artin groups.
In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups $G_1$ and $G_2$ of an infinite co-volume Kleinian group $G \subset \Isom(\mathbf{H}^3)$ having $Λ(G_1) = Λ(G_2)$ are commensurable. In particular, it is proved that any finitely generated subgroup $H$ of a Kleinian group $G \subset \Isom(\mathbf{H}^3)$ with $Λ(H) = Λ(G)$ is of finite index if and only if $H$ is not a virtually fiber subgroup.
We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.
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。
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.
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 …
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
This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space for n greater than 3. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of the hyperbolic 3-space.
We discuss which Kleinian groups are commensurable with Kleinian groups generated by rotations, with particular emphasis on Kleinian groups that arise from Dehn surgery on a knot.
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 show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. This implies, in particular, that there exist only finitely many conjugacy classes of cocompact two generated arithmetic Kleinian groups. The proof of the main result is based on a generalized Gromov--Guth inequality and bounds for the hyperbolic and tube volumes of the quotient orbifolds.
We prove that every finitely presentable group G arises as the fundamental group of an orientable 3-complex obtained from a hyperbolic link complement, by coning each boundary torus of the link exterior to a distinct point. We define the closed-link-genus, clg(G), of any finitely presentable group G, which completely characterizes fundamental groups of closed orientable 3-manifolds: clg(G)=0 if and only if G is the fundamental group of a closed orientable 3-manifold. Moreover clg(G) gives an upper bound for the concept `genus(G)' of genus defined earlier by Aitchison and Reeves, and in turn is…
We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.
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。
This is a summary of the material for 3 lectures on geometrically finite and infinite Kleinian groups delivered by the author at a workshop held at Tata Institute of Fundamental Research in April 2014.
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.
We give a full proof to Agol's announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.
We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do "superintegral" such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from Q-arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles pi/m for finitely many m. We settle two questions f…
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.
We prove that there are only finitely many arithmetic Kleinian maximal reflection groups.
Let $Γ$ be a non-elementary Kleinian group and $H<Γ$ a finitely generated, proper subgroup. We prove that if $Γ$ has finite co-volume, then the profinite completions of $H$ and $Γ$ are not isomorphic. If $H$ has finite index in $Γ$, then there is a finite group onto which $H$ maps but $Γ$ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circleof ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, e.g. limit groups. We const…
Brownian motion and orbit counting of Kleinian groups
本文研究了凯莱群的发散性与随机游走复发性之间的关系，并通过建立增长率联系，构造了发散型凯莱群。
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.
In this paper we give a necessary and sufficient condition in which a sequence of Kleinian punctured torus groups converges. This result tells us that every exotically convergent sequence of Kleinian punctured torus groups is obtained by the method due to Anderson and Canary. Thus we obtain a complete description of the set of points at which the space of Kleinian punctured torus groups self-bumps. We also discuss geometric limits of sequences of Bers slices.
We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.
We give a brief overview of the current state of the study of the deformation theory of Kleinian groups. The topics covered include the definition of the deformation space of a Kleinian group and of several important subspaces; a discussion of the parametrization by topological data of the components of the closure of the deformation space; the relationship between algebraic and geometric limits of sequences of Kleinian groups; and the behavior of several geometrically and analytically interesting functions on the deformation space.
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.
We show that the space of Kleinian punctured torus groups is not locally connected.
The purpose of this paper is to construct an example of a 2-knot wildly embedded in $\mathbb{S}^{4}$ as the limit set of a Kleinian group. We find that this type of wild 2-knots has very interesting topological properties.
We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35.
We show that $Γ< \textbf{SU}(3,1)$ is a non-elementary complex hyperbolic Kleinian group in which $tr(γ) \in \R$ for all $γ\in Γ$ if and only if $Γ$ is conjugate to a subgroup of $\textbf{SO}(3,1)$ or $\textbf{SU}(1,1)\times\textbf{SU}(2)$.
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 consider non-elementary Kleinian groups Γ, without invariant plane, generated by an elliptic and a hyperbolic element with their axes lying in one plane. We find presentations and a complete list of orbifolds uniformized by such Γ.
This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal positive number $λ>0$, such that any finitely-generated non-elementary Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups. We will generalize our previous technologies given in \cite{HS} to prove our general result. Our result can be consider as …
A positive integer $m$ will be called a {\it finitistic order} for an element $γ$ of a group $Γ$ if there exist a finite group $G$ and a homomorphism $h:Γ\to G$ such that $h(γ)$ has order $m$ in $G$. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian group admit a given integer $m>2$ as a finitistic order.
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.
Any action of a group $Γ$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $Γ$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic $3$-manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a $2$ dimensional subspace of bounded cohomology. Our techniques apply to…
We study orbital functions associated to Kleinian groups through the heat kernel approach developed in \cite{artmoiheatcounting1}.
For a torsion free Kleinian group $Γ$ without parabolics, we consider the decomposition of the limit set $L(Γ)$ into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on $L(Γ)$ when $L(Γ)=S^2_\infty$.
Let $Γ$ be a nonelementary discrete subgroup of $\mathrm{Sp}(n,1)$. We show that if the trace skew-field of $Γ$ is commutative, then $Γ$ stabilizes a copy of complex hyperbolic subspace of quaternionic hyperbolic $n$-space.
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…
In this paper we prove that there exists a positive number $λ>0$, such that any 2-generated Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups.
In the paper `Automorphic functions for a Whitehead-complement group', [Osaka J Math 43 (2006) 63-77] Matsumoto, Nishi and Yoshida constructed automorphic functions on real 3-dimensional hyperbolic space for a Kleinian group called the Whitehead-link-complement group. For a Kleinian group (of the first kind), no automorphic function/form has been studied before. In this note, their motivation is presented with a historical background.
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。
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.
In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the ``original knot'' fibers over the circle then the wild knot $Λ$ also fibers over the circle. As a consequence, the universal covering of $\mathbb{S}^{3}-Λ$ is $\mathbb{R}^{3}$. We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold.
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 $PSL(2,\mathbb{H})$
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 $\mathrm{U}(2,1)$.
We show that there exist infinitely many commensurability classes of finite volume hyperbolic 3-manifolds whose fundamental group contains a subgroup which is locally free but not free. The main technical tool is the fact that a collection of hyperbolic 3-manifolds of bounded volume contains infinitely many commensurability classes. The result then by an application of Thurston's hyperbolization theorem for Haken 3-manifolds.
In this paper we construct infinitely many wild knots, $\mathbb{S}^{n}\hookrightarrow\mathbb{S}^{n+2}$, for $n=2,3,4$ and 5, each of which is a limit set of a geometrically finite Kleinian group. We also describe some of their properties
We consider a finitely generated torsion free Kleinian group $H$ and a random walk on $H$ with respect to a symmetric nondegenerate probability measure $μ$ with finite support. When $H$ is geometrically infinite without parabolics or when $H$ is Gromov hyperbolic with parabolics, we prove that the Patterson-Sullivan measure is singular with respect to the harmonic measure coming from $μ$.
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 $ρ$.
Cusps, Kleinian groups and Eisenstein series
本文研究完备双曲流形中全秩尖点相关的 Eisenstein 级数，证明给定 Kleinian 群时每个全秩尖点对应特定上同调类，并通过计算交织算子证明不同尖点产生线性无关的类。
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…
Let M be a non-elementary convex cocompact hyperbolic 3 manifold and delta the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure provided that delta>1.
We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being non-empty implies that the intersection of the subgroups is non-trivial. We present examples to show that a conjecture of Susskind, stating that the intersection of the sets of conical limit points of subgroups $Φ$ and $Θ$ of a Kleinian group $Γ$ is contained in the limit set of …
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