Yong Hou
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.
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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 …
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}.
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 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 $<1+ε$.
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.
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…
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.
Real points of Schottky space ${\mathcal S}_{g}$ are in correspondence with extended Kleinian groups $K$ containing, as a normal subgroup, a Schottky group $Γ$ of rank $g$ such that $K/Γ\cong {\mathbb Z}_{2n}$ for a suitable integer $n \geq 1$. These kind of groups are called extended ${\mathbb Z}_{2n}$-Schottky groups of rank $g$. In this paper, we provide a structural decomposition theorem, in terms of Klein-Maskit's combination theorems, of these kind of groups.
Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite…
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.
Geometrically infinite Kleinain groups have nonconical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the nonconical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the nonconical limit set is positive.
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…
An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodies. A group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. We provide a structural description of these type of groups and, as a consequence, we obtain sharp upper bounds for the s…
Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups
本文研究Fuchsian Schottky群上有限程随机游动的调和测度，证明其维数严格低于对应极限集的Hausdorff维数，揭示了维数下降现象。
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。
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。
Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=<A,B>}. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary Kleinian group whose Jorgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jorgensen numbers from above in the diagonal slice of Schottky space.
Let $Λ$ be the limit set of a conformal dynamical system, i.e. a Kleinian group acting on either finite- or infinite-dimensional real Hilbert space, a conformal iterated function system, or a rational function. We give an easily expressible sufficient condition, requiring that the limit set is not too much bigger than the radial limit set, for the following dichotomy: $Λ$ is either a real-analytic manifold or a fractal in the sense of Mandelbrot (i.e. its Hausdorff dimension is strictly greater than its topological dimension). Our primary focus is on the infinite-dimensional case. An important…
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。
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.
Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Lambda(M) be the supremum of the bottom eigenvalue of the Laplacian of N, where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that Lambda(M)=D(M)(2-D(M)) if M is not handlebody or a thickened torus. We characterize exactly when Lambda(M)=1 and D(M)=1 in terms of the characteristic submanifold of the incompressib…
Hausdorff Dimension of non-conical and Myrberg limit sets
本文针对负曲率空间上的群作用，建立非锥形极限集Hausdorff维数极大性的多种判定条件，并证明Myrberg极限集的Hausdorff维数等于临界指数，证实Falk-Matsuzaki猜想。
Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds
本文介绍了Riley slice及其背景，并将Keen和Series的工作扩展到同构于$\mathbb{Z}_p*\mathbb{Z}_q$的Kleinian群的单复维模空间。
We prove that the hitting measure is singular with respect to Lebesgue measure for any random walk on a cocompact Fuchsian group generated by translations joining opposite sides of a symmetric hyperbolic polygon. Moreover, the Hausdorff dimension of the hitting measure is strictly less than 1. A similar statement is proven for Coxeter groups. Along the way, we prove for cocompact Fuchsian groups a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.
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。
We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/Γ$ where $X$ is a contractible Riemannian manifold and $Γ<\Isom(X)$ is a discrete subgroup, typically with infinite co-volume. The existence depends on the $L^2$-Betti numbers of $Γ$, its subgroups and of a uniform lattice of $\Isom(X)$. As an application, we show the existence of a uniform positive lower bound on the Cheeger constant of any manifold of the form $\H^4/Γ$ where $\H^4$ is real hyperbolic 4-space and $Γ<\Isom(\H^4)$ is discrete and isomorphic to a subgroup of the fundamental group…
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