Wild Knots as limit sets of Kleinian Groups

$N$维串珠项链与受 Schottky 群不变的高维野结

$N$-dimensional beaded necklaces and higher dimensional wild knots, invariant by a Schottky group

本文基于$n$维结和串珠项链，利用共形Schottky群构造了具有Cantor集野点的高维野结，证明其纤维化性质，并研究了循环分支覆盖，推广了已有结果。

嵌入梅尔森海绵中的野结

Wild knots embedded in the Menger Sponge

本文通过递归构造了无数非等价的嵌入梅尔森海绵中的野结，并能控制其在通常康托尔集中的野点集。还展示了由克莱因群作用动态定义的野结可被同胚到海绵中。

Jørgensen Number and Arithmeticity

A Jørgensen group is a non-elementary Kleinian group that can be generated by two elements for which equality holds in Jørgensen's Inequality. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of cocompact arithmetic Jørgensen groups. The paper also defines and computes the Jørgensen number of several non-cocompact Kleinian groups including some two-bridge knot and link groups.

野生纽结的一个非分类结果

A Non-classification Result for Wild Knots

该研究运用描述集合论方法，证明了野生纽结分类问题的难度严格高于可数结构的分类问题，给出了相关非分类结果。

构造具有指定几何极限的纽结

Constructing knots with specified geometric limits

本文针对几何有限、驯顺双曲3流形，给出构造显式纽结族使其补空间几何收敛于给定流形的构造性证明，并将完全增广链环类推广至Kleinian群设置。

Some examples of dynamically defined ambient homogeneous wild knots in higher dimensions

In this paper we consider the Kleinian groups acting conformally on the sphere $\mathbb{S}^{n+2}$ $(1\leq{n}\leq5)$ which have as limit sets wild spheres $K^n$ which were constructed in \cite{BHV} and prove that $K^n$ is ambient homogeneous. In other words, given two points $p,\,\,q\in{K}$ there exists a homeomorphism $ψ:\mathbb{S}^{n+2}\rightarrow\mathbb{S}^{n+2} $ such that $ψ(K)=K$ and $ψ(p)=q$.

Generating pairs of 2-bridge knot groups

We study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts: - Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes of generating pairs. - For any natural number N there is a closed hyperbolic 3-manifold whose fundamental group has N distinct Nielsen classes of generating pairs. - Two pairs of elements of a fundamental group of an HNN-extension are Nielsen equivalent iff they are so for the obvious reasons.

Profinite rigidity, Kleinian groups, and the cofinite Hopf property

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…

关于双曲3空间等距群的quandle

On the quandles of isometries of the hyperbolic 3-space

本文通过构造与Kleinian群及其非平凡元素相关联的quandle $Q(Γ, γ)$，将纽结quandle推广到双曲纽结，并建立了到$\mathrm{PSL}(2, \mathbb{C})$共轭quandle的具有离散像的单射同态。

双曲流形与折叠流形的单复维模空间

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。

Involutions of negatively curved groups with wild boundary behavior

We consider pairs (X,Y) where X is a compact, locally CAT(-1) space, and Y is a totally geodesic subspace. The inclusion induces an embedding of the boundaries at infinity of the universal covers; we focus on the case where these are spheres whose dimensions differ by 2. We show that if the embedding is tame, then it is unknotted. We give examples of pairs for which the embedding is knotted (and can be realized as the fixed point set of an involution). We also provide a criterion for knottedness of tame codimension two spheres in high dimensional (>5) spheres. Corresponding results for delta-h…

Two-Dimensional Knots and Representations of Hyperbolic Groups

We describe relations between hyperbolic geometry and codimension two knots or, more exactly, between varieties of conjugacy classes of discrete faithful representations of the fundamental groups of hyperbolic n-manifolds M into $\operatorname{SO}^{\circ} (n+2,1)$ and (n-1)-dimensional knots in the (n+1)-sphere. This approach allows us to discover a phenomenon of non-connectedness of these varieties for closed n-manifolds M, $n\geq 3$, with large enough number of disjoint totally geodesic surfaces, to construct quasisymmetric infinitely compounded "Julia" knots $K\subset S^{n+1}$ which are eve…

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。