Commensurability and locally free Kleinian groups

Examples of relative deformation spaces that are not locally connected

We provide an infinite family of pared manifolds whose relative deformation spaces of hyperbolic structures on these manifolds are not locally connected. This is a natural extension of the recent result of Bromberg that shows the space of Kleinian punctured torus groups is not locally connected.

Cannon-Thurston Maps,i-bounded Geometry and a theorem of McMullen

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.

Cannon-Thurston Maps for Surface Groups

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.

The Asymptotic distribution of circles in the orbits of Kleinian 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…

Local topology in deformation spaces of hyperbolic 3-manifolds

We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.

Totally Geodesic Spectra of Quaternionic Hyperbolic Orbifolds

In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give criteria for when the totally geodesic subspaces of such an orbifold determine its commensurability class. We give a parametrization of the commensurability classes of finite volume quaternionic hyperbolic orbifolds in terms of arithmeti…

环面计数与Kleinian群的自连接

Torus counting and self-joinings of Kleinian groups

该论文针对Kleinian群自连接下的一类d维环面堆积，建立了小体积环面的计数与等分布渐近公式，推广了Oh-Shah的圆堆积结果。

Locally Equivalent Correspondences

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same covolume and pro-congruence completion. We also make effective a f…

随机3-流形的均匀模型与短曲线

Uniform models and short curves for random 3-manifolds

本文提出了两种在满足特定拓扑条件的3-流形上构造双曲度量的方法，以渐近概率1适用于随机Heegaard分裂，并利用Kleinian群形变理论证明了几何不变量的增长率和随机3-流形的算术性等结果，且不依赖几何化猜想。

Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces

In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. We use a combination of techniques from algebraic groups and quadratic forms to prove several results about these spaces.

Geometry of the complex of curves II: Hierarchical structure

This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian representations. The authors' previous result that the complex is delta-hyperbolic was hard to apply because the complex is not locally finite; in this paper some tools are developed for overcoming this problem, and a combinatorial mechanism introduced which describes sequences of elementary moves in the graph of markings on a surface. These tools are applied to …

Baumslag-Solitar复形中的无挠格

Torsion-Free Lattices in Baumslag-Solitar Complexes

该研究分类了 Baumslag-Solitar 复形自同构群中存在不可公度无挠格的整数对，并构造无穷多可公度类，同时刻画了 Leighton 性质成立的情形。

图离散群与刚性

Graphically discrete groups and rigidity

本文引入群的图离散性概念，证明顶点群为图离散群的群图具强刚性，包括自由积的作用刚性与特定双曲群图的拟等距刚性，还给出例子并指出其非可公度不变性。