On the distribution of primitive subgroups of $\mathbb{Z}^{d}$ of large covolume

Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.

关于无限余体积离散子群的奇异性猜想的一个注记

A note on the singularity conjecture for infinite covolume discrete subgroups

本文研究了半单李群上生成无限余体积离散子群的随机游走，证明了在性质(T)或秩为1等条件下Furstenberg边界平稳测度与Lebesgue测度类的奇异性，并给出了奇异性的充分条件。

高秩无限余体积离散子群的 Furstenberg 测度的奇异性

Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank

本文研究了具有 Property (T) 的半单 Lie 群中无限余体积离散子群上的对称随机游走，证明了此时 Furstenberg 边界上的命中测度具有奇异性，并指出这是高秩情形下的首个此类结果。

The gap distribution of directions in some Schottky groups

We prove the existence and some properties of the limiting gap distribution functions for the directions of orbits of some infinite covolume subgroups of $Isom(\mathbb{H}^2)$ in the Poincaré disk.

Non-Divergence of Unipotent Flows on Quotients of Rank One Semisimple Groups

Let $G$ be a semisimple Lie group of rank $1$ and $Γ$ be a torsion free discrete subgroup of $G$. We show that in $G/Γ$, given $ε>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than $ δ$ for $1-ε$ proportion of the time for some $δ>0$. The result also holds for any finitely generated discrete subgroup $Γ$ and this generalizes Dani's quantitative nondivergence theorem \cite{D} for lattices of rank one semisimple groups. Furthermore, for a fixed $ε>0$ there exists an injectivity radius $δ$ such that for any unipotent trajectory $\{u_tx\}_{t\in …

Effective joint equidistribution of primitive rational points on expanding horospheres

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems -- an approach which does not lend itself to effective bounds. We implement a strategy based on spectral theory, Fourier analysis and Weil's bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical…