The fundamental inequality for cocompact Fuchsian groups

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.

Fundamental inequality for hyperbolic Coxeter and Fuchsian groups equipped with geometric distances

We prove that the hitting measure is not equivalent to the Lebesgue measure for a large class of nearest-neighbour random walks on hyperbolic reflection groups and Fuchsian groups.

Structure of attractors for boundary maps associated to Fuchsian groups

We study dynamical properties of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups. These maps are defined on the unit circle (the boundary of the Poincaré disk) by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity point and show that for a family of such maps the cycle property holds: the orbits coincide after finitely many steps. We also show that for an open set of discontinuity points the associated two-dimensional natural extension maps possess global attractors with finite …

Groups, drift and harmonic measures

In this short note we will describe an old problem and a new approach which casts light upon it. The old problem is to understand the nature of harmonic measures for cocompact Fuchsian groups. The new approach is to compute numerically the value of the drift and, in particular, get new results on the dimension of the measure in some new examples.

尖点 Hitchin 分量的压力度量

Pressure metrics for cusped Hitchin components

本文研究尖点 Hitchin 分量，针对几何有限 Fuchsian 群到 PSL(d,R) 的表示构造压力度量，并在 d=3 时给出与 Hilbert 长度相关的新压力度量。

与余紧三角群相关联的Bowen-Series映射的连续变形

Continuous deformation of the Bowen-Series map associated to a cocompact triangle group

本文确定了余紧三角群中满足扩张性质的精确符号集合，验证了Bowen-Series基本域在该集合中具有该性质，并自然关联了四个连续变形的圆函数族，证明了其非周期性与满射性等价，有限Markov性与双曲不动点等价。

Entropy and drift for Gibbs measures on geometrically finite manifolds

We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density.

圆周的分段线性作用的双曲性与障碍

Hyperbolicity and obstructions to PL actions of the circle

该研究探讨作用于圆周的分段线性同胚的 acylindrically 双曲群的结构，证明其为虚自由群等三类，推广直角 Artin 子群性质，还得出高维双曲流形基本群无法作用于圆周的结论。

Effective discreteness radius of stabilisers for stationary actions

We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $βr^δ$ for some explicit constants $β, δ> 0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a nove…