Harmonic quasi-isometries of pinched Hadamard surfaces are injective

Geometric finiteness in negatively pinched Hadamard manifolds

In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete subgroups $Γ$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every discrete geometrically infinite isometry subgroup $Γ$ has a set of nonconical limit points with the cardinality of the continuum.

Harmonic quasi-isometric maps III :quotients of Hadamard manifolds

In a previous paper, we proved that a quasi-isometric map $f:X\longrightarrow Y$ between two pinched Hadamard manifolds $X$ and $Y$ is within bounded distance from a unique harmonic map. We extend this result to maps $f:Γ\backslash X\longrightarrow Y$, where $Γ$ is a convex cocompact discrete group of isometries of $X$ and $f$ is locally quasi-isometric at infinity.

Hadamard 流形中的乒乓论证

Ping-pong in Hadamard manifolds

本文针对负压缩 Hadamard 流形建立定量 Tits 二择一结果，证明两个非椭圆等距生成的非初等离散群含词长一致有界的秩 2 自由子群，并在双曲情形给出凸余紧性。

Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distance

For a pinched Hadamard manifold $X$ and a discrete group of isometries $Γ$ of $X$, the critical exponent $δ_Γ$ is the exponential growth rate of the orbit of a point in $X$ under the action of $Γ$. We show that the critical exponent for any family $\mathcal{N}$ of normal subgroups of $Γ_0$ has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients $Γ_0/Γ$. The key tool is to analyse the spectrum of transfer operators associated to subshifts of finite type, for which we obtain a result of independent interest.

Sur la complétude de certaines variétés pseudo-riemanniennes localement symétriques

We prove that some symetric semi-riemannian manifolds do not admit a proper domain which is divisible by the action of a discrete group of isometries. In other words, if a closed semi-riemannian manifold is locally isometric to such a model, and if its developing map is injective, then the manifold is actually geodesically complete, and therefore isometric to a quotient of the whole model space. Those results extend, under additional hypothesis (the injectivity of the developing map), the theorems of Carriere and Klingler stating that closed lorentz manifolds of constant curvature are geodesic…