Holography and the geometry of certain convex cocompact hyperbolic 3-manifolds

Projective Anosov representations, convex cocompact actions, and rigidity

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different)…

Simpson's Theory and Superrigidity of Complex Hyperbolic Lattices

We attack a conjecture of J. Rogawski: any cocompact lattice in $S U (2, 1)$ for which the ball quotient $X = B^2 / Γ$ satisfies $b_1 (X) = 0$ and $H^{1, 1} (X) \cap H^2 (X, \bbq) \approx \bbq$ is arithmetic. We prove the Archimedian suprerigidity for representation of $Γ$ is $S L (3, \bbc)$.

Towards the trace formula for convex-cocompact groups

We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this subgroup is an invariant distribution on the group, and we state precise conjecture about its Fouriertransform. We apply our trace formula to resolvent kernels. This yields functional equations of Selberg zeta functions and information about their singularities. The paper leaves …

Resonances for Symmetric Tensors on Asymptotically Hyperbolic Spaces

On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric tensors, a similar result is proven for (convex cocompact) quotients of hyperbolic space.

渐近测地超曲面与双曲流形的基本群

Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds

研究双曲流形中渐近测地超曲面的性质，证明其基本群虚特殊且整数线性，进一步构造渐近测地且强填充的超曲面，部分回答Al Assal-Lowe的问题。

余紧负曲率流形的McKean刚性与\

McKean Rigidity for Cocompact Negatively Curved Manifolds and the \(p\)-Laplacian

该研究针对截面曲率不超过-1的闭黎曼流形，证明其万有覆叠的底部谱达到McKean下界当且仅当万有覆叠为常曲率-1的双曲空间，还推广了p-基本音的相关刚性结果，通过转化缺陷为平稳测度并利用热核正性完成证明。

Geometric limits of knot complements

We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain balls of arbitrarily large radius. We also show that a complete hyperbolic 3--manifold with two convex cocompact ends cannot be a geometric limit of knot complements in $\BS^3$.

双曲三维流形的体积与Chern-Simons不变量的重整化

The renormalization of volume and Chern-Simons invariant for hyperbolic 3-manifolds

本文通过等距叶状结构渐近分析，重整化凸余紧双曲三维流形的Chern-Simons不变量，证明其指数发散项由Weitzenböck联络的挠率2-形式积分给出，并得到复体积的渐近行为及主系数的新复值量。

Renormalized volume and the volume of the convex core

We obtain upper and lower bounds on the difference between the renormalized volume and the volume of the convex core of a convex cocompact hyperbolic 3-manifold which depend on the injectivity radius of the boundary of the universal cover of the convex core and the Euler characteristic of the boundary of the convex core. These results generalize results of Schlenker obtained in the setting of quasifuchsian hyperbolic 3-manifolds.

双曲群的子群、有限性性质与复双曲格

Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

本文证明最简单类型余紧复双曲算术格的足够深有限指数子群存在到整数群的大量同态，其核具有类型F_{m-1}而非F_m，从而构造出双曲群的许多有限展示非双曲子群，回答了Brady的长期问题，并给出非球面Kähler流形Singer猜想的一个特殊情形证明。

Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry

The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\mathrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud and Kassel, called $\mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ which can be understood as a Lorentzian version of a famous Theorem of…

具有圆形谢尔宾斯基极限集的双曲三维流形中的水平圆

Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets

本文对具有圆形谢尔宾斯基极限集的几何有限双曲三维流形，分类了标架丛上水平圆流的轨道闭包，证明水平圆闭包为恰当浸入子流形，推广了凸余紧情形下的已知结果。

Hodge theory on hyperbolic manifolds of infinite volume

Let $Y=Γ\backslash H^n$ be a quotient of the hyperbolic space by the action of a discrete convex-cocompact group of isometries. We describe certain spaces of $Γ$-invariant currents on the sphere at infinity of $H^n$ with support on the limit set of $Γ$. These spaces are finite-dimensional. The main result identifies the cohomology of $Y$ with a quotient of such spaces. We explain in which sense this result generalizes the classical Hodge theorem for compact quotients. We obtain analogous results for the cohomology groups $H^p(Γ,F)$, where $F$ is a finite-dimensional representation of the full …

Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume

To a complex projective structure $Σ$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|ϕ_Σ\|_\infty$ and $\|ϕ_Σ\|_2$ of the quadratic differential $ϕ_Σ$ of $Σ$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination.…

Additive continuity of the renormalized volume under geometric limits

We study the infimum of the renormalized volume for convex-cocompact hyperbolic manifolds, as well as describing how a sequence converging to such values behaves. In particular, we show that the renormalized volume is continuous under the appropriate notion of limit. This result generalizes previous work in the subject.