Towards the trace formula for convex-cocompact groups

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

Krasnov (arXiv: hep-th/0005106) identified the renormalized volume of a Schottky 3-manifold with the action of the Liouville theory on the conformal infiinity. We try to compute the renormalized volume in terms of more transparent geometric quantities.

The Yamabe operator and invariants on octonionic contact manifolds and convex cocompact subgroups of ${\rm F}_{4(-20)}$

An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the OC Yamabe operator, and apply it to construct a conformally invariant tensor. It becomes an OC metric if the OC positive mass conjecture is true. We also show the connected sum of two scalar positiv…

On Conformal Qc Geometry, Spherical Qc Manifolds and Convex Cocompact Subgroups of ${\rm Sp}{(n+1,1)}$

Conformal qc geometry of spherical qc manifolds are investigated. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal qc transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. On a scalar positive spherical qc manifold, we can construct the Green function of the qc Yamabe operator, which can be applied to construct a conformally invariant tensor. It becomes a spherical qc metric if the qc positive mass conjecture is true. Conformal qc geometry of spher…

Ruelle and Quantum Resonances for Open Hyperbolic Manifolds

We establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds between the spectrums of the geodesic flow and the Laplacian acting on natural tensor bundles. This extends previous work detailing the correspondence for cocompact quotients.

The classification of Kleinian groups of Hausdorff dimensions at most one and Burnside's conjecture

In this paper we provide the complete classification of convex cocompact Kleinian group of Hausdorff dimensions less than $1.$ In particular, we prove that every convex cocompact Kleinian group of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. The result implies that the converse of Burside's conjecture \cite{Burside} is true: All non-classical Schottky groups must have Hausdorff dimension $\ge1$. The prove of the theorem relies on the result of Hou \cite{Hou}.

Energy of Twisted Harmonic Maps of Riemann Surfaces

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_ρ$ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation $ρ$ of $π_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that the energy function $E_ρ$ is proper for any convex cocompact representation of the fundamental group. More generally, if $ρ$ is a discrete embedding onto a normal subgroup of a convex cocompact group $Γ$, then $E_ρ$ defines a proper function on the quotient $\Teich/Q$ where $Q$ …

Anosov subgroups: Dynamical and geometric characterizations

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture "rank one behavior" of Anosov subgroups and are direct generalizations of rank one equivalents to conve…

几乎弗uchsian表示在PU(2,1)中的研究

Almost-Fuchsian representations in PU(2,1)

本文研究了PU(2,1)中非最大Toledo不变量的凸-紧致表示，通过构造全纯且几乎全等距的极小曲面，展示了高 genus 下的构造方法及主要贡献。

等变Guillemin迹公式

An equivariant Guillemin trace formula

针对紧流形上的真、余紧群作用，提出Guillemin迹公式的等变推广，先发展分布迹的等变版本，为构造等变Ruelle动力ζ函数提供基础。

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.…

Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group

We prove Patterson's conjecture about the singularities of the Selberg zeta function associated to a convex-cocompact, torsion free group acting on a hyperbolic space.

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.

Wave 0-Trace and length spectrum on convex co-compact hyperbolic manifolds

For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $Γ$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. By writing this trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of $Γ$ is greater than $n/2$.

${\rm SO}(p,q+1)$-高阶 Teichmüller 空间中的体积、熵与直径

Volume, entropy, and diameter in ${\rm SO}(p,q+1)$-higher Teichmüller spaces

本文研究闭p-流形群到SO(p,q+1)的凸余紧表示的伪黎曼体积、熵与直径，给出熵与体积乘积的仅依赖抽象群几何的一致下界，证明熵的上界为p-1且等号情形可刻画，并得到有界直径下的有限性与紧性结果。