Related papers: Large covers and sharp resonances of hyperbolic su…

Let $G$ be $\SO(n,1)$ or $\SU(n,1)$ and let $\Gamma\subset G$ denote an arithmetic lattice. The hyperbolic manifold $\Gamma\backslash \calH$ comes with a natural family of covers, coming from the congruence subgroups of $\Gamma$. In many…

On an asymptotically hyperbolic manifold (X,g), we show that the resolvent resonances coincide, with multiplicities, with the poles of the renormalized scattering operator, except for the special points n/2-k (with k>0 integer) where an…

For a geometrically finite hyperbolic surface of infinite volume we write down the spectral decomposition for the Laplacian on 1-forms, generalize the Kudla and Millson's construction of hyperbolic Eisenstein series and other related…

Let X be a Hadamard manifold and $\Gamma$ a discrete group of isometries of X which contains an axial isometry without invariant flat half plane. We study the behavior of conformal densities on the geometric limit set of $\Gamma$ in order…

Let $\Gamma < G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(\Gamma\backslash G)$ admits a strong spectral…

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we…

We use model theory to study relative profinite rigidity of $3$-manifold groups and show that given any residually finite group $\Gamma$ with finite character variety and single-cusped finite volume hyperbolic $3$-manifold $M$, cofinitely…

Given a negatively curved compact Riemannian surface $X$, we give an explicit estimate, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of $X$. The…

For geometrically finite hyperbolic manifolds $\Gamma\backslash H^{n+1}$, we prove the meromorphic extension of the resolvent of Laplacian, Poincar\'e series, Einsenstein series and scattering operator to the whole complex plane. We also…

Let $\calM_1$ and $\calM_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\calM_1)$ and $L^2(\calM_2)$ (respectively, multiplicities of lengths of closed geodesics in…

We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}:…

Let $\Gamma$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $\Gamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta(\Gamma)$ is…

In analogy with the spectral theory of geometrically finite hyperbolic manifolds, we initiate the study of resonances on geometrically finite (q+1)-regular graphs of groups. We prove the meromorphic continuation of the resolvent of the…

We show that for acylindrically hyperbolic groups $\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\rho$ of $\Gamma$ in a (nonzero) uniformly convex Banach space the vector space…

We prove asymptotically isometric, coarsely geodesic metrics on a toral relatively hyperbolic group are coarsely equal. The theorem applies to all lattices in SO(n,1). This partly verifies a conjecture by Margulis. In the case of hyperbolic…

We consider sequences $(X_n)_{n\in \mathbb{N}}$ of coverings of convex cocompact hyperbolic surfaces $X$ with Euler characterictic $\chi(X_n)$ tending to $-\infty$ as $n\to \infty.$ We prove that for $n$ large enough, each $X_n$ has an…

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$…

Motivated by the analysis of Schr\"odinger operators with periodic potentials we consider the following abstract situation: Let $\Delta_X$ be the Laplacian on a non-compact Riemannian covering manifold $X$ with a discrete isometric group…

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = \Gamma\backslash{\bf H}^2$ associated with…

For a noncompact complex hyperbolic space form of finite volume $X=\mathbb{B}^n/\Gamma$, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification $\overline{X}$ of $X$ similar to the…