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

Given a convex co-compact hyperbolic surface $X=\Gamma\backslash \mathbb{H}^2$, we investigate the resonance spectrum $\mathcal{R}_j$ of the laplacian $\Delta_j$ on large finite abelian covers $X=\Gamma_j\backslash \mathbb{H}^2$, where…

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma\backslash \mathbb{H}^2$ be the associated hyperbolic surface. Conditional on the generalized Riemann hypothesis for quadratic $L$-functions, we establish a…

Let $\Lambda$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface $X=\Lambda \backslash\H$ has infinitely many resonances in…

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=\Gamma\backslash\mathbb{H}$. Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We say that…

Let $X$ be a convex co-compact hyperbolic surface and let $\delta$ be the Hausdorff dimension of the limit set of the underlying discrete group. We show that the density of the resonances of the Laplacian in strips ${\sigma\leq \re(s) \leq…

Let $X=X_1 \times X_2$ be a direct product of two rank-one Riemannian symmetric spaces of the noncompact type. We show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension, the resolvent of the…

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings…

On geometrically finite hyperbolic manifolds $\Gamma\backslash H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In…

We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $\epsilon>0$, with probability tending to one as $n\to\infty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in…

Let $ X = \Gamma\setminus \mathbb{H} $ be a non-elementary geometrically finite hyperbolic surface and let $ \delta $ denote the Hausdorff dimension of the limit set $ \Lambda(\Gamma) $. We prove that for every $ \varepsilon > 0 $ the…

Let $X=G/K$ be a Riemannian symmetric space of the noncompact type and restricted root system $BC_2$ or $C_2$ (except $G=SO_0(p,2)$ with $p>2$ odd). The analysis of the meromorphic continuation of the resolvent of the Laplacian of $X$ is…

We present the Laplace operator associated to a hyperbolic surface $\Gamma\setminus\mathbb{H}$ and a unitary representation of the fundamental group $\Gamma$, extending the previous definition for hyperbolic surfaces of finite area to those…

Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…

Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number \lambda_0…

Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly…

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL} (2,\mathbb{Z})$. We establish a uniform and explicit lower bound of the second eigenvalue of the Laplace-Beltrami operator of congruence coverings of the hyperbolic surface $\Gamma…

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of…

We study boundary representations of hyperbolic groups $\Gamma$ on the (compactly embedded) function space $W^{\log,2}(\partial\Gamma)\subset L^2(\partial\Gamma)$, the domain of the logarithmic Laplacian on $\partial\Gamma$. We show that…

We show that every word hyperbolic, surface-by-(noncyclic) free group Gamma is as rigid as possible: the quasi-isometry group of Gamma equals the abstract commensurator group Comm(Gamma), which in turn contains Gamma as a finite index…

Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong…