Number variance for arithmetic hyperbolic surfaces

International audience

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in certain range, growing linearly in the degree of the invariant trace field, the volume of the $R$-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order at most $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic $3$-manifolds: We show that there are constants $A,B$ such that every such manifold $M$ is homotopy equivalent to a simplicial complex with at most $A\mathrm{Vol}(M)$ vertices, all of degrees bounded by $B$.

Length spectra of periodic orbits are investigated for some chaotic dynamical systems whose quantum energy spectra show unexpected statistical properties and for which the notion of arithmetical chaos has been introduced recently. These systems are defined as the unconstrained motions of particles on two-dimensional surfaces of constant negative curvature whose fundamental groups are given by number theoretical statements (arithmetic Fuchsian groups). It is shown that the mean multiplicity of lengths l of periodic orbits grows asymptotically like c.e12//l,I to infinity ., Moreover, the constant c (depending on the arithmetic group) is determined.

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in $\textrm{PSL}(2,{\mathbb{R}})^n$ . We identify the boundary with the sphere ${\mathbb{P}}(({\mathcal{ML}})^n)$ , where $\mathcal{ML}$ is the space of measured geodesic laminations on $\Sigma$ . In the case $n=2$ , we give a geometric interpretation of the boundary as the space of homothety classes of ${\mathbb{R}}^2$ -mixed structures on $\Sigma$ . We associate to such a structure a dual tree-graded space endowed with an ${\mathbb{R}}_+^2$ -valued metric, which we show to be universal with respect to actions on products of two $\mathbb{R}$ -trees with the given length spectrum.

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.

Quantization of a generic chaotic 3D billiard with smooth boundary. I. Energy level statistics

In this paper it is shown that one can choose an arbitrarily large number of inconjugate elements of the group Z/2Z*Z/2Z*Z/2Z which have the property that, under all representations of the group in SU(2,1) as a discrete complex hyperbolic ideal triangle group, the elements are hyperbolic and correspond to closed geodesics of equal length on the associated complex hyperbolic surface. This is an analogue of the geometric fact that the multiplicity of the length spectrum of a Riemann surface is never bounded or the equivalent algebraic phenomenon that an arbitrarily large number of conjugacy classes in a free group can have the same trace under all representations in SL(2,R ).

A number of questions related to the length spectrum of surfaces are discussed and in particular the existence of pairs of surfaces which though not isometric are isospectral. Here by isospectral we mean that a pair of bodies have the same distribution of chord lengths. In the Euclidean setting, we study isospectral convex dodecagons found by Mallows and Clark in the 1970's. Starting from their idea, we give constructions for isospectral pairs of hyperbolic surfaces that have no common cover. Since the work of Mallows and Clark is probably unfamiliar to readers with a background in topology/hyperbolic geometry we include expository material on other related topics about the distribution of chord lengths.