Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

Property (T) and rigidity for actions on Banach spaces

We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when 1. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement

Generic representations of abelian groups and extreme amenability

If $G$ is a Polish group and $\Gamma$ is a countable group, denote by \Hom(\Gamma, G) the space of all homomorphisms $\Gamma \to G$. We study properties of the group \cl{\pi(\Gamma)} for the generic \pi \in \Hom(\Gamma, G), when $\Gamma$ is abelian and $G$ is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on $\Gamma$, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic \cl{\pi(\Gamma)}; in the other two, we show that the generic \cl{\pi(\Gamma)} is extremely amenable. We also show that if $\Gamma$ is torsion-free, the centralizer of the generic $\pi$ is as small as possible, extending a result of King from ergodic theory.Comment: Version

Kazhdan Constants Associated with Laplacian on Connected Lie Groups.

. Let G be a finite dimensional connected Lie group. Fix a basis fX i g i=1;\Delta\Delta\Delta;n of the Lie algebra g and form the associated Laplace operator \Delta = \Gamma P 1in X 2 i in the enveloping algebra U (g) . Let ß be a strongly continuous unitary representation of G ; let dß(\Delta) be the closure of the essentially self-adjoint operator dß(\Delta) . We show that ß almost has invariant vectors if and only if 0 belongs to the spectrum of dß(\Delta). From this, we deduce that G has Kazhdan&apos;s property (T ) if and only if there exists ffl ? 0 such that, for any unitary representation without non zero fixed vectors, one has ffl ! minfSp(dß(\Delta))g . This answers positively a question of Y. Colin de Verdi`ere. It also allows us to define new Kazhdan constants, that we compare to the classical ones. 1. Introduction In 1967, Kazhdan introduced property (T ), a fixed point property of unitary representations for locally compact groups. More precisely, Definition 1.1. Let G b..