Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group $\mathrm{Iso}\left(X\right)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_b^2(\Gamma ,ℝ)$, $H_b^2(\Gamma ,{\ell }^p\left(\Gamma \right))$$(1<...$

We prove that the natural map $H_{\text{b}}^2\left(\Gamma \right)\to H^2\left(\Gamma \right)$ from bounded to usual cohomology is injective if $\Gamma$ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for $\Gamma$: the stable commutator length vanishes and any $C^1$–action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating $H^{*}\text{b}\left(\Gamma \right)$ to the continuous bounded cohomology of the ambient group...

Let $G$ be the Lie group ${ℝ}^2⋉{ℝ}^{+}$ endowed with the Riemannian symmetric space structure. Let $X_0,X_1,X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\Delta =-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i{\Delta }^{-1/2}$ and $S_i={\Delta }^{-1/2}{X}_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second-order Riesz transforms $T_{ij}=X_i{\Delta }^{-1}{X}_j$ are bounded from $H^1$ to $L^1$, while the transforms $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ and $R_{ij}=X_i{X}_j{\Delta }^{-1}$, for $i,j=0,1,2$, are not.

We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices.

On étudie les morphismes d’un groupe infini discret $\Pi$ dans un groupe de Lie $G$ contenu dans le groupe des difféomorphismes de la droite réelle. À un tel morphisme $H$, on associe deux ensembles de “bouts” de $\Pi$ “dans la direction” $H$. On calcule le nombre de bouts dans plusieurs situations. Dans le cas particulier où $\Pi$ est de type fini et où $G$ est le groupe des translations, $\Pi$ n’a qu’un bout dans la direction $H$ si, et seulement si, ils vérifient la propriété de Bieri-Neumann-Strebel.

We study certain $\mathrm{𝔰𝔩}(2,ℂ)$-actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $(𝔤,𝔭)$, $({𝔤}^{\text{'}},{𝔭}^{\text{'}})$ of Lie algebras and their parabolic subalgebras.