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.

The study of diffeomorphism group actions requires methods of infinite dimensional analysis. Really convenient tools can be found in the Frölicher - Kriegl - Michor differentiation theory and its geometrical aspects. In terms of it we develop the calculus of various types of one parameter diffeomorphism groups in infinite dimensional spaces with smooth structure. Some spectral properties of the derivative of exponential mapping for manifolds are given.