Notes on affine isometric actions of discrete groups

Abstract

Consider a lattice $\Gamma$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $\Gamma$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its restriction to $\Gamma$ is irreducible. We prove the existence of canonical irreducible affine isometric actions of $\Gamma$ associated to actions of $\Gamma$ on $\R$- trees. Using such actions we construct irreducible representations of semigroup of probabilistic measures on $\Gamma$ and construct the series of representations of the groups of diffeomorphisms of Riemann surfaces enumerated by the points of Thurston compactification of Teichm\"uller (Teichmuller) space.