Notes on affine isometric actions of discrete groups

Consider a lattice $\mathrm{\Gamma }$ in a group $G=S{L}_2(\mathbb{R}),SO(1,n),SU(1,n)$, $S{L}_2({\text{Q}}_p)$. We discuss actions of $\mathrm{\Gamma }$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its restriction to $\mathrm{\Gamma }$ is irreducible. We prove the existence of canonical irreducible affine isometric actions of $\mathrm{\Gamma }$ associated to actions of $\mathrm{\Gamma }$ on $\mathbb{R}$- trees. Using such actions we construct irreducible representations of semigroup of probabilistic measures on $\mathrm{\Gamma }$ 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.