Notes on affine isometric actions of discrete groups

TL;DR

This paper investigates affine isometric actions of discrete groups, especially lattices in Lie groups, establishing irreducibility properties, constructing canonical actions related to $ ext{R}$-trees, and developing new representations connected to Teichmüller theory.

Contribution

It demonstrates that irreducible affine isometric actions of Lie groups remain irreducible when restricted to lattices and constructs new irreducible actions linked to $ ext{R}$-trees and Teichmüller space.

Findings

Irreducibility of actions is preserved from groups to lattices.

Existence of canonical irreducible actions associated with $ ext{R}$-trees.

New series of representations of diffeomorphism groups related to Thurston compactification.

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.