[Paper Review] Notes on affine isometric actions of discrete groups

Yury A. Neretin|arXiv (Cornell University)|Dec 20, 1997

Random Matrices and Applications11 references3 citations

TL;DR

This paper studies affine isometric actions of discrete groups, particularly lattices in $SL_2( R)$, $SO(1,n)$, $SU(1,n)$, and $SL_2( Q_p)$. It proves that irreducible affine isometric actions of the ambient Lie group restrict to irreducible actions on the lattice, and constructs canonical irreducible affine isometric actions via actions on $ R$-trees, leading to new representations of diffeomorphism groups of Riemann surfaces indexed by Thurston boundary points.

ABSTRACT

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

Motivation & Objective

• To understand the structure of affine isometric actions of discrete subgroups (lattices) in semisimple Lie groups.
• To establish conditions under which irreducible actions of the ambient Lie group remain irreducible when restricted to a lattice.
• To construct canonical irreducible affine isometric actions of lattices using their actions on $ R$-trees.
• To apply these constructions to produce new unitary representations of semigroups of probability measures on the lattice.
• To extend these representations to the group of diffeomorphisms of Riemann surfaces, parameterized by points in the Thurston compactification of Teichmüller space.

Proposed method

• Analyzes the restriction of irreducible affine isometric actions of Lie groups $G = SL_2( R), SO(1,n), SU(1,n), SL_2( Q_p)$ to their lattices $\Gamma$.
• Uses the geometry of $ R$-trees to construct canonical affine isometric actions of $\Gamma$.
• Applies the theory of isometric actions on Hilbert spaces to derive unitary representations of the semigroup of finite measures on $\Gamma$.
• Leverages the dynamics of $\Gamma$-actions on $ R$-trees to define boundary actions and extend representations.
• Utilizes the Thurston compactification of Teichmüller space to parameterize representations of diffeomorphism groups.
• Employs techniques from infinite-dimensional Lie group theory and geometric group theory to analyze the structure of these representations.

Experimental results

Research questions

• RQ1 Under what conditions does an irreducible affine isometric action of a Lie group $G$ remain irreducible when restricted to a lattice $\Gamma \subset G$?
• RQ2 Can canonical irreducible affine isometric actions of a lattice $\Gamma$ be constructed from its action on an $ R$-tree?
• RQ3 How do actions on $ R$-trees lead to unitary representations of the semigroup of probability measures on $\Gamma$?
• RQ4 What is the role of the Thurston compactification of Teichmüller space in parameterizing representations of diffeomorphism groups?
• RQ5 Can affine isometric actions on Hilbert spaces be used to construct new representations of infinite-dimensional groups such as diffeomorphism groups?

Key findings

• The restriction of an irreducible affine isometric action of $G = SL_2( R), SO(1,n), SU(1,n), SL_2( Q_p)$ to a lattice $\Gamma$ remains irreducible.
• Canonical irreducible affine isometric actions of $\Gamma$ are constructed from its actions on $ R$-trees.
• These constructions yield irreducible unitary representations of the semigroup of finite measures on $\Gamma$.
• The paper constructs a continuous family of representations of the diffeomorphism group of a Riemann surface, indexed by points in the Thurston compactification of Teichmüller space.
• The representations are realized via affine isometric actions on Hilbert spaces, linking geometric group theory with infinite-dimensional representation theory.
• The results establish a bridge between dynamics on $ R$-trees, geometric structures on surfaces, and representation theory of diffeomorphism groups.