Superrigidity for dense subgroups of Lie groups and their actions on homogeneous spaces

TL;DR

This paper establishes W*-superrigidity for certain dense subgroups of Lie groups acting on homogeneous spaces, revealing that their von Neumann algebras uniquely encode the group actions and structures.

Contribution

It introduces a new cocycle superrigidity theorem for dense Lie group subgroups and demonstrates W*-superrigidity for actions of these groups, especially in hyperbolic geometry contexts.

Findings

Proves W*-superrigidity for dense subgroups of isometries of the hyperbolic plane.

Develops a new cocycle superrigidity theorem for dense Lie group subgroups.

Constructs countable type II_1 equivalence relations not realizable by free group actions.

Abstract

An essentially free group action of $\Gamma$ on $(X,\mu)$ is called W*-superrigid if the crossed product von Neumann algebra $L^\infty(X) \rtimes \Gamma$ completely remembers the group $\Gamma$ and its action on $(X,\mu)$. We prove W*-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. We also provide numerous countable type $II_1$ equivalence relations that cannot be implemented by an essentially free action of a group, both of geometric nature and through a wreath product construction.