# Maximal subgroups and von Neumann subalgebras with the Haagerup property

## TL;DR

This paper explores the structure of maximal subgroups and von Neumann algebras with the Haagerup property, providing explicit classifications and new insights into their maximality and intermediate structures.

## Contribution

It introduces methods to identify maximal Haagerup subgroups and subalgebras, especially within specific group actions, and answers open questions about their maximality.

## Key findings

- Maximal Haagerup subgroups inside $ ext{Z}^2
times SL_2( ext{Z})$ are classified.
- Explicit examples of maximal Haagerup subalgebras are constructed.
- Intermediate von Neumann algebras are shown to often arise from groups or group actions.

## Abstract

We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some remarks and examples concerning maximal non-(T) subgroups and subalgebras are also presented, and we answer two questions of Ge regarding maximal von Neumann subalgebras.