A remark on fullness of some group measure space von Neumann algebras

TL;DR

This paper extends the class of groups for which the associated group measure space von Neumann algebras are full, including groups like SL(3,Z) and lattices in simple Lie groups, demonstrating strong ergodicity and fullness.

Contribution

It broadens the known classes of groups with full von Neumann algebras, including non-biexact groups such as SL(3,Z) and certain lattice actions on homogeneous spaces.

Findings

Fullness of von Neumann algebras for new group classes

Strong ergodicity of actions on homogeneous spaces

Extension of results beyond biexact groups

Abstract

Recently C. Houdayer and Y. Isono have proved among other things that every biexact group $\Gamma$ has the property that for any non-singular strongly ergodic action $\Gamma\curvearrowright (X,\mu)$ on a standard measure space the group measure space von Neumann algebra $\Gamma\ltimes L^\infty(X)$ is full. In this note, we prove the same property for a wider class of groups, notably including $\mathrm{SL}(3,{\mathbb Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\Gamma\le G$, and any closed non-amenable subgroup $H\le G$, the non-singular action $\Gamma\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\Gamma\ltimes L^\infty(G/H)$ is full.