Malnormal subgroups of lattices and the Pukanszky invariant in group factors

TL;DR

This paper demonstrates that for certain lattices in semisimple real algebraic groups, the associated group factors contain a maximal abelian subalgebra with a Pukanszky invariant of infinity, revealing structural properties of these factors.

Contribution

It establishes the existence of a malnormal abelian subgroup in lattices of semisimple groups and links this to the structure of the associated von Neumann algebras.

Findings

Existence of malnormal abelian subgroups in lattices

Presence of a masa with Pukanszky invariant {∞} in group factors

Structural insight into von Neumann algebras of lattices

Abstract

Let $G$ be a connected semisimple real algebraic group. Assume that $G(\bb R)$ has no compact factors and let $\Gamma$ be a torsion-free uniform lattice subgroup of $G(\bb R)$. Then $\Gamma$ contains a malnormal abelian subgroup $A$. This implies that the $\tto$ factor $\vn(\Gamma)$ contains a masa $\fk A$ with Puk\'anszky invariant $\{\infty\}$.