Groups, Geometry, and Dynamics

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Published online first: 2021-08-03
DOI: 10.4171/GGD/614

Maximal subgroups and von Neumann subalgebras with the Haagerup property

Yongle Jiang[1] and Adam Skalski[2]

(1) Polish Academy of Sciences, Warszawa, Poland
(2) Polish Academy of Sciences, Warszawa, Poland

We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes \operatorname{SL}_2(\mathbb{Z})$ 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.

Keywords: Von Neumann algebras, Haagerup property, maximal subgroups/subalgebras

Jiang Yongle, Skalski Adam: Maximal subgroups and von Neumann subalgebras with the Haagerup property. Groups Geom. Dyn. Electronically published on August 3, 2021. doi: 10.4171/GGD/614 (to appear in print)