Maximal von Neumann subalgebras arising from maximal subgroups

TL;DR

This paper demonstrates the existence of maximal von Neumann subalgebras derived from maximal subgroups with infinite index, providing multiple approaches and applications to group actions.

Contribution

It introduces three methods to embed $LF_{ ext{infinity}}$ into $LF_2$ as a maximal subfactor, using maximal subgroups with infinite index.

Findings

Existence of maximal subgroups with infinite index.

Construction of maximal von Neumann subalgebras from group actions.

Multiple approaches to embedding $LF_{ ext{infinity}}$ as a maximal subfactor.

Abstract

Ge asked the question whether $LF_{\infty}$ can be embedded into $LF_2$ as a maximal subfactor. We answer it affirmatively by three different approaches, all containing the same key ingredient: the existence of maximal subgroups with infinite index. We also show that point stabilizer subgroups for every faithful, 4-transitive action on an infinite set give rise to maximal von Neumann subalgebras. Combining this with known results on constructing faithful, highly transitive actions, we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.