Maximal von Neumann subalgebras arising from maximal subgroups

Ge asked the question whether $L{F}_{\mathrm{\infty }}$ can be embedded into $L{F}_2$ 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.