Maximal von Neumann subalgebras arising from maximal subgroups

Ge (2003) asked the question whether LF infinity can be embedded in to LF2 as a maximal subfactor. We answer it affirmatively in 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. By combining this with the known results on constructing faithful, highly transitive actions, we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.