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Jun 15, 2019 at 20:11 comment added user6976 The fact that $N\times N$ is maximal if $F/N $ is simple indeed follows from Baumslag and Roseblade. By their tesult every subgroup containing $N\times N $ corresponds to a proper factor of $F/N $.
Aug 26, 2018 at 14:52 history edited YCor CC BY-SA 4.0
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May 12, 2016 at 19:51 comment added user6976 @Igor: I do not think they mention maximal subgroups there, but they do prove that F x F contains uncountably many pairwise non-isomorphic subgroups.
May 12, 2016 at 18:47 comment added Igor Rivin This is a theorem of Baumslag-Roseblade, 1984
May 12, 2016 at 17:54 vote accept CommunityBot moved from User.Id=6976 by developer User.Id=69903
May 12, 2016 at 16:23 history answered YCor CC BY-SA 3.0