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Aug 20, 2022 at 22:50 history edited HJRW CC BY-SA 4.0
Corrected terminology. Added Radu's example.
Aug 20, 2022 at 22:48 comment added HJRW @YCor: You're right, I had misremembered. "Almost simple" means something different. (en.wikipedia.org/wiki/Almost_simple_group) In a 1993 paper, Bhattacharjee used the term "nearly simple" for a slightly stronger notion. I suspect that Higman's group may be "nearly simple" in Bhattacharjee's sense, but will edit for now.
Aug 20, 2022 at 12:33 comment added M.Ramana Thank you so much for your nice answer.
Aug 20, 2022 at 11:35 vote accept M.Ramana
Aug 20, 2022 at 11:13 comment added Carl-Fredrik Nyberg Brodda Caprace & Radu recently found a “smaller” example of the form $F_5 \ast_{F_{25}} F_5$ and even $F_3 \ast_{F_{11}} F_3$.
Aug 20, 2022 at 9:13 comment added YCor Does the terminology "almost simple" for group with no nontrivial finite quotient really exist? Higman's group is so far from being simple...
Aug 19, 2022 at 21:14 history answered HJRW CC BY-SA 4.0