I am looking for a concrete example of an infinite simple group with two generators. Ideally, one generator has order 2, the other 3, but if there is a nice example without this requirement, it will be appreciated. Schupp proved that every countable group embeds into such a group, but I would like to see a concrete easy example (where it is easy to show simplicity).
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There was family of infinite finitely presented simple groups constructed by Graham Higman in
Higman, Graham Finitely presented infinite simple groups. Notes on Pure Mathematics, No. 8 (1974). Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974.
Unfortunately it does not seem to be particularly easy to get hold of that. They were later proved to be 2-generated in
Mason, David R. On the 2-generation of certain finitely presented infinite simple groups. J. London Math. Soc. (2) 16 (1977), no. 2, 229–231.
I remember attending some lectures by Higman in which he described his construction - they arise as groups of automorphisms of certain algebras, and I remember that it was possible for him to construct them and prove they were infinite and simple within two or three lectures. So their construction is orders of magnitude easier than that o Tarski Monsters!
Try the Tarski Monster (googling on this name will give you a wealth of literature), being a finitely generated infinite simple group, and generated by every two non-commuting elements.
R. Thompson's group V is an infinite simple finitely presented group admitting a presentation with 2 generators and 7 relations (generator orders are 2 and 6 in our paper (arXiv:1511.02123) with Martyn Quick).
The group V admits many generating sets, and I would not be surprised if you could find a generating pair having orders 2 and 3 respectively.