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Jun 18, 2016 at 3:17 comment added Andy Putman @luc guyot: I am aware that this says nothing interesting in the 2 generator case, which is why I said it was more of a comment than an answer.
Jun 17, 2016 at 22:30 comment added Luc Guyot As the rank of a closed surface group is two times the genus of the surface, we are left with the finite quotients of the fundamental group of a torus, i.e., the $2$-generated finite abelian groups. (For those groups, there is moreover no mystery about Nielsen equivalence, see e.g., Diaconnis-Graham, 1999.)
Aug 19, 2015 at 22:07 history answered Andy Putman CC BY-SA 3.0