Automorphisms of surface groups and free groups are fundamental in low-dimensional topology and have a close analogy with arithmetic groups like GLn(Z). A foundational question is whether the analogue of “congruence subgroups” are finitely generated. For the first level, this was proved by Dehn (1938); for the second level, it was proved by Johnson (1983). McCullough-Miller conjectured in 1986 that the third level should NOT be finitely generated, but this remained open until this year. (A proof was published in 2006, but turned out to be flawed.)

We introduce a new method for proving a group is finitely generated, and use this to disprove this conjecture for ALL levels: At every level, the congruence subgroups are finitely generated. Joint work with Mikhail Ershov and Andrew Putman, building on work of Mikhail Ershov and Sue He.