Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: New methods for finite generation

Seminar | November 21 | 3:45-4:45 p.m. | 939 Evans Hall

 Tom Church, Stanford University

 Department of Mathematics

Automorphisms of surface groups or free groups are fundamental in low-dimensional topology, and have a close analogy with arithmetic groups like \(GL_n(\mathbb Z)\). 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 (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, Sue He, and Andrew Putman. This talk will require no background, and should be accessible to math grad students in any field.