Andrew Putman

Representation stability and finite linear groups

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.

Cutting and pasting in the Torelli group

We introduce machinery to allow “cut-and-paste”-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives different groups depending on how the subsurfaces are embedded. We define a category TSur whose objects are surfaces together with a decoration restricting how they can be embedded into larger surfaces and whose morphisms are embeddings which respect the decoration. There is a natural “Torelli functor” on this category which extends the usual definition of the Torelli group on a closed surface. Additionally, we prove an analogue of the Birman exact sequence for the Torelli groups of surfaces with boundary and use the action of the Torelli group on the complex of curves to find generators for the Torelli group. For genus g 1 only twists about (certain) separating curves and bounding pairs are needed, while for genus gD 0 a new type of generator (a “commutator of a simply intersecting pair”) is needed. As a special case, our methods provide a new, more conceptual proof of the classical result of Birman and Powell which says that the Torelli group on a closed surface is generated by twists about separating curves and bounding pairs. 57S05; 20F05, 57M07, 57N05

Stability in the homology of congruence subgroups

The homology groups of many natural sequences of groups

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = 1

\{G_n\}_{n=1}^{\infty }

The second rational homology group of the moduli space of curves with level structures

{Gn}n=1∞ (e.g. general linear groups, mapping class groups, etc.) stabilize as

On the self-intersections of curves deep in the lower central series of a surface group

n \rightarrow \infty