The High-Degree Rational Cohomology of the Special Linear Group - Jenny Wilson

Stony Brook Mathematics Colloquium
May 18, 2023
Speaker: Jenny Wilson (University of Michigan)
Abstract:
In this talk I will describe some current efforts to understand the high-degree rational cohomology of SL_n(Z), or more generally the cohomology of SL_n(R) when R is a number ring. Although the groups SL_n(R) do not satisfy Poincare duality, they do satisfy a twisted form of duality, called (virtual) Bieri--Eckmann duality. Consequently, their high-degree rational cohomology groups are governed by an SL_n(R)-representation called the Steinberg module. The key to understanding these representations is through studying the topology of certain associated simplicial complexes. I will survey some results, conjectures, and ongoing work on the Steinberg modules, and the implications for the cohomology of the special linear groups. This talk includes work joint with Brück, Kupers, Miller, Patzt, Sroka, and Yasaki.