Fox derivatives, group cohomology, and higher property (T)

Abstract

I will focus on cohomology of finitely presented groups. I plan to investigate two conditions concerning it: vanishing and reducibility (for all unitary representations). These conditions are related to Kazhdan’s property (T): vanishing and reducibility coincide in degree one and are equivalent to this property. In 2020, Bader and Nowak gave a sufficient condition for vanishing and reducibility in two consecutive degrees. Using this condition, we were able to show reducibility in degree two for SL(3,Z), the special linear group of degree three. The key new tool we applied in the proof is the Fox calculus which can be used to compute group cohomology. This is the joint work with Marek Kaluba and Piotr Nowak.