High-dimensional cohomology of special linear and symplectic groups

Computing the cohomology of arithmetic groups is a fundamental and often difficult problem at the intersection of topology, group theory and number theory. In this talk, I will explain how one can use duality phenoma to compute the rational cohomology of the arithmetic groups SL_n(Z) and Sp_2n(Z) in "high" dimensions, i.e. close to their virtual cohomological dimension. Specifically, I will talk about joint work with Miller-Patzt-Sroka-Wilson in which we show that H^{n(n-1)/2 - 2}(SL_n(Z); Q) = 0 for n>3. This was previously unknown, but confirms a conjecture of Church-Farb-Putman. In ongoing work with Patzt-Sroka, we are also trying to adapt these techniques to the group Sp_2n(Z).