The High-Degree Rational Cohomology of the Special Linear Group

Explore the high-degree rational cohomology of SL_n(Z) and SL_n(R) for number rings R in this mathematics colloquium talk. Delve into the concept of virtual Bieri-Eckmann duality and its implications for the cohomology of special linear groups. Examine the Steinberg module, a crucial SL_n(R)-representation governing high-degree rational cohomology groups. Investigate the topology of associated simplicial complexes and their role in understanding these representations. Survey current results, conjectures, and ongoing research on Steinberg modules, including collaborative work with Brück, Kupers, Miller, Patzt, Sroka, and Yasaki. Gain insights into the complex mathematical structures underlying special linear groups and their cohomological properties.