Speaker
Jeremy Miller
Description
I will describe how Steinberg modules not only form a ring but in fact form a bi-algebra in a “duoidal” sense. This endows the homology of general linear groups with Steinberg module coefficients with the structure of a Hopf algebra (work joint with Ash and Patzt, independently work of Brown—Chan—Galatius—Payne). I will describe applications to the unstable cohomology of SLn(Z) and GLn(Z) due to Brown—Chan—Galatius—Payne and Brown—Hu—Panzer. These results resolve a question of Lee. This duoidal bi-algebra structure also has implications for the cohomology of congruence subgroups, such as work of Ash on injectivity of multiplication maps.
