Tropicalizations of locally symmetric varieties

We relate the top-weight rational cohomology of a locally symmetric variety to the cohomology of arithmetic groups associated to its rational boundary components. This relation is given in terms of a fundamental spectral sequence, whose applications to the cohomology of Siegel modular varieties and unitary modular varieties will be presented. By studying the combinatorics of the boundary, we are able to exhibit a Hopf algebra structure, with applications to the cohomology of arithmetic groups.

This is joint work with Madeline Brandt, Juliette Bruce, Melody Chan and Raluca Vlad.