Tropicalizations of locally symmetric varieties

A locally symmetric variety is a non-compact complex algebraic variety obtained as the quotient of a Hermitian symmetric domain by the action of an arithmetic group. I will start by reviewing the theory of toroidal compactifications of these varieties, originally due to Ash-Mumford-Rapoport-Tai. Building on this construction, we define the tropicalization of a locally symmetric variety to be a combinatorial object encoding the boundary strata of a toroidal compactification of the variety. I will discuss applications of this theory to the cohomology of moduli spaces and arithmetic groups, with an emphasis on the case of moduli of abelian varieties and general linear groups. Based on joint work with Assaf, Brandt, Bruce, and Chan.