Tropicalizations of Symmetric Spaces

Let D be a symmetric domain and G a discrete algebraic group acting on it. For example, one can take the 2-by-2 integral special linear group acting on the Siegel upper half-plane via Mobius transformations. We consider a certain (toroidal) compactification of the quotient space D/G, due to Mumford, and we describe its tropicalization, as follows. We construct a collection of polyhedral cones that are in dimension-reversing bijection with the boundary components of our compactification. You can think of these polyhedral cones as encoding the combinatorics of the boundary of our compactification. Finally, I will show how this tropical object can be used to deduce statements about the cohomology of the original space. The main example I will discuss will be the moduli space of tropical abelian varieties.

Based on joint work in progress with E. Assaf, M. Brandt, J. Bruce, and M. Chan.