Tropical geometry replaces polynomial algebra with piecewise-linear combinatorics. Addition becomes minimum; multiplication becomes addition. An algebraic curve becomes a graph. The tropicalization — the process of taking this combinatorial shadow — preserves certain information (intersection numbers, genus) while discarding others (precise coordinates, analytic structure). For algebraic curves, tropicalization is well understood: the tropical curve retains enough of the original's topology to compute enumerative invariants.
Locally symmetric varieties are not curves. They are quotients of symmetric spaces by arithmetic groups — objects of deep number-theoretic significance, including moduli spaces of abelian varieties with level structure. They carry arithmetic information that seems far removed from piecewise-linear combinatorics.
The paper tropicalizes them anyway. The combinatorial shadow of a locally symmetric variety retains enough structure to compute cohomological invariants of the corresponding arithmetic group. The tropical version — a polyhedral complex rather than a manifold — encodes the same topological information that the full algebraic object does, in the dimensions where the cohomology is concentrated.
The structural point: the combinatorial skeleton of an arithmetically structured space is unexpectedly rich. The information that tropicalization preserves is not just geometric (shapes, intersections) but arithmetic (group cohomology, level structure). The shadow is thinner than the object but not as thin as expected — it retains the arithmetic content alongside the topological content. The combinatorial machinery that works for curves extends, with effort, to objects of fundamentally higher complexity.