Tropicalizations of locally symmetric varieties

TL;DR

This paper rigorously studies tropicalizations of locally symmetric varieties and explores their applications to cohomology of moduli spaces and arithmetic groups, focusing on special unitary and abelian variety cases.

Contribution

It introduces a comprehensive framework for tropicalizations of locally symmetric varieties and applies it to new areas like cohomology of moduli spaces and arithmetic groups.

Findings

Tropicalizations provide new insights into the cohomology of moduli spaces.

Applications extend beyond tropical geometry to arithmetic group cohomology.

Detailed analysis of special unitary and abelian variety cases.

Abstract

This paper provides a rigorous study of tropicalizations of locally symmetric varieties. We give applications beyond tropical geometry, to the cohomology of moduli spaces as well as to the cohomology of arithmetic groups. We study two cases in detail: the special unitary case, and the case of level structures on the moduli space $\mathcal{A}_g$ of abelian varieties.