A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group

In his preprint https://arxiv.org/abs/1308.3813, Cartwright introduced the notion of a weak tropical complex in order to generalize the concepts of divisors and the Picard group on graphs from Baker and Norine's paper Riemann-Roch and Abel-Jacobi Theory on a Finite Graph. A tropical complex $\Gamma$ is a $\Delta$-complex equipped with certain algebraic data. Divisors in a tropical complex are formal linear combinations of ridges, and piecewise-linear functions on a tropical complex give rise in a natural way to divisors. Divisors that arise from PL-functions are called principal, and divisors that are locally principal are called Cartier. Two divisors that differ by a principal divisor are said to be linearly equivalent. The linear equivalence classes of Cartier divisors on a tropical complex $\Gamma$ form a group called the Picard group of $\Gamma$, by analogy to the definition of the Picard group of a variety in algebraic geometry. Every graph has a unique tropical complex structure. If $G$ and $H$ are graphs, and $\Gamma$ is a triangulation of their product, then $\Gamma$ has a weak tropical complex structure that is compatible with the tropical complex structures on $G$ and $H$. Thus, divisors on $\Gamma$ can be thought of as states in a higher-dimensional chip-firing game on $\Gamma$. Cartwright conjectured that the Picard groups of $\Gamma$, $G$, and $H$ were closely related. Let $Pic(\Gamma)$ be the tropical Picard group of $\Gamma$, and $Pic(G)$ and $Pic(H)$ be the tropical Picard groups of $G$ and $H$. Then, it was conjectured that there is a map $\gamma: Pic(G) \times Pic(H) \to Pic(\Gamma)$ that is always injective and is surjective if at least one of $G$ or $H$ is a tree. In this paper, we prove the conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.