# Sandpile monomorphisms and limits

## TL;DR

This paper explores tiling problems between convex polyforms and their implications for constructing monomorphisms between sandpile groups, providing new insights into the scaling limits of these groups on the plane.

## Contribution

It introduces a novel connection between tilings of polyforms and monomorphisms of sandpile groups, establishing the first framework for defining their scaling limits in the plane.

## Key findings

- Constructed monomorphisms between sandpile groups via tilings.
- Developed an exact sequence linking sandpile configurations and harmonic functions.
- Proposed a method to analyze subgroups of infinite sandpile groups using harmonic functions.

## Abstract

We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset\mathbb{R}^2$ with directed and uniquely colored edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we show that one can construct a monomorphism from the sandpile group $G_{\Gamma_1}=\mathbb{Z}^{\Gamma_1}/\Delta(\mathbb{Z}^{\Gamma_1})$ on the domain (graph) $\Gamma_1=\hat{P}_1\cap\mathbb{Z}^2$ to the respective group on $\Gamma_2=\hat{P}_2\cap\mathbb{Z}^2$. We provide several examples of infinite series of such tilings with polyforms converging to $\mathbb{R}^2$, and thus the first definition of scaling-limits for the sandpile group on the plane. Additional results include an exact sequence relating sandpile configurations to harmonic functions, an alternative formula for the order of the sandpile group based on a basis for the module of integer-valued harmonic functions, and three examples of how to prove the existence of (cyclic) subgroups for infinite families of sandpile groups by constructing appropriate integer-valued harmonic functions. The main open question concerns if the scaling-limits of the sandpile group for different sequences of polyforms converging to $\mathbb{R}^2$ are isomorphic.