# Comptes Rendus
Mathématique

Moritz Lang and Mikhail Shkolnikov

## Sandpile monomorphisms and limits

Volume 360 (2022), p. 333-341

Published online: 26 April 2022

https://doi.org/10.5802/crmath.291

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Algebra, Group theory / Algèbre, Théorie des groupes

# Sandpile monomorphisms and limits

Moritz Lang<sup>●, ∗</sup>,<sup>a</sup> and Mikhail Shkolnikov<sup>b</sup>

<sup>a</sup> University of Applied Sciences Technikum Wien, Höchstädtplatz 6, 1200 Wien, Austria

<sup>b</sup> Université de Genève, Section de mathématiques, route de Drize 7, villa Battelle, 1227 Carouge, Switzerland

E-mail: moritz.lang@technikum-wien.at (M. Lang)

**Abstract.** We introduce a tiling problem between bounded open convex polyforms $\hat{P} \subset \mathbb{R}^2$ with colored directed edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we construct a monomorphism from the sandpile group $G_{\Gamma_1} = \mathbb{Z}^{\Gamma_1} / \Delta(\mathbb{Z}^{\Gamma_1})$ on $\Gamma_1 = \hat{P}_1 \cap \mathbb{Z}^2$ to the one on $\Gamma_2 = \hat{P}_2 \cap \mathbb{Z}^2$. We provide several examples of infinite series of such tilings converging to $\mathbb{R}^2$, and thus define the limit of the sandpile group on the plane.

**2020 Mathematics Subject Classification.** 20K30, 60K35, 47D07.

Manuscript received 1st January 2021, revised 25th August 2021 and 24th October 2021, accepted 24th October 2021.

# 1. Introduction

## 1.1. Background

Let $\bar{\Gamma} = \Gamma \cup \{s\}$ be the vertices of a finite connected (multi-)graph with sink $s$. Denote by $\partial\Gamma$ the boundary of $\Gamma$ – the set of all vertices adjacent to the sink. The standard discrete graph Laplacian $\bar{\Delta}_{\Gamma}$ is then defined as the difference between the adjacency matrix of $\bar{\Gamma}$ and its degree/valency matrix. When we delete from $\bar{\Delta}_{\Gamma}$ the row and column corresponding to the sink, we obtain the reduced graph Laplacian $\Delta_{\Gamma}$. The sandpile group $G_{\Gamma}$ is then defined as the cokernel of $\Delta_{\Gamma}$ acting on $\mathbb{Z}^{\Gamma}$ [8], i.e.

$$G_{\Gamma} = \mathbb{Z}^{\Gamma} / \Delta_{\Gamma}(\mathbb{Z}^{\Gamma}).$$

The sandpile group was rediscovered several times. As a consequence, it is sometimes referred to as the critical group, as the graph Jacobian, or (sometimes) as the Picard group [2, 4, 6].

Of specific interest are the sandpile groups defined on finite connected domains of the standard square lattice $\mathbb{Z}^2$, i.e. on graphs $\bar{\Gamma}$ obtained from $\mathbb{Z}^2$ by contracting all vertices $\mathbb{Z}^2 \setminus (\mathbb{Z}^2 \cap P)$ outside of some bounded open set $P \subset \mathbb{R}^2$ to the sink. This is also the original setup of the sandpile model, a cellular automaton introduced by Bak, Tang and Wiesenfeld in 1987 [3] as the first and archetypical example of a system showing self-organized criticality. Shortly after

<footnote>* Corresponding author.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 1.** A) Dark-gray points represent the vertices of the standard square lattice $Z^2$, while the gray isosceles triangles correspond to M. B) The isosceles triangles belonging to the M-polyform $P_1$ are highlighted by a green background. The black points and lines represent the vertices and edges of the graph $\Gamma(P_1) = Z^2 \cap P_1$ defined by $P_1$. The sides of the M-polyform are directed and colored, exemplifying the definition of $P_1^{DC}$. C) DC-tiling of a M-polyform $P_2$ by four copies of the M-polyform $P_1^{DC}$ from (B). Tiles have a green background if they can be obtained from $P_1^{DC}$ by only translations and rotations, and blue otherwise. Note that the graph $\Gamma(P_2)$ also consists of the vertices lying on the common edges of pairs of tiles, and the corresponding edges (red points and lines).</figcaption></figure>

the introduction of this cellular automaton, Dhar showed that its recurrent configurations form a group isomorphic to $G_\Gamma$, and laid the foundation for its analysis [8, 9]. On such domains, the sandpile group provides connections between various mathematical fields, including fractal geometry [7, 14], graph theory and algebraic geometry [4, 5], tropical geometry [12, 13], domino tilings [10], and others.

Only for few infinite families of graphs, the structure of the respective sandpile groups has been determined. In particular, the decomposition of the sandpile group on domains of $Z^2$ is yet unknown. Further, results on the relationships between sandpile groups, in terms of homeomorphisms, on different domains of $Z^2$ are lacking<fnref n="1" />. This is in stark contrast to the role of the sandpile model as the archetypical example for self-organized criticality, given that this concept itself is based on the notion of scaling.

## 1.2. Sandpile monomorphisms (main result)

In this article, we analyze the relationships between sandpile groups defined on different domains of the standard square lattice $Z^2$. Specifically, given two domains $\Gamma_1, \Gamma_2 \subset Z^2$, $\Gamma_1 \subseteq \Gamma_2$, our goal is to understand under which conditions group monomorphisms from $G_{\Gamma_1}$ to $G_{\Gamma_2}$ exist.

For this, let M be the unique tiling of $\mathbb{R}^2$ by isosceles triangles with base length 1 and height $\frac{1}{2}$ such that each vertex of $(\mathbb{Z} + 0.5)^2$ coincides with the vertices of four triangles (Figure 1A). An

<footnote n="1">¹As a consequence of the burning algorithm [8], the restriction of a recurrent configuration to a sub-domain is recurrent again. Similarly, a recurrent configuration can be extended to a super-domain by e.g. embedding it into the minimally stable configuration. However, these and similar transformations only define morphisms in the category of sets (of recurrent configurations), but not in the category of (sandpile) groups.</footnote>
M-polyform $P \subset M$ then consists of a finite connected subset of triangles in $M$ (Figure 1B). Note that $M$ is not the usual triangular tiling of the plane, and that $M$-polyforms thus differ from the usual definition of polydiamonds. By a slight abuse of notation, we interpret each $M$-polyform $P$ to directly correspond to the open subset of $\mathbb{R}^2$ enclosed by its isosceles triangles, i.e. to the interior of $\bigcup_{m \in P} m$. To each $M$-polyform $P$, we then associate the domain $\Gamma(P) = \mathbb{Z}^2 \cap P$. This domain is obtained from the standard square lattice $\mathbb{Z}^2$ as described in the Introduction, i.e. by contracting all vertices $\mathbb{Z}^2 \setminus (\mathbb{Z}^2 \cap P)$ to the sink (Figure 1B). We interchangeably denote by $G_\Gamma$ and $G_P$ the sandpile groups defined on the domain $\Gamma = \Gamma(P)$. Denote by $P^{DC}$ the result of assigning directions and colors to the edges of an $M$-polyform $P$ such that each edge has a different color (Figure 1B). Given two $M$-polyforms $P_1$ and $P_2$, we say that $P_1$ DC-tiles $P_2$ if there exists a tiling $T^{P_1 \to P_2}$ of $P_2$ by copies of $P_1^{DC}$ (allowing all transformations which correspond to automorphisms of $M$), such that every common edge of two adjacent tiles in $T^{P_1 \to P_2}$ has the same color and direction (Figure 1C).

**Theorem 1.** Let $P_1$ and $P_2$ be two convex $M$-polyforms, and assume that $P_1$ DC-tiles $P_2$. Then, there exists a group monomorphism $G_{P_1} \to G_{P_2}$ from $G_{P_1}$ to $G_{P_2}$.

In the proof of this theorem (Section 3), we construct an explicit mapping $\mu(T^{P_1 \to P_2}) = (G_{P_1} \to G_{P_2})$ from DC-tilings to the corresponding group monomorphisms. Here, we only note two properties of this construction: (i) for two $M$-polyforms $P_1$ and $P_2$, there can exist more than one distinct DC-tiling of $P_2$ by $P_1$. For example, let the polyform $P$ describe a square with width $w$ and sides parallel to the standard axes of $\mathbb{R}^2$. Since the dihedral group $D_4$ of a square has order eight, there also exist eight different DC-tiling of $P$ by itself. For $w > 2$, $\mu$ maps each of these tilings to a different automorphism of $G_P$, which directly correspond to the action of the respective element of $D_4$ on $\Gamma(P)$. For $w = 2$, the domain $\Gamma(P)$ however consists of only a single vertex, and all eight tilings are mapped to the trivial automorphism; (ii) denote by $\hat{P}$ the result of extending a polyform $P$ by one triangle in $M$ adjacent to $P$ such that $\Gamma(\hat{P}) = \Gamma(P)$. Then, there exist no DC-tilings of $\hat{P}$ by $P$, or vice versa. However, since $G_P = G_{\hat{P}}$, the set of automorphisms is non-empty. We thus conclude that the mapping $\mu$ is in general neither injective nor surjective.

### 1.3. Limits of the sandpile group

Let **Pol** denote the poset of bounded convex $M$-polyforms, with $P_1 \subseteq_{DC} P_2$ if there exists a DC-tiling $T^{P_1 \to P_2}$ of the $M$-polyform $P_2$ by the $M$-polyform $P_1$ such that the position and orientation of one tile in $T^{P_1 \to P_2}$ directly corresponds to $P_1^{DC}$, i.e. $P_1^{DC} \in T^{P_1 \to P_2}$. We naturally identify **Pol** with its corresponding (small) category, with the (faithful) forgetful functor $U: \mathbf{Pol} \to \mathbf{Set}$ to the category of sets mapping each $M$-polyform to its corresponding open subset of $\mathbb{R}^2$ and $\subseteq_{DC}$ to set inclusions. Since the position and orientation of one tile uniquely identifies a DC-tiling (if it exists), the definition of **Pol** allows us to associate a DC-tiling $v(P_1 \subseteq_{DC} P_2) \in \{T^{P_1 \to P_2}\}$ to each morphism $P_1 \subseteq_{DC} P_2$, i.e. the unique DC-tiling satisfying $P_1^{DC} \in T^{P_1 \to P_2}$. We can then define the functor $F: \mathbf{Pol} \to \mathbf{Ab}$ from **Pol** to the category **Ab** of abelian groups, with $F(P) = G_P$ and $F(P_1 \subseteq_{DC} P_2) = \mu(v(P_1 \subseteq_{DC} P_2))$. To see that $F(id_P) = \text{id}_{G_P}$ and $F((P_2 \subseteq_{DC} P_3) \circ (P_1 \subseteq_{DC} P_2)) = (G_{P_2} \to G_{P_3}) \circ (G_{P_1} \to G_{P_2})$, we refer to the construction of the map $\mu$ in Section 3.

Of specific interest are infinite sequences $S = S_0 \subseteq_{DC} S_1 \subseteq_{DC} S_2 \dots$ of $M$-polyforms in **Pol** (identity and composed morphisms omitted), i.e. functors $S \in \mathbf{Pol}^\omega$ from the usual linear order $\omega = \{0, 1, \dots\}$ on the ordinal numbers to **Pol**. Trivially, each of these sequences, composed with the forgetful functor $U$, defines a direct limit $\lim US = \bigcup_i U(S_i) \subseteq \mathbb{R}^2$ (in the category of sets; **Pol** does not admit all filtered colimits), which we denote by $\hat{S}_\infty$. Furthermore, each sequence, composed with $F$, also defines a direct limit $\lim FS$, denoted either as $G_{\hat{S}_\infty}^S$ or, equivalently, by $G_{\Gamma(\hat{S}_\infty)}^S$. We interpret $G_{\Gamma(\hat{S}_\infty)}^S$ as the limit of the sandpile group for $\Gamma(S_i) \to \Gamma(\hat{S}_\infty)$ (with respect
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Depiction of a small part of the category **Polof** *M*-polyforms. Each shape represents a *M*-polyform *P*, while arrows represent morphisms *P*₁ ⊆ *DC* *P*₂ (identities and composed morphisms omitted). The position of the initial triangular *M*-polyform (black) is depicted by a gray background in each *M*-polyform. The composition of all non-bounded sequences *S* with the forgetful functor *U* in the depicted part of **Pol** has a direct limit of ℝ<fnref n="2" />.</figcaption></figure>

to the sequence *S*). In Figure 2, we depict the morphisms between four families of polyforms in **Pol**. The direct limit of each infinite sequence *S* ∈ **Pol*<sup>*w*</sup>* which only contains these polyforms and morphisms, with *S*<sub>*i*+1</sub> ≠ *S*<sub>*i*</sub> for all *i* ∈ *ω*, is given by ̂*S*<sup>*∞*</sup> = ℝ<sup>2</sup>, and thus Γ(*S*<sup>*∞*</sup>) = ℤ<sup>2</sup>. To our knowledge, the respective limits of the sandpile group *G*<sub>ℝ<sup>2</sup></sub><sup>*S*</sup> are the first<sup>2</sup> definitions of limits for the sandpile group on ℤ<sup>2</sup>.

If a given sequence *S* of *M*-polyforms is upper bounded, i.e. if there exists an *u* ∈ *ω* such that *US*<sub>*j*</sub> = *US*<sub>*u*</sub> = ̂*S*<sub>*∞*</sub> for all *j* ≥ *u*, it directly follows that *G*<sub>ℝ*∞*</sub><sup>*S*</sup> ≅ *G*<sub>*S*<sub>*u*</sub></sub>. Thus, for such upper bounded sequences, the limit of the sandpile group is completely determined (up to isomorphisms) by the upper bound, i.e. *F* preserves all finite direct limits. In such cases, we can drop the dependency of *G*<sub>ℝ*∞*</sub><sup>*S*</sup> on *S* and simply write *G*<sub>ℝ*∞*</sub>. We may ask if the same also holds for unbounded sequences:

**Question 2.** Let *S*<sup>*A*</sup>, *S*<sup>*B*</sup> ∈ **Pol**<sup>*w*</sup> be two (possibly unbounded) sequences of *M*-polyforms with common limit ̂*S*<sub>*∞*</sub> = lim<sub>→</sub> *US*<sup>*A*</sup> = lim<sub>→</sub> *US*<sup>*B*</sup>. Is *G*<sub>ℝ*∞*</sub><sup>*S*<sup>*A*</sup></sup> isomorphic to *G*<sub>ℝ*∞*</sub><sup>*S*<sup>*B*</sup>?</>

Let $\hat{\mathrm{Pol}}$ be the category with objects corresponding to all limits $\hat{S}_{\infty} = \lim \underline{\mathrm{US}}$ of sequences $S \in \mathrm{Pol}^{\omega}$ of polyforms, and morphisms $\hat{S}_{\infty}^{A} \subseteq \hat{S}_{\infty}^{B}$ if there exists a natural transformation $S^a \to S^b$ between two sequences $S^a, S^b \in \mathrm{Pol}^{\omega}$ with $\hat{S}_{\infty}^{A} = \lim \underline{\mathrm{US}}^a$ and $\hat{S}_{\infty}^{B} = \lim \underline{\mathrm{US}}^b$, i.e. if $S_i^a \subseteq \mathrm{DC} S_i^b$ for all $i \in \omega$. We interpret $\mathrm{Pol}$ to represent a full subcategory of $\hat{\mathrm{Pol}}$, with the object function of the (fully

<footnote n="2"><sup>2</sup>There exist several approaches to define sandpile models directly on $\mathbb{Z}^2$, which cope with the occurrence of infinite avalanches in various ways [1, 10, 15, 16]. The (weak) limits of the sandpile measures for some of these models concentrate on certain abelian groups. The relationship between these “groups of recurrent configurations of sandpile models on infinite graphs” to the sandpile groups on finite graphs is, to our knowledge, however unclear. Specifically, to our knowledge, it was not shown if these groups correspond to some notion of direct or projective limit in the category of groups.</footnote>
faithful) inclusion functor $I: \mathbf{Pol} \to \widehat{\mathbf{Pol}}$ given by $I(P) = \lim_{\to} U\delta P$, where $\delta: \mathbf{Pol} \to \mathbf{Pol}^\omega$ denotes the diagonal functor with $(\delta P)_i = P$ for all $i \in \omega$. Question 2 then asks if there exists a functor $\hat{F}: \hat{P} \to \mathbf{Ab}$ which preserves all direct limits, and for which $F$ factors as $\hat{F} \circ U$.

In case Question 2 can be answered in the affirmative, a unique limit $G_{\mathbb{Z}^2}$ (up to isomorphisms) of the sandpile group on $\mathbb{Z}^2$ would exist. In this case, the following would hold:

**Corollary 3.** Assume that $G_{\hat{S}_\infty}^{S^A} \cong G_{\hat{S}_\infty}^{S^B}$ whenever $\hat{S}_\infty = \lim_{\to} US^A = \lim_{\to} US^B$. Then, the limit of the sandpile group on $\mathbb{Z}^2$ is isomorphic to its limit on the upper-right quadrant of $\mathbb{Z}^2$, i.e. $G_{\mathbb{Z}^2} \cong G_{\mathbb{Z}^{\ge 0}}$.

This corollary utilizes that, in morphisms $P_1 \subseteq DC P_2$, the positions of the M-polyforms $P_1$ and $P_2$ are considered, and that thus two different morphisms can be mapped by $v$ to the same tiling $T^{P_1 \to P_2}$. This can be used to construct two sequences $S^A$ and $S^B$ such that there exists a natural isomorphism $FS^A \cong FS^B$, but for which $\hat{S}_\infty^A \neq \hat{S}_\infty^B$. Corollary 3 corresponds to choosing $S_0^A = S_0^B$ to be square-shaped M-polyforms with side length $w_0$, $S_{i+1}^A$ and $S_{i+1}^B$ to have side lengths $w_{i+1} = 5w_i$, $S_{i+1}^A$ to be positioned such that $S_i^A$ is in its center, and $S_{i+1}^B$ such that $S_i^B$ is at its bottom-left.

## 1.4. Relationship to harmonic functions and order of the sandpile group

We say that a domain $\Gamma \subseteq \mathbb{Z}^2$ is convex if there exists a convex open set $P \subseteq \mathbb{R}^2$ such that $\Gamma = P \cap \mathbb{Z}^2$. Different to before, we do not require $P$ to be an M-polyform anymore. We say that an R-valued function $H: \Gamma \to R, R \in \{\mathbb{Z}, \mathbb{Q}, \mathbb{R}\}$, is harmonic (on $\Gamma$) if $\Delta_\Gamma H(v) = 0$ for all vertices $v \in \Gamma_0$ in the interior $\Gamma_0 = \Gamma \setminus \partial\Gamma$ of the domain. Note that, different to certain other literature, neither the sink nor the outer boundary $\partial(\mathbb{Z}^2 \setminus \Gamma)$ is in the domain of $H$. The R-valued harmonic functions on $\Gamma$ form the R-module $\mathcal{H}_R^\Gamma$.

**Lemma 4.** For every finite convex domain $\Gamma \subset \mathbb{Z}^2$, $-\Delta_\Gamma : \mathcal{H}_G^\Gamma \cong G^\Gamma$ is an isomorphism from $\mathcal{H}_G^\Gamma = \{H \in \mathcal{H}_Q^\Gamma | \Delta_\Gamma H|_{\partial\Gamma} \in \mathbb{Z}^{\partial\Gamma}\} / \mathcal{H}_Z^\Gamma$ to the sandpile group $G_\Gamma$, with $\mathcal{H}_G^\Gamma$ the subgroup of the rational-valued harmonic functions $\mathcal{H}_Q^\Gamma$ with integer-valued Laplacians, modulo the integer-valued harmonic functions $\mathcal{H}_Z^\Gamma$. This isomorphism corresponds to the exact sequence

$$0 \longrightarrow G_{\Gamma} \longrightarrow \mathcal{H}_{Q}^{\Gamma} / \mathcal{H}_{Z}^{\Gamma} \longrightarrow (\mathbb{Q} / \mathbb{Z})^{\partial\Gamma} \longrightarrow 0.$$

This and the following lemma can be considered to be special cases of the general theory of toppling invariants introduced in [9]: By [7], every element of the sandpile group can be reached from the identity by only adding particles to the boundary of the domain. This allows to choose the (independent) toppling invariants generating the sandpile group such that their Laplacians only have support on the boundary $\partial\Gamma$ of the domain. This choice of toppling invariants (i.e. the exact sequence in Lemma 4) seems to be especially suited to further characterize the sandpile group as it e.g. significantly simplifies the proof of our main theorem. For this reason, an explicit construction of the isomorphism in Lemma 4 is provided in Section 2.

Denote by $\mathcal{B}_R^\Gamma = \{B_i\}_{i=1,...,|\partial\Gamma|}$ a basis for the module $\mathcal{H}_R^\Gamma$ of R-valued harmonic functions on a finite convex domain $\Gamma \subset \mathbb{Z}^2$. By definition, the Laplacian $\Delta_\Gamma H$ of every harmonic function $H \in \mathcal{H}_R^\Gamma$, and thus also of every basis function in $\mathcal{B}_R^\Gamma$, only has support at the boundary $\partial\Gamma$ of the domain. The Laplacian of every basis function in $\mathcal{B}_R^\Gamma$ can thus be restricted to $\partial\Gamma$ without information loss, and we refer to $\Delta_\Gamma \mathcal{B}_R^\Gamma = (\Delta_\Gamma B_1|_{\partial\Gamma},..., \Delta_\Gamma B_{|\partial\Gamma|}|_{\partial\Gamma}) \in R^{|\partial\Gamma| \times |\partial\Gamma|}$ as the potential matrix of $\Gamma$ (with respect to $\mathcal{B}_R^\Gamma$).

**Lemma 5.** Let $\Gamma \subset \mathbb{Z}^2$ be a finite convex domain, and $\mathcal{B}_Z^\Gamma$ be a basis for the module of integer-valued harmonic functions $\mathcal{H}_Z^\Gamma$ on $\Gamma$. Then, the order of the sandpile group $G_\Gamma$ is $|G_\Gamma| = |\det(\Delta_\Gamma \mathcal{B}_Z^\Gamma)|$.

As for the previous lemma, this result can be considered to represent a special case of the general theory of toppling invariants [9].
## 2. Harmonic functions (proofs of Lemmata 4 and 5)

Recall that, in the sandpile model, every recurrent configuration can be reached from the empty configuration (or any other configuration) by only adding particles to the boundary of the domain and “relaxing” the sandpile [7]. Since the elements of the sandpile group correspond to the equivalence classes of the recurrent configurations of the sandpile model (see Introduction), this can be restated as follows: for every element $C \in G^\Gamma$ of the sandpile group, there exist (infinitely many) functions $X \in \mathbb{Z}^\Gamma$ which only have support at the boundary $\partial\Gamma$ of the domain, and which satisfy that $[X] = C$, with $[\cdot] : \mathbb{Z}^\Gamma \to G_\Gamma$ the canonical projection map to the sandpile group. Since $\Gamma$ is assumed to be convex, there exist functions $B_X : \partial(\mathbb{Z}^2 \setminus \Gamma) \to \mathbb{Z}$ such that $X(v) = \sum_{w \sim v} B_X(w)$. When considering $B_X$ as a boundary condition, it follows, by the existence and uniqueness of solutions to the discrete Dirichlet problem on convex domains [15], that, for every such $X \in \mathbb{Z}^\Gamma$, there exists a unique rational-valued harmonic function $H_X \in \mathcal{H}_Q^\Gamma$ with $\Delta_\Gamma H_X = -X$.

The composition $[\cdot] \circ -\Delta_\Gamma$ of the discrete Laplacian with the canonical projection map then maps two harmonic functions $H_{X,1}, H_{X,2} \in \mathcal{H}_Q^\Gamma$, $\Delta_\Gamma H_{X,1}, \Delta_\Gamma H_{X,2} \in \mathbb{Z}^\Gamma$, to the same element of the sandpile group if and only if $-\Delta_\Gamma(H_{X,1} - H_{X,2}) \in \Delta_\Gamma(\mathbb{Z}^\Gamma)$. Since both $\Delta_\Gamma H_{X,1}$ and $\Delta_\Gamma H_{X,2}$ only have support at the boundary $\partial\Gamma$ of the domain, $H_{X,1} - H_{X,2}$ is thus an integer-valued harmonic function, which concludes our proof of Lemma 4.

We construct the inverse of $-\Delta_\Gamma : \mathcal{H}_Q^\Gamma \cong G_\Gamma$ in two steps. For every configuration $C \in G_\Gamma$, we first define the coordinates $\sigma_\Gamma : G_\Gamma \to (\mathbb{Q}/\mathbb{Z})^{\partial\Gamma}$,

$$ \sigma_\Gamma(C) \equiv -(\Delta\mathcal{B}_Z^\Gamma)^{-1}X \pmod{1}, $$

with respect to the basis $\mathcal{B}_Z^\Gamma$. Note that, for two different choices $X^\alpha, X^\beta \in \mathbb{Z}^\Gamma$, $-(\Delta\mathcal{B}_Z^\Gamma)^{-1}(X^\alpha - X^\beta) \in \mathbb{Z}^{\partial\Gamma}$, and that thus the coordinates $\sigma_\Gamma$ don't depend on the specific choice for $X$. This also implies that $\sigma_\Gamma$ correspond to toppling invariants as defined in [9].

In the second step, we then define the function $\phi_\Gamma : (\mathbb{R}/\mathbb{Z})^{\partial\Gamma} \to \mathcal{H}_R^\Gamma / \mathcal{H}_Z^\Gamma$, $\phi_\Gamma(s) = \sum_{i=1}^{\lvert\partial\Gamma\rvert} s_i B_i$. It is easy to check that the composition $\phi_\Gamma \circ \sigma_\Gamma : G_\Gamma \to \mathcal{H}_R^\Gamma / \mathcal{H}_Z^\Gamma$ is independent of the choice of the basis $\mathcal{B}_Z^\Gamma$, and that $-\Delta_\Gamma \phi_\Gamma(\sigma_\Gamma([X])) = [X]$. The latter implies that $\phi_\Gamma \circ \sigma_\Gamma$ is the inverse of $-\Delta_\Gamma$.

The isomorphism between the sandpile group $G_\Gamma$ and $\mathcal{H}_Q^\Gamma$ proposes to consider the sandpile group as a discrete subgroup of a continuous Lie group isomorphic to $\mathcal{H}_R^\Gamma / \mathcal{H}_Z^\Gamma$, to which we refer to as the extended sandpile group $\tilde{G}_\Gamma$ [14]. More precisely, the extended sandpile group is an extension of the torus $(\mathbb{R} \setminus \mathbb{Z})^{\partial\Gamma}$ by the usual sandpile group, and is defined by the exact sequence

$$ 0 \longrightarrow G_\Gamma \longrightarrow \tilde{G}_\Gamma \longrightarrow (\mathbb{R}/\mathbb{Z})^{\partial\Gamma} \longrightarrow 0. $$

In terms of the sandpile model, this Lie group is obtained by allowing each vertex $b \in \partial\Gamma$ in the boundary $\partial\Gamma$ of the domain to carry a real value $\tilde{C}(b) \in [0,4)$ of particles, while each vertex $v \in \Gamma_0$ in the interior $\Gamma_0 = \Gamma \setminus \partial\Gamma$ of the domain is still only allowed to carry an integer number of particles, i.e. $\tilde{C}(v) \in \{0,1,2,3\}$ (the toppling rules are kept unchanged) [14]. This definition lifts $\phi_\Gamma : (\mathbb{R}/\mathbb{Z})^{\partial\Gamma} \cong \tilde{G}_\Gamma$ to a group isomorphism, and a left-inverse of the inclusion map $G_\Gamma \to \tilde{G}_\Gamma$ is given by the floor function $[\cdot] : \tilde{G}_\Gamma \to G_\Gamma$. We thus naturally arrive at the function $f = [\cdot] \circ -\Delta_\Gamma \circ \phi_\Gamma : (\mathbb{R}/\mathbb{Z})^{\partial\Gamma} \to G_\Gamma$, $f(s) = -([\sum_{i=1}^{\lvert\partial\Gamma\rvert} s_i \Delta_\Gamma B_i])$, which justifies to interpret the usual sandpile group $G_\Gamma$ as the discretization of an $|\partial\Gamma|$-dimensional torus [14].

Due to the properties of the floor function, the preimage $f^{-1}(C)$ of an element $C \in G_\Gamma$ of the sandpile group under $f$ is connected. Denote by vol($f^{-1}(C)$) the volume of this preimage, with vol($\mathbb{R}/\mathbb{Z}^{\partial\Gamma}$) = 1. Since, for every $C \in G_\Gamma$, there exists a coordinate transformation $s \mapsto \tilde{s}$ such that $C$ has coordinates $\tilde{s} = 0$, we get that vol($f^{-1}(C)$) = vol($f^{-1}(0)$) = $\frac{1}{|G_\Gamma|}$ for all $C \in G_\Gamma$, with **0** the identity of the sandpile group. The preimage $f^{-1}(0)$ of the identity under $f$ forms a $|\partial\Gamma|$-parallelotope with edges $g_i$ given by $(\Delta\mathcal{B}_Z^\Gamma)g_i = e_i$, with $(e_i)_j = \delta_{ij}$ the $i^{th}$ unit vector and $\delta_{ij}$ the Kronecker delta. The volume of this parallelotope is vol($f^{-1}(0)$) = $|\det(\Delta\mathcal{B}_Z^\Gamma)^{-1}|$, and thus $|G_\Gamma| = |\det(\Delta\mathcal{B}_Z^\Gamma)|$, which proves Lemma 5.
**3. Construction of sandpile monomorphisms (proof of Theorem 1)**

Let $P_A$ and $P_B$ be two convex $M$-polyforms, and assume that there exists a $DC$-tiling $T^{P_A} \to P_B$ of $P_B$ by $P_A$. In this section, we then construct a monomorphism from the sandpile group on the domain $\Gamma_A = \Gamma(P_A) = \mathbb{Z}^2 \cap P_A$ to the sandpile group on $\Gamma_B = \Gamma(P_B) = \mathbb{Z}^2 \cap P_B$, and thus prove Theorem 1. Before starting this construction, we shortly state three properties of $DC$-tilings.

**Corollary 6.** The domains of different tiles do not overlap, i.e. $\Gamma_i \cap \Gamma_j = \emptyset$ for all $i \neq j$.

The domains of the tiles in general don't cover $\Gamma_B$. Specifically, all vertices of $\Gamma_B$ which lie directly on common edges (including their endpoints) of two tiles are not elements of any $\Gamma_i$ (red vertices in Figure 1). We refer to the set $\partial^T \Gamma_B = \Gamma_B \setminus \bigcup_i \Gamma_i$ of these vertices as the internal boundaries of the tiling. They separate the domains $\Gamma_i$ in the following sense:

**Corollary 7.** The removal of all vertices in $\partial^T \Gamma_B$ splits $\Gamma_B$ into the disconnected components $\{\Gamma_1, \Gamma_2, \dots, \Gamma_{|T^{P_A} \to P_B|}\}$.

By definition, each tile $P_i \in T^{P_A} \to P_B$ can be obtained from $P_A^{DC}$ by a combination of translations, rotations and reflections. If this is possible by using only translations and rotations, we assume the sign $s(P_i) = +1$ to the tile, and otherwise the sign $s(P_i) = -1$. This definition also induces signs $s(v) = s(\Gamma_i) = s(P_i)$ for the domains $\Gamma_i$ and vertices $v \in \Gamma_i$ belonging to the tiles. To the internal boundaries $\partial^T \Gamma_B$ and their vertices $b \in \partial^T \Gamma_B$, we assign the sign $s(\partial^T \Gamma_B) = s(b) = 0$. The relationship of each tile $P_i$ with the polyform $P_A$ (i.e. the translations, rotations and reflections mapping $P_A$ on $P_i$) corresponds to a function $\psi_i : \Gamma_A \to \Gamma_B$ which maps vertices $v_A \in \Gamma_A$ of the polyform onto their corresponding vertices $v_i$ of the tile. For two vertices $v, w \in \Gamma_B$, we then define the equivalence relation $\equiv_{DC}$ such that $v \equiv_{DC} w$ if there exists a $v_A \in \Gamma_A$ such that $v = \psi_i(v_A)$ and $w = \psi_j(v_A)$ for some tiles $P_i$ and $P_j$, or if both vertices are part of the internal boundaries, i.e. $v, w \in \partial^T \Gamma_B$. We denote by $[v]_{DC}$ the equivalence class of $v$ induced by $\equiv_{DC}$.

**Corollary 8.** Let $b \in \partial^T \Gamma_B$ be a vertex of the internal boundaries. Then, the number of neighbors of $b$ in every equivalence class $[v_B]_{DC}$, $v_B \in \Gamma_B$ carrying a positive sign is equal to the number of neighbors carrying a negative sign, i.e. $\sum_{\substack{v \in [v_B]_{DC} \\ v \sim b}} s(v) = 0$.

**Proof.** Assume that $b$ has at least one neighbor in $[v_B]_{DC}$; otherwise the corollary is trivially satisfied. Also, assume $v_B \notin \partial^T \Gamma_B$, since otherwise $s(v) = 0$ for all $v \in [v_B]_{DC}$, from which the corollary also trivially follows. Denote by $N(b, v_B) = \{v \in [v_B]_{DC} | v \sim b\}$ the set of neighbors of $b$ in the equivalence class of $v_B$. Every vertex can have maximally four neighbors, thus $|N(b, v_B)| \le 4$. Being part of the internal boundaries, $b$ must lie on at least one common edge (including endpoints) of two tiles $P_i \neq P_j$. These two tiles can be mapped onto one another by reflection on the common edge, and thus must have opposite signs. If a vertex $v \in \Gamma_i$ of $P_i$ is a neighbor of $b$, it follows that there must be a vertex $w \in \Gamma_j$ of $P_j$ which is also a neighbor of $b$, and which has opposite sign, i.e. $s(w) = -s(v)$. This excludes the case $|N(b, v_B)| = 1$, and proves the corollary for $|N(b, v_B)| = 2$. For $|N(b, v_B)| \in \{3, 4\}$, the structure of $M$ directly implies that $b$ has to lie on a common corner of three, respectively four, tiles. The corresponding internal angles of the tiles have to be smaller or equal to $360^\circ/3 = 120^\circ$, respectively $360^\circ/4 = 90^\circ$. The definition of $M$ only admits internal angles which are multiples of $45^\circ$ (Figure 1A). Thus, in both cases, only angles of $45^\circ$ or $90^\circ$ are possible. An angle of $45^\circ$ is only possible if all $v \in [v_B]$ lie on the internal boundaries, which implies $s(v) = 0$ (see above). If the angle is $90^\circ$, $|N(b, v_B)| = 3$ would imply that $P_B$ is not convex, which can thus be excluded. Finally, if the angle is $90^\circ$ and $|N(b, v_B)| = 4$, each of the four tiles to which these vertices belong must have exactly two adjacent tiles with opposite signs. $\square$

By Lemma 4, the sandpile group $G_\Gamma$ is isomorphic to $\mathcal{H}_G^\Gamma = \{H \in \mathcal{H}_Q^\Gamma | \Delta_\Gamma H|_{\partial_\Gamma} \in \mathbb{Z}^{\partial_\Gamma}\} / \mathcal{H}_Z^\Gamma$. To prove Theorem 1, it thus suffices to construct a monomorphism from $\mathcal{H}_G^{P_A}$ to $\mathcal{H}_G^{P_B}$ given
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Construction of sandpile monomorphisms. A) A harmonic function $H_A \in \mathcal{H}_G^{\Gamma_A}$ (left) corresponding to the element $[-\Delta_{\Gamma_A} H_A] \in G_{P_A}$ of the sandpile group (right) on a given $M$-polyform $P_A^{DC}$ (colored arrows indicate edges, green squares vertices of $\Gamma_A = \Gamma(P_A)$). B) The $M$-polyform $P_A$ from (A) DC-tiles the depicted $M$-polyform $P_B$. Depicted is the rational-valued function $\hat{H}_B$ (top-left), and the three integer-valued functions $X_1, X_2$ and $X_3$ (one for each tile) which cure the non-harmonicity of $\hat{H}_B$. C) The function $H_B = \hat{H}_B + X_1 + X_2 + X_3$ is harmonic everywhere and can be reinterpreted as an element of $\mathcal{H}_G^{\Gamma_B}$ (left). It corresponds to the element $[-\Delta_{\Gamma_B} H_B] \in G_{P_B}$ of the sandpile group on $P_B$ onto which $[-\Delta_{\Gamma_A} H_A] \in G_{P_A}$ is mapped by the monomorphism (right).</figcaption></figure>

a DC-tiling $T^{P_A \to P_B}$ of the $M$-polyform $P_B$ by $P_A$. For a harmonic function $H_A \in \mathcal{H}_G^{\Gamma_A}$, define the rational-valued function $\hat{H}_B \in \mathbb{Q}^{\Gamma_B}$ in the following way (Figure 3A&B): for each vertex $v \in \Gamma_i$ belonging to tile $P_i \in T^{P_A \to P_B}$, set $\hat{H}_B(v) = s(P_i) H_A(v_A)$ with $v_A \in \Gamma_A$ the unique vertex satisfying $\psi_i(v_A) = v$. Otherwise, that is if $v$ belongs to the internal boundaries, set $\hat{H}_B(v) = 0$. Because $\hat{H}_B(b) = 0$ for all vertices $b \in \partial^T \Gamma_B$ of the internal boundaries, Corollary 7 implies that $\Delta_{\Gamma_B} \hat{H}_B(v) = s(\Gamma_i) \Delta_{\Gamma_A} H_A(v_A)$ for all vertices $v \in \Gamma_i$ belonging to the domain of a tile $P_i$, with $\psi_i(v_A) = v$. This implies that the Laplacian of $\hat{H}_B$ is zero in the interior of the sub-domains $\Gamma_i$ of $\Gamma_B$, and integer-valued at their boundaries. From Corollary 8, on the other hand, it follows that $\Delta_{\Gamma_B} \hat{H}_B(b) = 0$ for every vertex $b \in \partial^T \Gamma_B$ of the internal boundaries. Thus, $\hat{H}_B$ is harmonic nearly everywhere, except at the vertices directly adjacent to (but not including) the internal boundaries, for which $\Delta_{\Gamma_B} \hat{H}_B$ is integer-valued.

The “harmonic deficit” of $\hat{H}_B$ can be cured, one tile at a time: for a given tile $P_i$, we can define an integer-valued function $X_i \in \mathbb{Z}^{\Gamma_B}$ whose Laplacian is zero everywhere in the interior of $\Gamma_B$, except for those vertices $v \in \partial \Gamma_i \setminus \partial \Gamma_B$ at the boundary of $\Gamma_i$ which are not also at the boundary of
$\Gamma_B$. Note that these vertices are all next to an internal boundary. For these vertices $v \in \partial\Gamma_i \setminus \partial\Gamma_B$, we require that $\Delta_{\Gamma_B} X_i(v) = -\Delta_{\Gamma_B} \hat{H}_B(v)$. For example, $X_i$ can be defined by the following iterative algorithm: first, set $X_i(v) = 0$ for all $v \in \Gamma_i$. Define $\Gamma^0 = \Gamma_i$ and $B^0 = \{v \in \Gamma_B \setminus \Gamma_i \mid \exists w \in \Gamma_i : v \sim w\}$. For every $v \in B^0$, choose an integer value $X_i(v)$ such that for every $w \in \partial\Gamma_i \setminus \partial\Gamma_B$, $\Delta_{\Gamma_B} \hat{H}_B(w) = -\Delta_{\Gamma_B} X_i(w)$. Note that, since $\Gamma_i$ is convex, the corresponding equation system always admits a solution (even though non-unique). Then, in every iteration $s = 1, 2, \dots$, set $\Gamma^s = \Gamma^{s-1} \cup B^{s-1}$ and $B^s = \{v \in \Gamma_B \setminus \Gamma^s \mid \exists w \in \Gamma^s : v \sim w\}$. Note that $\Gamma^s$ is convex again. Now, for every $v \in B^s$, choose an integer value $X_i(v)$ such that for every $w \in \partial\Gamma^s \setminus \partial\Gamma_B$, $\Delta_{\Gamma_B} X_i(w) = 0$. Again, this is possible due to convexity of $\Gamma^s$. Continue until all values of $X_i$ are defined. In general, there is some degree of freedom to choose the values of $X_i$ assigned in every step of this algorithm. However, note that the difference $X_i^\alpha - X_i^\beta$ of every two such possible choices $X_i^\alpha$ and $X_i^\beta$ is integer-valued harmonic.

Given the functions $X_i$, we define $H_B \in \mathbb{Q}^\Gamma$ by $H_B = \hat{H}_B + \sum_i X_i$. By construction, $H_B$ is harmonic everywhere and has an integer-valued Laplacian. We can thus reinterpret $H_B$ to be an element of $\mathcal{H}_{\mathbb{G}}^{\Gamma_B}$. It is then easy to see that the function $\xi : \mathcal{H}_{\mathbb{G}}^{P_A} \to \mathcal{H}_{\mathbb{G}}^{P_B}$, $\xi(H_A) = H_B$, is injective, and that it satisfies $\xi(H_B^1 + H_B^2) = \xi(H_B^1) + \xi(H_B^2)$. The function $\xi$ is thus a group monomorphism, and with the isomorphism $-\Delta_\Gamma : \mathcal{H}_{\mathbb{G}}^P \cong G_P$ from Lemma 4, we get that $\mu(T^{P_A \to P_B}) = \Delta_{\Gamma_B} \circ \xi \circ (\Delta_{\Gamma_A})^{-1} : G_{P_A} \to G_{P_B}$ is the group monomorphism stated to exist in Theorem 1.

Acknowledgements

The authors are grateful to Nikita Kalinin for carefully reading the manuscript.

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