Necessary conditions for tiling finitely generated amenable groups

TL;DR

This paper explores necessary conditions for tiling finitely generated amenable groups with Wang tiles, extending known conditions from free groups and a5^2 to a broader class of groups, and establishing their equivalence.

Contribution

It generalizes and unifies necessary tiling conditions from free groups and a5^2 to all finitely generated amenable groups, confirming a conjecture.

Findings

Necessary conditions are equivalent for tilings of free groups and a5^2.

These conditions are necessary for tilings of any finitely generated amenable group.

The paper confirms a conjecture by Jeandel regarding tiling conditions.

Abstract

We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group. Piantadosi gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group. We then consider two other conditions: the first, also given by Piantadosi, is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al., is a necessary condition to decide if a set of Wang tiles gives a tiling of $\mathbb Z^2$. We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel.

Figures

Equations

$$\forall x\in {\mathcal{A}}^G,\forall h\in G,({\sigma }_h(x_g){)}_{g\in G}=(x_{h^{-1}g}{)}_{g\in G}$$

$$Y=Y_{\mathcal{F}}=\{x\in {\mathcal{A}}^G\mid \forall P⊏x:P\notin \mathcal{F}\}.$$

$$X_T=\{(x_g)\in T^G|\forall g\in G,s\in S\cup S^{-1},x_g(s)=x_{gs}(s^{-1})\}$$

$$\forall a,b\in\mathcal{A},\qquad a\to b\text{ in }\Gamma_{i}\Longleftrightarrow\left\{\begin{array}[]{ccc}1_{G}&\to&a\\ g_{i}&\to&b\end{array}\right.\notin\mathcal{F}.$$

$$|w{|}_a=\#\{i\mid w_i=a,1\le i\le |w|\}.$$

$$\forall a\in \mathcal{A},|w{|}_a=\sum _{\omega \in \mathcal{SC}(\Gamma )}{\lambda }_{\omega }|\omega {|}_a.$$

$$\forall x\in X,\forall g\in G,\Phi (x{)}_g=\phi ({\sigma }_{g^{-1}}(x){|}_H),$$

$$\forall a\in \mathcal{A},\sum _{j=1}^{\#\mathcal{SC}({\Gamma }_1)}{x}_{1,j}|{\omega }_1^j{|}_a=\sum _{j=1}^{\#\mathcal{SC}({\Gamma }_2)}{x}_{2,j}|{\omega }_2^j{|}_a=\cdots =\sum _{j=1}^{\#\mathcal{SC}({\Gamma }_d)}{x}_{d,j}|{\omega }_d^j{|}_a.$$

$$x_{1,1}$$

$$\forall g\in S,\forall c\in \mathcal{C},\sum _{{\tau }_i\in c_g}{x}_i=\sum _{{\tau }_j\in c_{g^{-1}}}{x}_j$$

$$(g_1,a)$$

$$(g_1,b)$$

$$(g_1,c)$$

$$\forall c\in \mathcal{C},\sum _{{\tau }_i\in c_{g_1}}{x}_i=\sum _{{\tau }_j\in c_{g_1^{-1}}}{x}_j,$$

$${\Psi }_1^c:\{{\tau }_i^k:{\tau }_i\in c_{g_1},1\le k\le x_i\}\to \{{\tau }_{i^{\prime }}^{k^{\prime }}:{\tau }_{i^{\prime }}\in c_{g_1^{-1}},1\le k^{\prime }\le x_{i^{\prime }}\}$$

$${\mathrm{stab}}_{\sigma }(\Phi (x))$$

$$=\{g\in G_2:\forall h\in G_2,x_{\pi (h)\pi (g)}=x_{\pi (h)}\}$$

$$\tilde{\pi }:G_2/{\pi }^{-1}({\mathrm{stab}}_{\sigma }(x))\to G_1/{\mathrm{stab}}_{\sigma }(x)$$

$$G=\bigcup _n{S}_n\text{and}\forall g\in G,\frac{\#(S_{n}g△S_n)}{\#S_n}\underset{n\to \infty }{\to }0,$$

$$\#\{h\in S_n\cap S_n{g}^{-1}:x_h\in c_g\}=\#\{h\in S_n\cap S_{n}g:x_h\in c_{g^{-1}}\}$$

$$\forall g\in S,\forall c\in \mathcal{C},\left|\sum _{{\tau }_i\in c_g}{x}_i^n-\sum _{{\tau }_j\in c_{g^{-1}}}{x}_j^n\right|\le \frac{\#S_{n}g△S_n}{\#S_n}+\frac{\#S_n{g}^{-1}△S_n}{\#S_n}.$$

$$\forall g\in S,\forall c\in \mathcal{C},\sum _{{\tau }_i\in c_g}{x}_i=\sum _{{\tau }_j\in c_{g^{-1}}}{x}_j,$$

$$\forall i\leq n,\left\{\begin{array}[]{l}\text{if }w_{i}=g_{j}\text{, then }\Gamma_{j}\text{ has an edge }i-1\to i;\\ \text{if }w_{i}=g_{j}^{-1}\text{, then }\Gamma_{j}\text{ has an edge }i\to i-1.\\ \end{array}\right.$$

Full text

Piantadosi [19] gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group.

We then consider two other conditions: the first, also given by Piantadosi [19], is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al. [9], is a necessary condition to decide if a set of Wang tiles gives a tiling of $\mathbb{Z}^{2}$.

We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel [14].

MCS classification : Primary 37B50; Secondary 37B10, 05B45.

Keywords : Symbolic dynamics, tilings, groups, periodicity, amenability, domino problem

1 Introduction

$\mathbb{Z}^{2}$-subshifts of finite type (SFT) are a set of colorings of the $2$-dimensional lattice $\mathbb{Z}^{2}$, or tilings, defined by a finite set of local restrictions. There are various equivalent ways to express the restrictions, such as the Wang tiles formalism introduced by Hao Wang [21]. This formalism was introduced to study the domino problem: given as input a set of restrictions (e.g. a set of Wang tiles), is there an algorithm that decides whether there is a tiling of $\mathbb{Z}^{2}$ that respects those restrictions?

R. Berger [7] showed that the domino problem is undecidable. The proof depends heavily on notions of periodicity and aperiodicity, more precisely on the existence of a set of Wang tiles that only tile $\mathbb{Z}^{2}$ in a strongly aperiodic manner. This is in stark contrast with the situation on $\mathbb{Z}$ where the domino problem is decidable thanks to a graph representation [17].

There has been a recent interest in symbolic dynamics on more general contexts, such as where the lattice $\mathbb{Z}^{2}$ is replaced by the Cayley graph of an infinite, finitely generated group. Using again the existence of strongly aperiodic SFTs, the domino problem was shown to be undecidable, apart from $\mathbb{Z}^{d}$, on some semisimple Lie groups [18], the Baumslag-Solitar groups [2], the discrete Heisenberg group (announced, [20]), surface groups [10, 1], semidirect products on $\mathbb{Z}^{2}$ [6] or some direct products [4], polycyclic groups [13], some hyperbolic groups [11]…It is decidable on free groups [19] and on virtually free groups [3], and it is conjectured that these are the only groups where the domino problem is decidable (Conjecture 7.1 below).

As a consequence, outside of free and virtually free groups, one can not expect to find simple necessary and sufficient conditions for admitting a valid tiling. However, heuristics can be very useful when making an exhaustive search for SFTs with desired properties; necessary conditions in particular allow fast rejection of most empty SFTs. For example, a transducer-based heuristic was used in the search for the smallest set of Wang tiles that yield a strongly aperiodic $\mathbb{Z}^{2}$-SFT [15]. It is also of theoretical interest to understand how the group properties impact necessary conditions.

1.1 Statements of results

We first consider a necessary and sufficient condition introduced by Piantadosi for an SFT on the free group to admit a valid tiling [19]. It is well-known that an SFT on a finitely generated group can only admit a tiling if the “corresponding” SFT on the free group does, so this becomes a necessary condition on an arbitrary f.g. group (Corollary 4.3).

The next two stronger conditions were introduced by Piantadosi (to decide if an SFT admits a strongly periodic tiling of the free group) and by Chazottes-Gambaudo-Gautero [9] in a more general context of tiling the plane by polygons, but which is necessary for an SFT to admit a tiling of $\mathbb{Z}^{2}$ [16]. We prove that the two conditions are equivalent (Theorem 3.11), and that they form a necessary condition for an SFT to admit a valid tiling on any amenable group (Theorem 5.3), confirming a remark of Jeandel ([14], Section 3.1).

Finally, we provide for any non-free finitely generated group a counterexample that satisfies all conditions but does not provide a valid tiling.

2 Preliminaries

2.1 Symbolic dynamics on groups

In the whole article $G$ is an infinite, finitely generated group with unit element $1_{G}$. We write $G=\langle\mathcal{S}\mid\mathcal{R}\rangle$ where $\mathcal{S}=\{g_{1},\dots,g_{d}\}$ is a finite set of generators and $\mathcal{R}=\{r_{1},\dots,r_{m},\dots\}\subset(S\cup S^{-1})^{\ast}$ is a (possibly infinite) set of relations. By convention $r\in\mathcal{R}$ means that $r=1_{G}$.

For instance:

• •

the free group $\mathbb{F}_{d}$ is the group on $d$ generators with no relations;

• •

$\mathbb{Z}^{2}=\langle\{g_{1},g_{2}\}\mid g_{1}g_{2}g_{1}^{-1}g_{2}^{-1}\rangle$.

Let $\mathcal{A}$ be a finite set endowed with the discrete topology; denote its cardinality $\#\mathcal{A}$. Let $\mathcal{A}^{G}=\{(x_{g})_{g\in G}\,|\,\forall g\in G:x_{g}\in\mathcal{A}\}$ be the set of all functions from $G$ to $\mathcal{A}$ endowed with the product topology. Given a finite subset $F\subset G$, an element $P\in\mathcal{A}^{F}$ is called a pattern and $F=\operatorname{supp}(P)$ its support; the set of all patterns is denoted $\mathcal{A}^{\ast}$.

$\mathcal{A}^{G}$ is a compact space called the $G$-full shift. It is a symbolic dynamical system under the following $G$-action, called the $G$-shift:

[TABLE]

We call $G$-subshift a closed shift-invariant subset $Y\subset\mathcal{A}^{G}$.

A pattern $P\in\mathcal{A}^{F}$ is said to appear in a configuration $x\in\mathcal{A}^{G}$ (and we write $P\sqsubset x$) if there exists $g\in G$ such that $\sigma_{g}(x)|_{F}=P$.

Given a set of forbidden patterns $\mathcal{F}\subset\mathcal{A}^{\ast}$, we can define the corresponding $G$-subshift:

[TABLE]

Every $G$-subshift can be defined in this way using a set of forbidden patterns. When a subshift can be defined by a finite set of forbidden patterns, we say it is a $G$-subshift of finite type ($G$-SFT). If furthermore the set of forbidden patterns can be chosen so that every pattern in $\mathcal{F}$ has support of the form $\{1_{G},g_{i}\}$ where $g_{i}\in\mathcal{S}$ for some set of generators $\mathcal{S}$, we say it is a $G$-nearest-neighbor subshift of finite type ($G$-NNSFT). Notice that this definition depends on the choice of $\mathcal{S}$ which is usually clear in the context.

For example, If we consider $G=\mathbb{Z}$ with generator $+1$, $\mathcal{A}=\{0,1\}$ and $\mathcal{F}=\{11\}$ we obtain a $\mathbb{Z}$-NNSFT, the golden mean shift, a classical example in symbolic dynamics.

Definition 2.1** (Weakly & strongly aperiodic).**

For a configuration $x\in\mathcal{A}^{G}$, we define the orbit of the element $x$ under the shift action as $\operatorname{orb}_{\sigma}(x)=\{\sigma_{g}(x)|g\in G\}$ and the set of elements on $G$ that fix the configuration $x$ by $\operatorname{stab}_{\sigma}(x)=\{g\in G|\sigma_{g}(x)=x\}$. A configuration $x\in\mathcal{A}^{G}$ is

strongly periodic

if $\operatorname{stab}_{\sigma}(x)$ has finite index or, equivalently, if $\operatorname{orb}_{\sigma}(x)$ is finite;

strongly aperiodic

if $\operatorname{stab}_{\sigma}(x)=\{1_{G}\}$.

weakly periodic

if it is not strongly aperiodic;

weakly aperiodic

if it is not strongly periodic.

More generally, a subshift $X\subset\mathcal{A}^{G}$ is weakly/strongly aperiodic if every configuration on $X$ is weakly/strongly aperiodic.

Example 2.2*.*

In $G=\mathbb{Z}^{2}$,

• •

the configuration $x$ such that $x_{g}=0$ for all $g$ is strongly periodic;

• •

the configuration $x$ such that $x_{g_{1}^{n}}=0$ for all $n$, and $x_{g}=1$ otherwise, is weakly periodic and weakly aperiodic;

• •

the configuration $x$ such that $x_{(0,0)}=0$, and $x_{g}=1$ otherwise, is strongly aperiodic.

2.2 Wang tiles, NNSFT and graphs

Definition 2.3** (Wang tiles, Wang subshifts).**

Let $G=\langle S\mid\mathcal{R}\rangle$ be a finitely generated group and $\mathcal{C}$ a finite set of colors. A Wang tile on $\mathcal{C}$ and $S$ is a map $S\cup S^{-1}\rightarrow\mathcal{C}$.

Given a set $T$ of Wang tiles, the corresponding $G$-Wang subshift is defined as:

[TABLE]

We call the elements in $X_{T}$ $G$-Wang tilings.

Notice that the definition of a Wang tile depends only on the chosen set of generators, so that the same Wang tile can be used for $\mathbb{F}_{2}$ and $\mathbb{Z}^{2}$, for example.

Take any $G$-NNSFT $X$ on alphabet $\mathcal{A}$, where $G=\langle\{g_{1},\dots,g_{d}\}\mid\mathcal{R}\rangle$ is an arbitrary finitely generated group. Let $\mathcal{F}$ be a set of forbidden patterns with each support of the form $\{1_{G},g_{i}\}$.

We associate to $X$ a set of $d$ graphs $\Gamma_{1},\dots,\Gamma_{d}$, where the set of vertices is $\mathcal{A}$ for all $\Gamma_{i}$, and

[TABLE]

By definition of a $G$-NNSFT, it follows that a configuration $x$ belongs to $X$ if, and only if, $x_{h}\to x_{hg_{i}}$ is an edge in $\Gamma_{i}$ for all $h\in G$ and all $1\leq i\leq d$.

Definition 2.4** (Cycles).**

A cycle on a graph $\Gamma$ is a path - with possible edge and vertex repetitions - that starts and ends on the same vertex. A cycle through the vertices $a_{1},\ldots,a_{n}a_{1}$, with $a_{i}\in\mathcal{A}$, is denoted $\overline{a_{1},\ldots,a_{n}}$.

A cycle is simple if it does not contain any vertex repetition. Denote $\mathcal{SC}(\Gamma)$ the set of simple cycles on $\Gamma$, which is a finite set.

Remark 2.5*.*

In graph theory, cycles are sometimes called closed walks, in which case cycle means simple cycle. We decided to follow Piantadosi’s conventions [19] for convenience.

Let $w$ be a cycle and $a\in\mathcal{A}$. We define:

[TABLE]

In any cycle, the path between the closest repetitions is a simple cycle. By removing this simple cycle and iterating the argument, we can see that any cycle $w$ can be decomposed into simple cycles, in the sense that there are integers $\lambda_{\omega}$ for $\omega\in\mathcal{SC}(\Gamma)$ such that:

[TABLE]

We say that two $G$-subshifts $X,Y\subset\mathcal{A}^{G}$ are (topologically) conjugate if there is a shift-commuting homeomorphism $\Phi$ (that is, $\Phi\circ\sigma_{g}=\sigma_{g}\circ\Phi$ for all $g\in G$) such that $\Phi(X)=Y$. An shift-commuting homeomorphism (or conjugacy) corresponds to a reversible cellular automaton: there is a finite subset $H\subset G$ and a local rule $\varphi:\mathcal{A}^{H}\to\mathcal{A}$ such that

[TABLE]

and $\Phi^{-1}$ is itself a cellular automaton.

Proposition 2.6**.**

For any set of generators, each $G$-SFT is conjugate to a $G$-NNSFT and each $G$-NNSFT is conjugate to a $G$-Wang subshift.

This is folklore. A detailed proof for the SFT - NNSFT part can be found in [5] (Propositions 1.6 and 1.7), and a proof of the NNSFT - Wang subshift part in [12].

Since the conjugacy from a $G$-Wang subshift to a $G$-NNSFT can be chosen letter-to-letter (i.e. $H=\{1_{G}\}$ in the definition), it is easy to see that the conjugacy does not depend on $G$, so we could say that a set of graphs and a set of Wang tiles are conjugate.

Proposition 2.7**.**

Let $X$ and $Y$ be two conjugate $G$-subshifts. $X$ admits a valid tiling if and only if $Y$ admits a valid tiling. The same is true for weakly/strongly (a)periodic tilings.

3 Piantadosi’s and Chazottes-Gambaudo-Gautero’s conditions

3.1 State of the art on the free group and $\mathbb{Z}^{2}$

The first two condition were introduced by Piantadosi in the context of symbolic dynamics on the free group $\mathbb{F}_{d}$.

Definition 3.1** ([19]).**

A family of graphs $\Gamma=\{\Gamma_{i}\}_{1\leq i\leq d}$ on alphabet $\mathcal{A}$ satisfies condition $(\star)$ if and only if there is some nonempty $\mathcal{A}^{\prime}\subset\mathcal{A}$ with a coloring function $\Psi:\mathcal{\mathcal{A}}^{\prime}\times\mathcal{S}\to\mathcal{\mathcal{A}}^{\prime}$ such that, for any color $a\in\mathcal{\mathcal{A}}^{\prime}$ and any generator $g_{i}\in S$, $a\to\Psi(a,g_{i})\text{ is an edge in }\Gamma_{i}$.

Theorem 3.2** ([19]).**

Let $X$ be a $\mathbb{F}_{d}$-NNSFT on alphabet $\mathcal{A}$. $X$ is nonempty if and only if the corresponding set of graphs satisfies condition $(\star)$.

This theorem provides a decision procedure for Domino problem in free groups of any rank: find a subalphabet such that every letter admits a valid neighbourg in the subalphabet for every generator.

Definition 3.3** ([19]).**

Consider a family of graphs $\Gamma=\{\Gamma_{i}\}_{1\leq i\leq d}$ and $\mathcal{SC}(\Gamma_{i})=\{\omega^{j}_{i}\}_{1\leq j\leq\#\mathcal{SC}(\Gamma_{i})}$ the set of simple cycles for each graph $\Gamma_{i}$.

We denote by $(\star\star)$ the following equation on $x_{i,j}$:

[TABLE]

We say that the graph family satisfies condition $(\star\star)$ if equation $(\star\star)$ is not empty (e.g. all graphs contain at least a cycle) and admits an nontrivial positive solution.

Remark 3.4*.*

We formulated the previous condition in terms of simple cycles (using the formalism from Theorem 3.6 instead of Theorem 3.4 in [19]) because they form a finite set, making it easier to prove formally when the condition is not satisfied.

Theorem 3.5** ([19], Theorem 3.6).**

A $\mathbb{F}_{d}$-NNSFT contains a strongly periodic configuration if and only the associated family of graphs satisfies condition $(\star\star)$.

Example 3.6*.*

We illustrate Piantadosi’s conditions on the following example:

[math]1$$2[math]1$$2$$\Gamma_{1}:$$\Gamma_{2}:

The corresponding $\mathbb{F}_{2}$-NNSFT admits a tiling, because it satisfies condition $(\star)$ on alphabet $\mathcal{A}^{\prime}=\mathcal{A}$. However, it does not admit a periodic tiling: the simple cycles of $\Gamma_{1}$ are (up to shifting) $\{\overline{012}\}$ and the simple cycles of $\Gamma_{2}$ are $\{\overline{1},\overline{2}\}$, so Equation $(\star\star)$ is:

[TABLE]

which obviously doesn’t admit a solution. As we will see later, the corresponding $\mathbb{Z}^{2}$-NNSFT doesn’t admit any tiling.

Remark 3.7*.*

For example, if all graphs $\Gamma_{i}$ share a common cycle $w$, then condition $(\star\star)$ admits a solution and therefore the corresponding $\mathbb{F}_{d}$-NNSFT contains a periodic configuration.

Definition 3.8** ([9]).**

Let $T$ be a set of Wang tiles on colors $\mathcal{C}$ and set of generators $S$. For each $g\in S\cup S^{-1}$ and each color $c\in\mathcal{C}$, define $c_{g}$ the subset of Wang tiles $\tau_{i}\in T$ such that $\tau_{i}(g)=c$. We call $(\star\star)^{\prime}$ the following equation :

[TABLE]

We say that $T$ satisfies condition $(\star\star)^{\prime}$ if Equation $(\star\star)^{\prime}$ admits a positive nontrivial solution.

Theorem 3.9** ([9]).**

If a set $T$ of Wang tiles admits a valid tiling of $\mathbb{Z}^{2}$, then it satisfies condition $(\star\star)^{\prime}$.

This condition and result was introduced in [9], but a much easier presentation in our context is given in [16].

Example 3.10*.*

Example 3.6 is conjugate to the following set of Wang tiles:

0\mapsto\tau_{0}$$1\mapsto\tau_{1}$$2\mapsto\tau_{2}$$\tau_{0}$$a$$b$$b$$a$$\tau_{1}$$b$$a$$c$$a$$\tau_{2}$$c$$b$$a$$b

Equation $(\star\star)^{\prime}$ becomes the following, where next to each equation is the corresponding generator and color:

[TABLE]

This equation does not admit a positive nontrivial solution, so the corresponding $\mathbb{Z}^{2}$-Wang subshift is empty.

3.2 Conditions $(\star\star)$ and $(\star\star)^{\prime}$ are equivalent

Although conditions $(\star\star)$ and $(\star\star)^{\prime}$ were introduced in very different contexts (periodic tilings of the free group and tilings of the Euclidean plane, respectively), it turns out that they are equivalent. The fact that $(\star\star)$ is a condition on graphs (NNSFTs) and $(\star\star)^{\prime}$ is a condition on sets of Wang tiles (Wang subshifts) is only cosmetic since Proposition 2.6 lets us go from one model to the other.

Theorem 3.11**.**

Let T be a set of Wang tiles over the set of colors $\mathcal{C}$ and the set of generators $S$.

$T$* satisfies condition $(\star\star)^{\prime}$ if, and only if, the associated graphs satisfy condition $(\star\star)$.*

Proof.

$(\Leftarrow)$ Let $(x_{i,j})$ be a nonnegative solution to equation $(\star\star)$. For every tile $\tau_{i}$, put $x_{i}=\sum_{j=1}^{\#\mathcal{SC}(\Gamma_{1})}x_{1,j}|\omega_{1}^{j}|_{\tau_{i}}$.

Because each simple cycle of $\Gamma_{1}$ is a cycle, it contains as many tiles in $c_{g_{1}}$ as in $c_{g_{1}^{-1}}$; that is, $\sum_{\tau_{i}\in c_{g_{1}}}|\omega_{1}^{j}|_{\tau_{i}}=\sum_{\tau_{j}\in c_{g_{1}^{-1}}}|\omega_{1}^{j}|_{\tau_{i}}$. Summing over all simple cycles $\omega_{1}^{j}$, we get $\sum_{\tau_{i}\in c_{g_{1}}}x_{i}=\sum_{\tau_{i}\in c_{g_{1}^{-1}}}x_{j}$.

Since $(x_{i,j})$ is a solution to Equation $(\star\star)$, we also have $x_{i}=\sum_{j=1}^{\#\mathcal{SC}(\Gamma_{n})}x_{n,j}|\omega_{n}^{j}|_{\tau_{i}}$ for every $n$, so the same argument shows that $(x_{i})$ is a nonnegative solution of equation $(\star\star)^{\prime}$.

$(\Rightarrow)$ Because equation $(\star\star)^{\prime}$ admits a solution, it admits a rational solution, and therefore an integer solution. Let $(x_{i})$ be an integer, nonnegative solution of equation $(\star\star)^{\prime}$.

For the generator $g_{1}$, consider the graph $\Gamma_{1}$ obtained by the letter-to-letter conjugacy of Proposition 2.6: concretely, it is the graph on vertices $\{\tau_{i}\}_{1\leq i\leq n}$ with $\tau_{i}\to\tau_{j}\Leftrightarrow\exists c\in\mathcal{C},\tau_{i}\in c_{g_{1}}\text{ and }\tau_{j}\in c_{g_{1}^{-1}}$.

We define an auxiliary graph $G_{1}$ on vertices $\{\tau_{i}^{k}\}_{1\leq i\leq n,1\leq k\leq x_{i}}$ (that is, $x_{i}$ copies for each tile $\tau_{i}$) as follows.

Because

[TABLE]

we can fix an arbitrary bijection :

[TABLE]

and put an edge $\tau_{i}^{k}\to\tau_{i^{\prime}}^{k^{\prime}}$ if and only if $\Psi_{1}^{c}(\tau_{i}^{k})=\tau_{i^{\prime}}^{k^{\prime}}$ for some $c\in\mathcal{C}$. Because each vertex has indegree and outdegree 1, it is a (not necessarily connected) Eulerian graph and admits a finite set of cycles covering every vertex exactly once.

Notice that by construction, if $G_{1}$ has an edge $\tau_{i}^{k}\to\tau_{i^{\prime}}^{k^{\prime}}$, then $\Gamma_{1}$ has an edge $\tau_{i}\to\tau_{i^{\prime}}$. Therefore each cycle of $G_{1}$ can be sent on a cycle in $\Gamma_{1}$ though the projection $\tau_{i}^{k}\mapsto\tau_{i}$. In this way, project the finite set of cycles obtained above and decompose them into simple cycles of $\Gamma_{1}$. Denote $x_{1,j}$ the total number of each simple cycle $\omega_{1}^{j}$ obtained in this way.

Because each tile $\tau_{i}$ was present in $G_{1}$ as a vertex in $x_{i}$ copies, we have for every $i$: $\sum_{j=1}^{\#\mathcal{SC}(\Gamma_{1})}x_{1,j}|\omega_{1}^{j}|_{\tau_{i}}=x_{i}$.

Now apply the same argument for each generator $g_{2}\dots g_{n}$ and the variables $(x_{i,j})$ thus obtained are a solution to equation $(\star\star)$. ∎

4 Necessary conditions for tiling arbitrary groups

Since the above conditions apply on sets of Wang tiles or set of graphs, they actually are conditions on a family of $G$-SFT where $G$ range over all groups with a fixed number of generators. The following proposition relates the properties of these SFT. It can be found (under a different form) in [8] (Proposition 10 and remark below)

Proposition 4.1**.**

Let $G_{1}=\langle\{g_{1}\dots g_{d}\}|\mathcal{R}\rangle$, $G_{2}=\langle\{g_{1}\dots g_{d}\}|\mathcal{R}^{\prime}\rangle$ be finitely generated groups, with $\mathcal{R}^{\prime}\subset\mathcal{R}$. Consider the canonical surjective morphism $\pi:G_{2}\rightarrow G_{1}$ defined by $\pi(g_{i})=g_{i}$, $\forall 1\leq i\leq d$. Let $\Phi:\mathcal{A}^{G_{1}}\to\mathcal{A}^{G_{2}}$ be defined by $\Phi(x)_{g}=x_{\pi(g)}$. Let $X_{1}$ and $X_{2}$ be the corresponding $G_{1}$-NNSFT and $G_{2}$-NNSFT respectively, such that $X_{2}$ has the same local rules that $X_{1}$.

We have:

If $x$ is a valid tiling for $X_{1}$ then $\Phi(x)$ is a valid tiling for $X_{2}$. 2. 2.

If $x$ is weakly periodic then $\Phi(x)$ is weakly periodic. In particular, if $X_{1}$ admits a weakly periodic tiling, then $X_{2}$ admits a weakly periodic tiling. 3. 3.

If $x$ is weakly aperiodic then $\Phi(x)$ is weakly aperiodic. In particular, if $X_{1}$ admits a weakly aperiodic tiling, then $X_{2}$ admits a weakly aperiodic tiling.

The strong properties are not preserved by $\Phi$, but of course the image of a strongly (a)periodic tiling remains weakly (a)periodic. Stronger versions with different hypotheses can be found in [8, 14].

Proof.

Since $X_{2}$ is an NNSFT, it is enough to check that, for all $h\in G_{2}$ and all $1\leq i\leq d$, $\Phi(x)_{h}\to\Phi(x)_{hg_{i}}$ is an edge in $\Gamma_{i}$, that is to say, that it is not a forbidden pattern for $X_{2}$. By definition of $\Phi$, $\Phi(x)_{h}=x_{\pi(h)}$ and $\Phi(x)_{hg_{i}}=x_{\pi(h)\pi(g_{i})}=x_{\pi(h)g_{i}}$. Because $x$ is a valid tiling for $X_{1}$, we have that $x_{\pi(h)}\to x_{\pi(h)g_{i}}$ is an edge in $\Gamma_{i}$, which proves the result. 2. 2.

If $x$ is a weakly periodic tiling in $X_{1}$, then $\operatorname{stab}_{\sigma}(x)$ is nontrivial by definition. We have:

[TABLE]

Since $\pi$ is surjective, this means that $\pi(\operatorname{stab}_{\sigma}(\Phi(x)))=\operatorname{stab}_{\sigma}(x)$. $\operatorname{stab}_{\sigma}(x)$ is nontrivial so $\operatorname{stab}_{\sigma}(\Phi(x))=\pi^{-1}(\operatorname{stab}_{\sigma}(x))$ is nontrivial as well. 3. 3.

If $x$ is a weakly aperiodic tiling in $X_{1}$, then $\operatorname{stab}_{\sigma}(x)$ does not have finite index. The canonical morphism $\pi:G_{2}\to G_{1}$ yields a morphism on the quotient:

[TABLE]

and $\tilde{\pi}$ is surjective since $\pi$ is surjective. Remember that $\operatorname{stab}_{\sigma}(\Phi(x))=\pi^{-1}(\operatorname{stab}_{\sigma}(x))$ by the previous point. Since $\operatorname{stab}_{\sigma}(x)$ does not have finite index, $G_{1}/\operatorname{stab}_{\sigma}(x)$ is infinite, so $G_{2}/\pi^{-1}(\operatorname{stab}_{\sigma}(x))$ is infinite as well, and $\operatorname{stab}_{\sigma}(\Phi(x))=\pi^{-1}(\operatorname{stab}_{\sigma}(x))$ does not have finite index.

∎

Remark 4.2*.*

In the last proposition, the converse of the point (1) does not hold. For instance, if consider $G=\mathbb{Z}^{2}=\langle g_{1},g_{2}\mid g_{1}g_{2}g_{1}^{-1}g_{2}^{-1}\rangle$. Example 3.6 provided an example of a set of graphs that satisfied condition $(\star)$ (so the corresponding $\mathbb{F}_{2}$-NNSFT admits a valid tiling) but does not satisfies the conditions $(\star\star)$ (so the corresponding $\mathbb{Z}^{2}$-NNSFT does not admit any valid tiling).

To understand why, notice that $ker(\pi)$ contains $g_{1}g_{2}g_{1}^{-1}g_{2}^{-1}$, so if a tiling $x\in\mathcal{A}^{\mathbb{F}_{2}}$ is such that $x_{1_{\mathbb{F}_{2}}}\neq x_{g_{1}g_{2}g_{1}^{-1}g_{2}^{-1}}$, then $\Phi^{-1}(x)=\emptyset$. If this happens for all $x\in X_{2}$ then $X_{1}$ is empty.

Corollary 4.3**.**

Let $\Gamma_{1},\dots,\Gamma_{d}$ be a set of graphs that does not satisfy the condition $(\star)$. Then the corresponding $G$-NNSFT is empty for an arbitrary group $G$ with $d$ generators.

Proof.

If there was a valid tiling in $G=\langle g_{1},\dots,g_{d}\mid\mathcal{R}\rangle$ then, applying Proposition 4.1, we would obtain a tiling on $\mathbb{F}_{d}=\langle g_{1},\dots,g_{d}\mid\emptyset\rangle$, which is in contradiction with Theorem 3.2. ∎

5 Necessary conditions for tiling amenable groups

Definition 5.1** (Følner sequence).**

Let $G$ be a finitely generated group. A Følner sequence for $G$ is a sequence of finite subsets $S_{n}\subset G$ such that: F

[TABLE]

where $S_{n}g=\{hg\ :\ h\in S_{n}\}$ and $A\triangle B=(A\backslash B)\cup(B\backslash A)$ is the symmetric difference.

In the previous definition, it is easy to see that the second condition only has to be checked for $g$ in a finite generating set. The set $S_{n}g\triangle S_{n}$ can be understood as the border of $S_{n}$, so an element of a Følner sequence must have a small border relative to its interior.

Definition 5.2** (Amenable group).**

A finitely generated group $G$ is amenable if it admits a Følner sequence.

A few examples :

• •

$\mathbb{Z}^{d}$ is amenable and a Følner sequence is given by $S_{n}=[-n,n]^{d}$. Indeed, if $(g_{i})_{1\leq i\leq d}$ is the canonical set of generators, then $\#S_{n}=(2n+1)^{d}$ and $\#(S_{n}+g_{i})\triangle S_{n}=2\cdot(2n+1)^{d-1}$.

• •

$\mathbb{F}_{d}$ for $d\geq 2$ is not amenable. In particular, the balls $S_{n}$ of radius $n$ - that is, reduced333with no $g_{i}^{-1}g_{i}$ or $g_{i}g_{i}^{-1}$ factors words of length $\leq n$ on the set of generators $(g_{i})_{1\leq i\leq d}$ - are not a Følner sequence. Indeed, one can easily check that $\#S_{n}=\Omega(d^{n})$ and $\#(S_{n}g_{i}\triangle S_{n})=\Omega(d^{n})$.

The following theorem was conjectured in [14], Section 3.1.

Theorem 5.3** (Heuristic for tiling an amenable group).**

Let $G$ be a finitely generated amenable group, $S$ a finite set of generators, and $T$ a set of Wang tiles.

If there is a tiling of $G$ with the tiles $T$, then condition $(\star\star)$ (or equivalently $(\star\star)^{\prime}$) is satisfied.

Proof.

Let $x\in T^{G}$ be a tiling of $G$ and $S_{n}$ be a Følner sequence for $G$. Using notations from Definition 3.8, for a color $c\in\mathcal{C}$ and a generator $g\in S$, $c_{g}$ is the set of tiles $\tau$ such that $\tau(g)=c$.

For any $h\in S_{n}\cap S_{n}g^{-1}$, we have $x_{h}\in c_{g}\Leftrightarrow x_{hg}\in c_{g^{-1}}$ (and in this case, $hg\in S_{n}\cap S_{n}g$). This means that, for all $c\in\mathcal{C},g\in S$ and $n\in\mathbb{N}$:

[TABLE]

so in particular $|\#\{h\in S_{n}\ :\ x_{h}\in c_{g}\}-\#\{h\in S_{n}\ :\ x_{h}\in c_{g^{-1}}\}|\leq\#S_{n}g\triangle S_{n}+\#S_{n}g^{-1}\triangle S_{n}$.

For each tile $\tau_{i}$, let $x^{n}_{i}=\frac{\#\{h\in S_{n}\ :\ x_{h}=\tau_{i}\}}{\#S_{n}}$. The previous computation implies that:

[TABLE]

Notice that the right-hand side tends to [math] as $n$ tends to infinity by definition of a Følner sequence. Consider the sequence of vectors $((x_{i}^{n})_{i})_{n\in\mathbb{N}}$ and, by compacity, let $(x_{i})$ be any limit point of this sequence. Since $\sum_{i}x^{n}_{i}=1$ for all $n$ by definition, $\sum_{i}x_{i}=1$ as well, and we have

[TABLE]

so $(x_{i})$ is a nontrivial solution to Equation $(\star\star)$. Condition $(\star\star)^{\prime}$ follows by Theorem 3.11.∎

6 Counterexamples

It is clear that none of the $(\star)$, $(\star\star)$ or $(\star\star)^{\prime}$ conditions can be a sufficient condition to admit a $\mathbb{Z}^{d}$-tiling, since it would be a decision procedure for the Domino problem; this argument applies to any group where the Domino problem is undecidable. For completeness, we provide explicit counterexamples for any non-free finitely generated group.

Theorem 6.1**.**

Let $G$ be an arbitrary finitely generated group. If $G$ is not free, then there exists a Wang tile set that satisfies the three conditions $(\star)$, $(\star\star)$ and $(\star\star)^{\prime}$ and such that the corresponding $G$-Wang subshift is empty.

Proof.

Write $G=\langle g_{1}\dots g_{d}\mid\mathcal{R}\rangle$, and take $r_{1}:w_{1}\dots w_{n}\in\mathcal{R}$, with $w_{1}\dots w_{n}$ a reduced word on generators $g_{1}\dots g_{d}$ (no generator is next to its inverse).

We build a family of graphs $\Gamma_{d}$ on vertices $\{0,\dots,n\}$ with the following edges:

[TABLE]

Notice that every vertex has indegree and outdegree at most $1$ and we did not create any cycle in the process, so we can complete every $\Gamma_{j}$ to be isomorphic to a $n$-cycle graph $C_{n}$.

Now we define a set of $n+1$ Wang tiles on $n+1$ colors $\{0\dots n\}$ as follows. Tile $\tau_{i}$ has the following colors: for all $j$, $g_{j}^{-1}\to i$ and $g_{j}\to k$ if there is an edge $\tau_{i}\to\tau_{k}$ in $\Gamma_{j}$.

Example 6.2*.*

For $\mathbb{Z}^{2}$, we have $r_{1}:g_{1}g_{2}g_{1}^{-1}g_{2}^{-1}=1$. Therefore $\Gamma_{1}$ contains $0\to 1$ and $3\to 2$, and $\Gamma_{2}$ contains $1\to 2$ and $4\to 3$. One possible completion for $\Gamma_{1}$ and $\Gamma_{2}$ is the following:

\tau_{0}$$\tau_{1}$$\tau_{4}$$\tau_{3}$$\tau_{2}$$\Gamma_{1}:$$\tau_{0}$$\tau_{1}$$\tau_{2}$$\tau_{4}$$\tau_{3}$$\Gamma_{2}:

and the corresponding set of Wang tiles:

$\tau_{0}$[math]1$$1[math]\tau_{1}$$1$$4$$2$$1$$\tau_{2}$$2[math]4$$2$$\tau_{3}$$3$$2[math]3$$\tau_{4}$$4$$3$$3$$4

This tiling satisfies condition $(\star\star)^{\prime}$ since we can assign the same weight $\frac{1}{n}$ to each tile.

It is clear that a tiling $x$ of $G$ using tiles $\tau_{0},\dots\tau_{n}$ must contain every tile. Assume w.l.o.g that $x_{1}=\tau_{0}$. By construction we must have $x_{w_{1}}=\tau_{1}$, $x_{w_{1}w_{2}}=\tau_{2}$, and by an easy induction $x_{w}=\tau_{n}$. But since $w=1$ in $G$, we have $\tau_{0}=x_{1}=x_{w}=\tau_{n}$, a contradiction. Therefore there is no tiling of $G$ using tiles $\tau_{0},\dots\tau_{n}$. ∎

7 Conclusion

We would like to mention the two following conjectures that relate the fact of admitting a valid (periodic) tiling and the underlying group structure:

Conjecture 7.1** ([3]).**

A finitely generated group has a decidable domino problem if and only if it is virtually free.

Conjecture 7.2** ([8]).**

A finitely generated group has an SFT with no strongly periodic point if and only if it is not virtually cyclic.

In both cases, the “if” direction is proven and the “only if” direction is open.

Acknowledgements

The first author would like to thank Pascal Vanier and Emmanuel Jeandel for providing access to a preprint of [16] and help in understanding [9]. We are grateful to an anonymous referee for many helpful remarks on a previous version.