3.1 Definition

For the purposes of this section, we define a graph $\Gamma $ as a tuple $(V_{\Gamma }, E_{\Gamma })$ , where $V_{\Gamma }$ is the set of vertices and $E_{\Gamma }\subseteq V_{\Gamma }^2$ is a set of edges, such that the graph is locally finite. We also associate the graph with two functions $\mathfrak {i},\mathfrak {t}:E_{\Gamma }\to V_{\Gamma }$ that give the initial and terminal vertex of an edge, respectively. Given an edge $e\in E_{\Gamma }$ , we denote by $\bar {e}$ the edge pointing in the opposite direction to e, that is, $\mathfrak {t}(\bar {e}) = \mathfrak {i}(e)$ and $\mathfrak {i}(\bar {e})=\mathfrak {t}(e)$ .

The main interest of this object is its fundamental group. As its name suggests, this group is obtained through a precise definition of paths on the graph of groups. Luckily there is an explicit expression for the fundamental group, which allows us to skip the formal definition. A complete treatment of the concept can be found in [Reference Lyman42, Reference Serre49].

Theorem 3.2. Let $T\subseteq \Gamma $ be a spanning tree. The group $\pi _1(\Gamma , \mathcal {G}, T)$ is isomorphic to a quotient of the free product of the vertex groups, with the free group on the set $E_{\Gamma }$ of oriented edges. That is,

where R is the normal closure of the subgroup generated by the following relations:

• • $\alpha _{\bar {e}}(h)e = e\alpha _{e}(h)$ , where e is an oriented edge of $\Gamma $ , $h\in G_{e}$ ;

• • $\bar {e} = e^{-1}$ , where e is an oriented edge of $E_{\Gamma }$ ;

• • $e = 1$ if e is an oriented edge of T.

We will omit $\mathcal {G}$ as the context allows it. Furthermore, the fundamental group does not depend on the spanning tree, up to isomorphism.

Let us look at how traditional operations of geometric group theory are viewed as fundamental groups of graph of groups. The amalgamated free product $G\ast _{K}H$ is viewed as the fundamental group $\pi _1(\Gamma _1)$ :

Similarly, an HNN-extension $G\ast _{\phi }$ can be seen as the fundamental group $\pi _1(\Gamma _2)$ :

with H a subgroup of G, $\alpha _e = \text {id}$ and $\alpha _{\bar {e}} = \phi :H\to \phi (H)$ an isomorphism. In this sense, the concept of graph of groups can be seen as the natural generalization of these concepts.

This article is primarily concerned with a specific class of graph of groups.

As their name suggest, GBS groups were introduced as a generalization of Baumslag–Solitar groups. This class also contains all torus knot groups as well as $\mathbb {Z}$ , $\mathbb {Z}^2$ and the fundamental group of the Klein bottle. For an extensive introduction to this class, see [Reference Forester26, Reference Forester27, Reference Levitt40].

3.2 Torus knot groups

The $(n,m)$ -torus knot group is given by the presentation

$$ \begin{align*}\Lambda(n,m) = \langle a,b \ | \ a^nb^{-m}\rangle.\end{align*} $$

As mentioned above, they are a particular case of the generalized Baumslag–Solitar groups given by the amalgamated free product $\mathbb {Z}\ast _{\mathbb {Z}}\mathbb {Z}$ , along with the inclusions $1\mapsto n$ and $1 \mapsto m$ . Their name comes from the fact that they are the knot groups of torus knots [Reference Rolfsen47].

Remark that $\Lambda (n,m)\simeq \Lambda (m,n)$ . Also, if n or m is equal to one, then $\Lambda (n,m) \simeq \mathbb {Z}$ and the group is therefore abelian. In fact, these are the only cases where the group is amenable as a consequence of the following proposition.

This fact is deduced from the short exact sequence

$$ \begin{align*}1 \to \mathbb{Z} \to \Lambda(n,m) \to \mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}\to 1.\end{align*} $$

As we will later see, these groups are part of a larger subclass of GBS groups having this property.

This is a consequence of the fact that these groups contain isomorphic copies of $\mathbb {Z}^2$ , namely $\langle a^n, ba\rangle $ .

3.3 Baumslag–Solitar groups

Baumslag–Solitar groups were introduced, as we now know them, in [Reference Baumslag and Solitar12] to provide examples of non-Hopfian groups, although cases of them were defined some years prior by Higman in [Reference Higman31]. They are simple cases of HNN-extensions, and have provided discriminating examples in both combinatorial and geometric group theories. Concerning the study of SFTs on groups, they are also of particular interest since as one-relator groups, they have decidable word problem. Since they are also 1-ended, they fall under the scope of Conjecture 1.

Baumslag–Solitar groups are defined by the presentation:

$$ \begin{align*}BS(m,n) = \langle a, t \mid t^{-1}a^m t = a^n \rangle.\end{align*} $$

The first things to note are that $BS(1,1) = \mathbb {Z}^2$ and $BS(m,n) \simeq BS(-m,-n)$ . Baumslag–Solitar groups may behave radically differently: the groups $BS(1,n)$ are solvable and amenable while the $BS(m,n)$ groups with $m,n>1$ contain free subgroups and are consequently non-amenable. This dichotomy is also present in the classification of Baumslag–Solitar groups up to quasi-isometry. On the one hand, groups $BS(1,n)$ and $BS(1,n')$ are quasi-isometric if and only if n and $n'$ have a common power [Reference Farb and Mosher25], and in this case, the two groups are even commensurable. On the other hand, groups $BS(m,n)$ and $BS(m',n')$ are quasi-isometric as soon as $2\leq m < n$ and $2\leq m'<n'$ [Reference Whyte50].

3.4 Weakly aperiodic SFTs on groups generated from graph structures

One case that does not fall within the hypothesis of Proposition 3.6 is when all vertices of the graph have $\mathbb {Z}$ as their vertex group, which is known not to admit any weakly aperiodic SFT. However, a careful study shows that in this case, weakly aperiodic SFT can nevertheless be constructed unless the group is $\mathbb {Z}$ itself.

Proof. Let G be a GBS group with its corresponding graph of groups $\Gamma $ . Because G is not $\mathbb {Z}$ , at least one edge, $e\in E_{\Gamma }$ , satisfies $\alpha _{e}\not \equiv \pm 1$ . If this edge is a loop, from the previous remarks, we know that G contains a non- $\mathbb {Z}$ Baumslag–Solitar group. These groups are known to admit weakly aperiodic SFT [Reference Aubrun and Kari4], so we are done. Similarly, if the edge is in the spanning tree $T\subseteq \Gamma $ such that $G = \pi _{1}(\Gamma , T)$ , then G contains a knot group $\Lambda (n,m)$ , which admits a weakly aperiodic SFT by virtue of containing $\mathbb {Z}^2$ as a subgroup.

The last case is when all edges in the spanning tree satisfy $\alpha _{e'}\equiv \pm 1$ and there are no loops. Let e be an edge such that $\alpha _{e},\alpha _{\bar {e}}\neq \pm 1$ , and $v,u$ its end points. Because T is spanning, we know that $v,u\in V_{T}$ , and therefore if $G_v = \langle a \rangle $ and $G_u = \langle b \rangle $ , we have that in G, $a = b^{\pm 1}$ . Then, the relation given by the edge e is

$$ \begin{align*}a^{\alpha_{e}(1)}e = eb^{\alpha_{\bar{e}}(1)} \! \iff a^{\alpha_{e}(1)}e = ea^{\pm\alpha_{\bar{e}}(1)}.\end{align*} $$

This means G contains the non- $\mathbb {Z}$ Baumslag–Solitar group $BS(\alpha _{e}(1), \pm \alpha _{\bar {e}}(1))$ , which, as mentioned before, admits a weakly aperiodic SFT.

The same proof scheme can be used for the class of Artin groups, which are another example of groups generated from an underlying graph structure.

Let $\Gamma = (V, E, \unicode{x3bb} )$ be a labelled graph with labels $\unicode{x3bb} : E\to \{2,3, \ldots \}$ . We define the Artin group of $\Gamma $ through the following presentation:

$$ \begin{align*} A(\Gamma) := \langle V \ | \ \underbrace{abab...}_{\unicode{x3bb}(e)} = \underbrace{baba...}_{\unicode{x3bb}(e)} \ \text{ for all } e = (a,b)\in E\rangle. \end{align*} $$

Let $\Gamma _n$ be the graph of two vertices a and b and the edge connecting them labelled by n. Artin groups defined as $A(\Gamma _n)$ are known as dihedral. Notice that $A(\Gamma _2) \simeq \mathbb {Z}^2$ . Furthermore, for $n\geq 3$ , $A(\Gamma _n)$ is virtually $\mathbb {F}_m\times \mathbb {Z}$ for some $m\geq 2$ (see [Reference Crisp20]). This fact also follows from the proof of the next proposition.

4.1 State of the art

The existence of an aperiodic SFT is a geometric property of groups, at least for finitely presented ones, as stated below.

The hypothesis of finite presentation is essential here. For example, from [Reference Barbieri8], we know that the Grigorchuk group, which is finitely generated but not finitely presented, admits a strongly aperiodic SFT. Nevertheless, the Grigorchuk group has uncountably many groups which are quasi-isometric to it. This means that at least one of them has undecidable word problem and, therefore, does not admit a strongly aperiodic SFT by [Reference Jeandel32]. This result can also be seen as a consequence of the fact that being recursively presented is not a quasi-isometry invariant.

However, we do have results when two finitely generated groups are commensurable. Two groups are said to be commensurable if they have finite index subgroups which are isomorphic.

Commensurability implies quasi-isometry, but there exists many examples where the converse does not hold. For instance, the groups $BS(m,n)$ and $BS(p,q)$ are quasi-isometric whenever $1<n<m$ and $1<p<q$ , but are not commensurable if $(m,n)$ are co-prime and $(p,q)$ are co-prime [Reference Casals-Ruiz, Kazachkov and Zakharov15, Reference Whyte50].

4.2 Lifting minimal aperiodic subshifts

The goal of this section is to lift a minimal strongly aperiodic SFT from a normal subgroup of finite index to the whole group. To do this, we make use of the locked shift, as introduced by Carroll and Penland [Reference Carroll and Penland14].

Let A be a finite alphabet, G a finitely generated group with N a finitely generated normal subgroup. We define the subshift

$$ \begin{align*}\text{Fix}_A(N) = \{x\in A^{G}\mid \sigma^n(x) = x \text{ for all } n\in N\}.\end{align*} $$

Definition 4.1. For a finite index normal subgroup N, we define the N-locked subshift L as $\textrm {Fix}_{R}(N)\cap \Sigma $ , where R is a set of right coset representatives with $1\in R$ , and $\Sigma $ is the subshift defined by the the finite set of forbidden patterns

$$ \begin{align*}\{p:\{1,r\}\to R \mid r\in R\setminus\{1\}, \ p(1) = p(r) \}.\end{align*} $$

Proof. If we take $S = \{s_1, \ldots , s_m\}$ as a set of symmetric generators for N, we see that $\text {Fix}_R(N)$ is an SFT by the set of forbidden rules given by

$$ \begin{align*}\{p:\{1,s_i\}\to R \mid s_i\in S, \ p(1)\neq p(s_i)\}.\end{align*} $$

Therefore, L is also an SFT. To see it is non-empty, we define $y\in R^{G}$ by $y_{nr} = r$ . If we take $n'\in N$ ,

$$ \begin{align*}\sigma^{n'}(y)_{nr} = y_{n^{\prime -1}n r} = r = y_{nr}.\end{align*} $$

Thus, $y\in \text {Fix}_R(N)$ . Next, we take $r'\in R$ and see that

$$ \begin{align*}y_{nr} = \sigma^{n^{-1}}(y)_r = y_r \neq y_{rr'} = \sigma^{n^{-1}}(y)_{rr'} = y_{nrr'}.\end{align*} $$

This way, $y\in \Sigma $ , and therefore $y\in L$ . Finally, if we take $x\in L$ and $g = nr\in G$ such that $\sigma ^{g}(x)=x$ , then

$$ \begin{align*}x = \sigma^{nr}(x) = \sigma^{r(r^{-1}nr)}(x) = \sigma^{r}(x).\end{align*} $$

With this, $x_1 = x_{r^{-1}r} = \sigma ^r(x)_r = x_r$ . Because $x\in \Sigma $ , $r=1$ and thus $g = n\in N$ .

We now have all the ingredients to prove the result.

Proof. Let $X\subseteq A^{H}$ be a minimal strongly aperiodic SFT over H. Given a set R of right coset representatives with $1\in T$ , we define the G-subshifts

$$ \begin{align*}\hat{X} = \{y\in A^G\mid \text{ there exists } x\in X \text{ for all }(h,r)\in H\times R, \ y_{hr} = x_h \},\end{align*} $$

and $Y = \hat {X}\times L$ , where L is the H-locked shift. Let us see that Y is the subshift for which we are looking.

• • Aperiodicity. Suppose there is a $y\in Y$ and $g\in G$ such that $\sigma ^{g}(y) = y$ . Due to Lemma 4.4, we know that $g\in H$ . Then, for some $r\in R$ and $x\in X$ ,

  $$ \begin{align*}x_{h} = y_{hr} = y_{g^{-1}hr} = x_{g^{-1}h},\end{align*} $$
  that is, $x = \sigma ^{g}(x)$ which contradicts the aperiodicity of X.

• • Minimality. Let us take $y, y'\in Y$ along with $x,x'\in X$ their corresponding X configurations. By the minimality of X, there exists a sequence $\{h_n\}_{n\in \mathbb {N}}\subseteq H$ such that $\sigma ^{h_n}(x)\to x'$ . Then, for $(h,r)\in H\times R$ ,

  $$ \begin{align*}\sigma^{h_n}(y)_{hr} = y_{h_n^{-1}hr} = x_{h_n^{-1}h} = \sigma^{h_n}(x)_h \to x^{\prime}_{h} = y^{\prime}_{hr}.\end{align*} $$
  Thus, $\sigma ^{h_n}(y)\to y'$ .

4.3 Strong aperiodicity for GBS

The quasi-isometric structure of GBS groups is well understood: they are classified according to the following result.

GBS groups that fall into the first category are called unimodular. In [Reference Jeandel32], Jeandel shows that $\mathbb {F}_2\times \mathbb {Z}$ admits a strongly aperiodic SFT through the use of Kari-like tiles. Nevertheless, this construction is not minimal and is only valid on $\mathbb {F}_2\times \mathbb {Z}$ and not all $\mathbb {F}_n\times \mathbb {Z}$ for $n\geq 3$ . As explained in §4.2, for a group G, containing a finite index subgroup that admits a minimal strongly aperiodic SFT is not enough to get a minimal strongly aperiodic SFT on G. Fortunately, the statement of Theorem 4.6 can be sharpened, as stated in the following remark.

In §6, we provide an example of a minimal strongly aperiodic SFT on $\mathbb {F}_n\times \mathbb {Z}$ for all $n\geq 2$ . The case of groups quasi-isometric to $BS(2,3)$ is also treated in this paper: in §7, we explain how to construct a strongly aperiodic SFT on $BS(2,3)$ . Groups $G = BS(1,n)$ for some $n>1$ are already known to possess a minimal strongly aperiodic SFT [Reference Aubrun and Schraudner7]. In total, we are able to construct strongly aperiodic SFTs for all GBS.

6.1 The flow SFT

Let us begin by introducing the flow shift over $\mathbb {F}_n\times \mathbb {Z}$ , which we will denote as $Y_{f}$ . We define this shift from tiles representing different directions. Let $S = \{s_1, \ldots , s_{n}\}$ be a set of generators of $\mathbb {F}_n$ and $\mathbb {Z} = \langle t\rangle $ . We will define the flow shift over the alphabet . We can interpret these tiles as pointing in the direction specified by a generator or its inverse.

To define $Y_{f}$ , we demand that a configuration $y\in A^{\mathbb {F}_n\times \mathbb {Z}}$ satisfies

$$ \begin{align*}y_{g} = s \! \implies\!\! \begin{cases}y_{gs}\neq s^{-1} \\ y_{gs'} = s^{\prime -1} \quad \text{for all } s'\in A\setminus\{s\}.\\ y_{gt} = s \end{cases} \end{align*} $$

Notice that fixing a tile at the identity completely determines the tiling of the $2n-1$ subtrees of $\mathbb {F}_n$ the tile does not point towards. Then, this leaves $2n-1$ possible tiles for the unspecified neighbour. In addition, the last rule makes sure that each $\mathbb {Z}$ -coset contains the same tile.

This allows us to describe each configuration with an infinite word W (see Figure 3). Given $y\in Y_f$ , we recursively define $W(y)\in A^{\mathbb {N}}$ by setting and setting .

Figure 3 If we look at the configuration in $\mathbb {F}_2 = \langle a, b\rangle $ as shown, the prefix of its infinite word is given by $bab$ .

Due to the local rules, this correspondence between configurations and infinite words effectively creates a bijection W between $Y_{f}$ and $\partial _{\infty }\mathbb {F}_{n}$ , the boundary of $\mathbb {F}_n$ .

6.2 The structure of a path-folding SFT

Let $X\subseteq B^{\mathbb {Z}^{2}}$ be an horizontally expansive, strongly aperiodic SFT on $\mathbb {Z}^2$ ; for instance, the SFT detailed in Proposition 5.5. Without loss of generality, we assume X to be a nearest neighbour SFT. We want to ‘fold’ each configuration along the path defined by the infinite word of a configuration in $Y_{f}$ . Let Z be the subshift of $B^{\mathbb {F}_n\times \mathbb {Z}}\times Y_f$ , given by the following set of allowed patterns.

• • For each valid pattern H of support $\{(0,0), (1,0)\}$ in X, we define the pattern P of support $\{1, t\}$ by

  $$ \begin{align*}P(1) = (H_{(0,0)}, d), \quad P(t) = (H_{(1,0)}, d),\end{align*} $$
  where $d\in A$ .

• • For each valid pattern V of support $\{(0,0), (0,1)\}$ in X, we define the patterns Q of support $\{1, s\}$ by

  $$ \begin{align*}Q(1) = (V_{(0,0)}, s), \quad Q(s) = (V_{(1,0)}, s'),\end{align*} $$
  where $s'\in A\setminus \{s^{-1}\}$ .

Proposition 6.2. The configurations in Z have the following structure:

$$ \begin{align*}x\otimes y: wt^{i} \to (x_{(i,j)}, \ y_w),\end{align*} $$

where $w\in \mathbb {F}_n$ , $x\in X$ , $y\in Y_{f}$ defined by the word W, with

$$ \begin{align*}j = 2\max\{|w'| : w'\sqsubseteq w \wedge w'\sqsubseteq W\} - |w|.\end{align*} $$

Proof. Let us have $y\in Y_f$ and $x\in X$ . We begin by showing that $x\otimes y\in Z$ . We know the second coordinate satisfies the allowed patterns by definition, so we must look at the first.

Let $g = wt^{i}\in \mathbb {F}_n\times \mathbb {Z}$ , then

$$ \begin{align*}x\otimes y(g) = (x_{(i,j)}, \ y_g),\end{align*} $$

as in the definition. We begin by looking at the support $\{1, t\}$ . We have that $gt = wt^{i+1}$ . Because w does not change when adding t, j does not change and $y_{wt^{i+1}} = y_{wt^i}$ . Therefore,

$$ \begin{align*}\{x\otimes y(g), x\otimes y(gt)\} = \{(x_{(i,j)}, y_w), (x_{(i+1, j)}, y_{w})\}\end{align*} $$

is allowed for all $g\in \mathbb {F}_n\times \mathbb {Z}$ . For patterns of support $\{1,s\}$ , with $s\in S\cup S^{-1}$ , we have $gs = wst^{i}$ . Let us denote $u = \text {arg}\max \{|w'| : w'\sqsubseteq w \wedge w'\sqsubseteq W\}$ , $j= 2|u| - |w|$ and $\pi _1(x\otimes y)_{gs}= x_{(i,j')}$ . If it happens that $y_w = s$ , we have two cases:

• • $u = w$ . Then, by applying s, we continue on the configurations path, i.e. $ws\sqsubseteq W$ . Therefore, $j' = j + 1$ ;

• • $u \sqsubset w$ . Then, because of the local rules defining $Y_f$ , the last letter in w must be $s^{-1}$ . Then, $|ws| = |w| - 1$ and therefore $j' = j+1$ .

This means $\{x\otimes y(g), x\otimes y(gs)\} = \{(x_{(i,j)}, y_w), (x_{(i, j+1)}, y_{w})\}$ is allowed. If, however, $y_w \neq s$ , we have that

$$ \begin{align*} \text{arg}\max\{|w'| : w'\sqsubseteq ws \wedge w'\sqsubseteq W\} = u. \end{align*} $$

Thus, $j' = 2|u| - |ws| = j -1$ . Once again, this means $\{x\otimes y(g), x\otimes y(gs)\} = \{(x_{(i,j)}, y_w), (x_{(i, j+1)}, y_{w})\}$ is allowed. We conclude that $x\otimes y\in Z$ .

Now, let us have $z\in Z$ . We can easily obtain $y\in Y_f$ linked to a word W through the recursive method mentioned above. To find x, we begin by setting

$$ \begin{align*} x_{(i,0)} = \pi_1(z_{t^{i}}) \quad \text{for all } i\in\mathbb{Z}. \end{align*} $$

Next, we define the path function $\rho :\mathbb {Z}\to G$ as follows:

$$ \begin{align*}\rho_W(j) = \begin{cases} W_0 \ldots W_j & \text{if } j\geq 0, \\ (W_0)^{j} & \text{if } j < 0. \end{cases} \end{align*} $$

We continue looking at our configuration y by defining the group elements $\{g_{i,j}\}_{i,j\in \mathbb {Z}}$ as $g_{i,j} = \rho _W(j)t^{i}$ .

Finally, we set

$$ \begin{align*}x_{(i,j)} = \pi_1(z_{g_{i,j}}).\end{align*} $$

Claim. $x\in X$ .

Let us take a look at two cases.

• • $\text {There exists } (i,j)\in \mathbb {Z}^2$ : $\{x_{(i,j)}, x_{(i+1,j)}\}$ is forbidden in X. This would mean that the pattern $\{z_{g_{i,j}}, z_{g_{i,j}t}\}$ would be forbidden in Z, which is a contradiction.

• • $\text {There exists } (i,j)\in \mathbb {Z}^2$ : $\{x_{(i,j)}, x_{(i,j+1)}\}$ is forbidden in X.

  Notice that $g_{r,n+1} = g_{r,n}W_{n+1}$ . This would mean that the pattern $\{z_{g_{i,j}}, z_{g_{i,j}W_{n+1}}\}$ would be forbidden in Z, which is a contradiction.

Claim. $z = x\otimes y$ .

Because of the way y was obtained, it suffices to check the first coordinate. Let us have $g = wt^i$ and $\pi _1(x\otimes y)_g = x_{(i,j)}$ as in the proposition statement. In addition, let

$$ \begin{align*}u = \text{arg}\max\{|w'| : w'\sqsubseteq w \wedge w'\sqsubseteq W\},\end{align*} $$

and $N = |u|$ . As we have seen, this means that

$$ \begin{align*}\pi_1(z_{h}) = x_{(0, N)},\end{align*} $$

because $u = \rho _W(N)$ . Now, if we have $w = u w_0 \ldots w_m$ , we can see that $y_u$ is not in the direction of the flow. Thus, we can deduce from the allowed local rules that the second coordinate of x must decrease by 1 when applying $w_0$ . Now, because X is expansive, we know that x is the only configuration with the pattern $x|_{\mathbb {Z}\times \{N\}}$ on $\mathbb {Z}\times \{N\}$ . This allows us to say

$$ \begin{align*}\pi_{1}(z_{hw_0}) = x_{(0, N-1)}.\end{align*} $$

By repeating the same argument for $w_{1}$ up to $w_{m}$ , we obtain

$$ \begin{align*}\pi_{1}(z_{w}) = \pi_{1}(z_{hw_0\ldots w_{m}}) = x_{(0, \ N-(m-N))} = x_{(0, \ j)}\end{align*} $$

and we can conclude

$$ \begin{align*}\pi_1(z_g) = \pi_1(z_{wt^{i}}) = x_{(i,j)}.\\[-34pt] \end{align*} $$

6.3 Minimality

We would like to see if properties from the aperiodic SFT X over $\mathbb {Z}^2$ can be lifted to our new aperiodic subshift Z. In particular, we are interested in preserving minimality. A $\mathbb {Z}^2$ -SFT of the sought after characteristics is shown to exist in Proposition 5.5.

The idea is as follows. First, we show that the flow shift $Y_f$ is minimal. The idea here is, for configurations defined by words $W'$ and W, to shift the first configuration progressively obtaining configurations whose defining word is $W_0W_1 \ldots W_n e_n W'$ , where $e_n$ is an error term of length 1. Second, we couple this minimality with that of X to establish the sought after result.

Lemma 6.4. Let $y,y'\in Y_f$ be two configurations defined by the words W and $W'$ , respectively. Then, there exists a sequence $\{g_n\}_{n\in \mathbb {N}}$ in $\mathbb {F}_n\times \mathbb {Z}$ such that

$$ \begin{align*} \lim_{n\to\infty}\sigma^{g_n^{-1}}(y') = y, \end{align*} $$

and $|g_{n+1}| = |g_n| + 1$ for all $n\in \mathbb {N}$ .

Proof. We would like to find $g_n$ such that $W(\sigma ^{g_n^{-1}}(y')) {\kern-1pt}={\kern-1pt} W_0 \ldots W_n e_n W'$ , with $|e_n| {\kern-1pt}={\kern-1pt} 1$ . We add the error term so we avoid forbidden flow patterns (we must avoid $W_n = (W^{\prime }_0)^{-1}$ at every point) and for the size of the new word to increase by exactly 1 at each step (see Figure 4). This term will disappear upon taking the limit.

Figure 4 The first three steps to go from the word $(ba)^{\infty }$ to $(b^{-1}a^{-1})^{\infty }$ , with error term $e_n = \ a^{-1}$ for all n. The original word is marked in red, the new one in blue and the error term in green (colour available online).

We begin by introducing the directions involved in the error term:

$$ \begin{align*}a_{i} = \begin{cases} s_i^{-1} & \text{if }\ W^{\prime}_0 = s_i, \\ s_i &\text{if not. }\end{cases}\end{align*} $$

Then, we define $g_0 = a_iW_0^{-1}$ for $W_0\in (S\cup S^{-1})\setminus \{s_i, s_i^{-1}\}$ . This way, we arrive at $W(\sigma ^{g_0^{-1}}(y')) = W_0e_0W'$ , where $e_0$ is the arrow we added as padding to avoid $W_0$ conflicting with $W^{\prime }_0$ . Next, we recursively define $g_n = a_i(W_0 \ \ldots \ W_n)^{-1}$ for $W_n\in (S\cup S^{-1})\setminus \{s_i, s_i^{-1}\}$ . This way, we have

$$ \begin{align*}W(\sigma^{g_n^{-1}}(y')) = W_0 \ \ldots\ W_n e_n W',\end{align*} $$

where $e_n$ is the error term of size 1. Therefore,

$$ \begin{align*}\lim_{n\to\infty}\sigma^{g_n^{-1}}(y') = y.\\[-34pt] \end{align*} $$

Proof. We prove that the SFT Z satisfies the statement of the theorem. Let us take two configurations $x'\otimes y'$ and $x\otimes y$ in Z. Because X is minimal, there exists a sequence $\{(i_n, j_n)\}_{n\in \mathbb {N}}$ in $\mathbb {Z}^2$ with $(j_n)_{n\in \mathbb {N}}$ increasing, such that

$$ \begin{align*}\lim_{n\to\infty}\sigma^{(i_n,j_n)}(x') = x.\end{align*} $$

Let $\{g_n\}_{n\in \mathbb {N}}$ be the sequence from Lemma 6.4, that is,

$$ \begin{align*}\lim_{n\to\infty}\sigma^{g_n^{-1}}(y') = y.\end{align*} $$

Let $M\in \mathbb {N}$ be such that $j_M\geq 2$ . Next, let $\{n_k\}_{k\geq M}$ be the increasing subsequence satisfying $n_k + 1 = j_{k}$ . Then,

$$ \begin{align*}\sigma^{(g_{n_k}t^{i_k})^{-1}}(x'\otimes y') = (x^{\prime}_{(-i_k,-j_k)}, W_0).\end{align*} $$

It follows that

$$ \begin{align*}\sigma^{(g_{n_k}t^{i_k})^{-1}}(x'\otimes y') = \sigma^{(i_k,j_k)}(x')\otimes \sigma^{g_{n_k}^{-1}}(y'),\end{align*} $$

and thus

$$ \begin{align*}\lim_{k\to\infty}\sigma^{(g_{n_k}t^{i_k})^{-1}}(x'\otimes y') = x\otimes y.\end{align*} $$

This shows that Z is minimal.

As a consequence, because unimodular groups contain $\mathbb {F}_n\times \mathbb {Z}$ as a finite index normal subgroup, Proposition 4.5 tells us that they admit minimal strongly aperiodic SFTs. In particular, both torus knot groups and $BS(n,n)$ admit this kind of subshift, as they are unimodular. This latter result improves [Reference Esnay and Moutot24].

7.1 The group $BS(2,3)$

Since $BS(2,3)$ is an HNN-extension, by Britton’s lemma, we get a normal form for elements of $BS(2,3)$ .

Proof. Britton’s lemma states that every element $g\in BS(2,3)$ has the form

$$ \begin{align*} g = a^{N}t^{e_1}a^{m_1} \ldots t^{e_n}a^{m_n}, \end{align*} $$

where $N\in \mathbb {Z}$ and $e_i=\pm 1$ , such that if $e_i = 1$ , then $m_i\in \{0,1\}$ , if $e_i = -1$ , then $m_i\in \{0,1,2\}$ , and we never have a subword of the form $t^{\pm } a^0 t^{\mp }$ .

Notice that if $g = a^{N}$ , it is already in the form for which we are looking. Next, if we have $g = a^{N}t$ , we can decompose $N = 2d + r$ , where $0\leq r < 2$ to change the order of the generators,

$$ \begin{align*} g = a^Nt = a^{2d + r}t = a^{r}t a^{3d}. \end{align*} $$

Analogously, if $g = a^{N}t^{-1}$ , we decompose $N = 3d + r$ with $0\leq r < 3$ and arrive at

$$ \begin{align*} g = a^Nt^{-1} = a^{3d + r}t^{-1} = a^{r}t^{-1} a^{2d}. \end{align*} $$

Finally, for an arbitrary g, we simply iterate the two preceding procedures to arrive at an expression for g in the sought after form.

7.2 The orbit coding construction

In this section, we briefly overview the key ideas in the construction originally found by Kari for $\mathbb {Z}^2$ [Reference Kari34] and then generalized to $BS(m,n)$ [Reference Aubrun and Kari4, Reference Aubrun and Kari6]. We start with an overview of the original construction on $\mathbb {Z}^2$ from a group theoretical point of view to set the scene for generalizations to $BS(m,n)$ . For details and proofs, we refer to the original article [Reference Kari34]. Consider the standard presentation $\langle a,t\mid at=ta\rangle $ for $\mathbb {Z}^2$ . The idea of Kari is to start with a rational piecewise affine map $f:I\subseteq \mathbb {R}\to \mathbb {R}$ such that all $x\in I$ are immortal, meaning that for every $k\in \mathbb {Z}$ , the kth iteration $f^k(x)$ lies inside I, and f is aperiodic. Then he defines an SFT $X_f$ such that:

1. (1) each configuration of the SFT encodes the orbit of an immortal point by f;

2. (2) within a configuration, each $\langle a\rangle $ -coset encodes a real number $x\in I$ . This is done thanks to the Beatty sequence $(B_k(x))_{k\in \mathbb {Z}}$ of x given by $B_k(x)=\lfloor (k+1)x\rfloor - \lfloor kx\rfloor $ , that is, a bi-infinite sequence that alternates between the two integers $\lfloor x\rfloor $ and $\lfloor x\rfloor +1$ , and that in average converges to x;

3. (3) the computation of f follows the t-direction. If the real number x is encoded on $t^k\cdot \langle a\rangle $ for some $k\in \mathbb {Z}$ , then the real number $f(x)$ is encoded on $t^{k+1}\cdot \langle a\rangle $ (see Figure 6 on the left). For $g=a^jt^k\in \mathbb {Z}^2$ , the integer $k\in \mathbb {Z}$ is called the height of g;

   

   Figure 6 On the left, the Cayley graph of $\mathbb {Z}^2=\langle a,t\mid at=ta\rangle $ where is pictured how an orbit for f is encoded. In the middle, the equivalent picture for $BS(2,3)=\langle a, t \mid t^{-1}a^2 t = a^3 \rangle $ . For these two pictures, edges corresponding to generator t in the Cayley graphs are pictured with a double arrow. On the right, edges with double arrows represent the direction given by the flow SFT on $BS(2,3)$ .

4. (4) the function f is computed locally from one coset to the next one. This local computation is exact up to some bounded error, in a way such that globally, the errors compensate and vanish making the global computation exact;

5. (5) the aperiodicity of each configuration is a consequence of the aperiodicity of f.

The main difficulty in this construction is to ensure that only finitely many letters are needed in the alphabet of the SFT. Since we choose a rational piecewise affine map, the coefficients used as colours on the Wang tiles are certainly also rational numbers. The nature of the different encodings (both of the real numbers and of the local computation) ensures that there are a finite amount of Wang tiles and the corresponding SFT $X_f$ is non-empty (see [Reference Kari, Durand-Lose and Margenstern35] for more details).

The same construction may be adapted to $BS(m,n)$ , as presented in [Reference Aubrun and Kari4, Reference Aubrun and Kari6]. There are some technicalities to synchronize the different sheets of $BS(m,n)$ , but the general idea is the same. The main difference with $\mathbb {Z}^2$ is that in this construction, the real number $f^k(x)$ is encoded not only on a unique $\mathbb {Z}$ -coset but on infinitely many of them. With the presentation $\langle a, t \mid t^{-1}a^m t = a^n \rangle $ , every coset $g\cdot \langle a\rangle $ with $g\in BS(m,n)$ encodes the real number $f^k(x)$ , provided that g can be represented by a word w on $\{a,a^{-1},t,t^{-1}\}$ such that $|w|_{t}-|w|_{t^{-1}}=k$ . Similar to $\mathbb {Z}^2$ , the number k plays the role of the height of g (see Figure 6 in the middle). The construction provides a weakly aperiodic SFT, but since a same real number $f^k(x)$ is encoded on infinitely many $\langle a\rangle $ -cosets, this SFT is not strongly aperiodic.

We modify the construction for $BS(m,n)$ so that the computation of f no longer follows the generator t, but rather a direction given by a flow SFT similar to the flow of §6.1 (see Figure 6 on the right). Figure 6 sums up how Kari’s construction on $\mathbb {Z}^2$ , the construction on $BS(m,n)$ as presented in [Reference Aubrun and Kari6] and our construction on $BS(m,n)$ are similar, but also how our construction differs from that of [Reference Aubrun and Kari6] and thus provides strong aperiodicity instead of weak aperiodicity only.

7.3 A flow SFT on $BS(2,3)$

Consider the alphabet $A=\{ t, at, t^{-1}, at^{-1}, a^2t^{-1}\}$ and the SFT $Y_{\text {flow}}\subset A^{BS(2,3)}$ defined by the following local rules: for every group element $g\in BS(2,3)$ and every configuration $y\in Y_{\text {flow}}$ :

• • $y_g=y_{g\cdot a^2}$ if $y_g\in \{t,at\}$ ;

• • $y_g=y_{g\cdot a^3}$ if $y_g\in \{t^{-1},at^{-1}, a^{2}t^{-1}\}$ ;

• • if $y_g=u\in A$ , then for every $v\in A\setminus \{u\}$ , we have $y_{g\cdot v^{-1}}=v$ .

This SFT can be equivalently, and in a more visual way, defined through the finite patterns with support $\{1,a,a^2,t, ta,ta^2,ta^3\}\cup A$ as pictured in Figure 7.

Figure 7 The allowed patterns $p^{(1)}$ to $p^{(5)}$ for the flow SFT on $BS(2,3)$ . For more readability, the outgoing edges are pictured in red (colour available online).

Notice that for each flow configuration $y\in Y_{\text {flow}}$ and for every $g\in BS(2,3)$ , the restriction of y to $g\cdot a^{\mathbb {Z}}$ is necessarily periodic, with this period being either $a^2$ or $a^3$ . More precisely, the coset $g\cdot a^{\mathbb {Z}}$ is $a^2$ -periodic if $y_g\in \{t,at\}$ and $a^3$ -periodic if $y_g\in \{t^{-1},at^{-1},a^2t^{-1}\}$ . Consequently, we may represent y just by a flow on the Bass–Serre tree of $BS(2,3)$ , that is to say, an edge colouring of the complete tree of degree $5$ where each vertex has a single outgoing arrow and four incoming arrows (see Figure 8).

Figure 8 A configuration in the flow SFT on $BS(2,3)$ , pictured on the Bass–Serre tree only. Starting from the upper-left corner, this configuration is represented by a word with prefix $tta^2t^{-1}$ .

In the same fashion as in §6.1, we can express flow configurations from $Y_{\text {flow}}$ as infinite words.

7.4 Embedding $BS(2,3)$ into $\mathbb {R}^2$ along a flow configuration

We also define an embedding of $BS(2,3)$ into $\mathbb {R}^2$ driven by a flow configuration $y\in Y_{\text {flow}}$ , denoted by $\Phi _y:BS(2,3)\to \mathbb {R}^2$ , that is first recursively defined on finite words w on the alphabet $B=\{ a,t,a^{-1},t^{-1}\}$ .

We define $\Phi _y(w)=( \alpha (w),h_y(w))$ recursively, coordinate by coordinate. Let $\varepsilon $ denote the empty word. The second coordinate is such that, for $u\in \{ t, at, a^2t, t^{-1}, at^{-1}\}$ ,

$$ \begin{align*} h_y(\varepsilon) &= 0,\\ h_y(w.u) &=h_y(w)+1\quad\text{if }y_w=u,\\ &=h_y(w)-1\quad\text{otherwise},\\ h_y(w.a) &= h_y(w) = h_y(w.a^{-1}). \end{align*} $$

This coordinate $h_y(g)$ represents how far the $\langle a\rangle $ -cosets of a group element g is from $\langle a\rangle $ (the $\langle a\rangle $ -coset of the identity) if we follow the flow configuration y from the identity. In our construction, this corresponds to the simulated height in the original Kari’s SFT, and we call it the y-height of g in what follows. The first coordinate is

$$ \begin{align*} \alpha(\varepsilon) &= 0,\\ \alpha(w.t) &=\alpha(w.t^{-1})=\alpha(w),\\ \alpha(w.a) &=\alpha(w)+{\big(\tfrac{2}{3}\big)}^{\beta(w)},\\ \alpha(w.a^{-1}) &=\alpha(w)-{\big(\tfrac{2}{3}\big)}^{\beta(w)}, \end{align*} $$

where $\beta (w):=\parallel w\parallel _t = |w|_t - |w|_{t^{-1}}$ counts the contribution of the generator t to w. This first coordinate $\alpha (w)$ is exactly the first coordinate of the $\Phi $ embedding given in [Reference Aubrun and Kari6]. The difference with $\Phi _y$ lies in the second coordinate, $h_y(w)$ , that no longer follows the generator t but the path induced by the flow configuration y instead.

Proof. We prove this by induction on the size of the normal form of Lemma 7.1. Since $h_y(w.a^{\pm 1}) = h_y(w)$ , we can get rid of the last term, $a^k$ , in the writing of the normal form; it does not contribute to $h_y$ . Assume every $h_y$ is well defined for all group elements that can be written with n letters from alphabet $A=\{ t, at, t^{-1}, at^{-1}, a^2t^{-1}\}$ . Let $g\in BS(2,3)$ be an element with normal form $w\in A^{n+1}$ . Denote $g'$ the group element with normal form $w_0\ldots w_{n-1}\in A^n$ . Then,

$$ \begin{align*} h_y(g) &= h_y(w_0\ldots w_n). \end{align*} $$

There are two cases, depending on whether $w_n=y_{g'}$ or $w_n\neq y_{g'}$ . In the first case,

$$ \begin{align*} h_y(g) &= h_y(w_0\ldots w_n) +1\\ &= h_y(g') +1\quad\text{by induction hypothesis}, \end{align*} $$

and in the second case,

$$ \begin{align*} h_y(g) &= h_y(w_0\ldots w_n) -1\\ &= h_y(g') -1\quad\text{by induction hypothesis}, \end{align*} $$

so that $h_y(g)$ does not depend on the chosen word.

Proofs of analogous results for $\alpha $ and $\beta $ can be performed in a similar way. Again, following [Reference Aubrun and Kari6], we define $\unicode{x3bb} :BS(2,3)\to \mathbb {R}$ as

$$ \begin{align*} \unicode{x3bb}(g) = \tfrac{1}{2}\big(\tfrac{3}{2}\big)^{\beta(g)}\alpha(g). \end{align*} $$

Proof. The first point is a direct application of the rules that define $\beta $ . For the second point, we have that

$$ \begin{align*} \unicode{x3bb}(g\cdot ta^i) &= \frac{1}{2}\bigg(\frac{3}{2}\bigg)^{\beta(g\cdot ta^i)}\alpha(g\cdot ta^i)\\ &= \frac{1}{2}\bigg(\frac{3}{2}\bigg)^{\beta(g)+1}\alpha(g\cdot ta^i)\\ &= \frac{1}{2}\bigg(\frac{3}{2}\bigg)^{\beta(g)+1}\bigg(\alpha(g)+ i \cdot \bigg(\frac{2}{3}\bigg)^{\beta(g\cdot t)}\bigg)\\ &= \frac{3}{2}\unicode{x3bb}(g) + \frac{i}{2}\bigg(\frac{3}{2}\bigg)^{\beta(g)+1}\bigg(\frac{2}{3}\bigg)^{\beta(g)+1},\\ \unicode{x3bb}(g\cdot ta^i) &= \frac{3}{2}\unicode{x3bb}(g)+ \frac{i}{2}.\\[-3.3pc] \end{align*} $$

7.5 A strongly aperiodic SFT on $BS(2,3)$

To construct an aperiodic SFT, we will add a new layer of tiles to the flow shift. These new tiles are Wang tiles for $BS(2,3)$ that encode a piecewise linear function.

Each tile consists of 7-tuple integers $s = (t_1, t_2, l, b_1, b_2, b_3, r)$ , as shown in Figure 9. Let $\tau $ be a set of these Wang tiles. We say that a configuration $z\in \tau ^{BS(2,3)}$ is a valid tiling if the colours of neighbouring tiles match. More explicitly, for every $g\in BS(2,3)$ , we must have

$$ \begin{align*} z_{g}(r) &= z_{g\cdot a^2}(\ell),\\ z_{g}(b_i) &= z_{g\cdot a^{i-1}t}(t_1)\quad\text{for }i=1,2,3,\\ z_{g}(b_i) &= z_{g\cdot a^{i-2}t}(t_2)\quad\text{for }i=1,2,3. \end{align*} $$

Figure 9 A Wang tile for $BS(2,3)$ .

We say that a Wang tile for $BS(2,3)$ computes a function $f:I\subset \mathbb {R}\to I$ if

$$ \begin{align*} f\bigg( \frac{t_1+t_2}{2}\bigg) + \ell = \frac{b_1+b_2+b_3}{3}+r. \end{align*} $$

If this equality holds for f, we say that the tile computes f along the generator t. If f is invertible and the tile computes $f^{-1}$ , we say that the tile computes f against the generator t.

Let us define the circle $I = [{1}/{10};5/2]/_{{1}/{10}\sim 5/2}$ . We introduce $T:I \to I$ the piecewise linear map defined by

This linear map is invertible with inverse

$$ \begin{align*} T^{-1}: x \mapsto \begin{cases} 10x & \text{if } x\in\hspace{3pt} ]\tfrac{1}{10};\tfrac{1}{4}[, \\[8pt] \tfrac{2}{5}x & \text{if } x\in [\tfrac{1}{4};\tfrac{5}{2}]. \end{cases} \end{align*} $$

It is not difficult to see that T admits immortal points, that is, real numbers x such that for every $k\in \mathbb {Z}$ , $T^k(x)$ lies in I. It is also easy to check, since $5$ and $2$ are coprime, that T is aperiodic, meaning that for every $x\in I$ if $T^k(x)=x$ for some integer $k\in \mathbb {Z}$ , then $k=0$ .

We do not use the same function as in [Reference Kari34] to construct a strongly aperiodic SFT on $\mathbb {Z}^2$ and in [Reference Aubrun and Kari4] to construct a weakly aperiodic SFT on $BS(3,2)$ , because it may cause trouble in our construction. Indeed, a careful observation of how tiles are built (see [Reference Aubrun and Kari5] for the bounds on the values for $\ell $ ) shows that the tileset corresponding to the piece of the function given by $x\mapsto \tfrac 23x$ is empty! It is safer to use a piecewise linear function where no multiplicative coefficient matches $\tfrac 23$ , and hence our choice for T. More generally, for $BS(m,n)$ , no multiplicative coefficient should match ${m}/{n}$ .

Thanks to the machinery presented in [Reference Aubrun and Kari4, Reference Aubrun and Kari6], we can define from the function T two tilesets: first, $\tau _T$ that computes T along t, then $\tau _{T^{-1}}$ that computes $T^{-1}$ along t—or equivalently computes T against t. We thus define the following quantities that depend on three parameters: a function f that can be either T or $T^{-1}$ in our case, a real number $x\in [ {1}/{10};\tfrac 52]$ and a group element $g\in BS(2,3)$ .

(7.1) $$ \begin{align} t_k(x,g) &= \lfloor (2\unicode{x3bb}(g)+k )x \rfloor - \lfloor (2\unicode{x3bb}(g)+(k-1) )x \rfloor \quad\text{for }k=1,2,\nonumber \\ b_k(f,x,g) &= \lfloor (3\unicode{x3bb}(g)+k )f(x) \rfloor - \lfloor (3\unicode{x3bb}(g)+(k-1) )f(x) \rfloor \quad\text{for }k=1,2,3, \\ \ell(f,x,g) &= \tfrac{1}{2}f(\lfloor 2\unicode{x3bb}(g)x \rfloor) - \tfrac{1}{3}\lfloor 3\unicode{x3bb}(g)f(x) \rfloor, \nonumber \\ r(f,x,g) &= \tfrac{1}{2}f(\lfloor (2\unicode{x3bb}(g)+2)x \rfloor) - \tfrac{1}{3}\lfloor (3\unicode{x3bb}(g)+3)f(x) \rfloor.\nonumber \end{align} $$

We gather $\tau _T$ and $\tau _{T^{-1}}$ , which are both finite by [Reference Aubrun and Kari5, Proposition 8], into a single tileset $\tau $ which is thus finite and combine it with the flow SFT $Y_{\text {flow}}$ to define the SFT $Y_T$ over the alphabet $A\times \tau $ as follows: every configuration $z\in Y_T$ , which we denote $z_g=(y_g,\tau _g)$ for every $g\in BS(2,3)$ , satisfies:

• • if $y_g=t$ , then $\tau _g\in \tau _T$ ;

• • if $y_g\in \{at,t^{-1},t^{-1},at^{-1},a^2t^{-1}\}$ , then $\tau _g\in \tau _{T^{-1}}$ .

These two conditions impose that the computation of iterates of T follows the flow: we put tiles that compute T on outgoing arrows and tiles that compute $T^{-1}$ on incoming arrows (see Figure 10).

Figure 10 The flow configuration drives the choice for computing T or $T^{-1}$ in the different sheets of $BS(2,3)$ .

Combining the formulae from equation (7.1) and the patterns from Figure 7, we can picture Wang tiles from $\tau $ as follows.

Proposition 7.6. The tile pictured on the left of Figure 11 computes T along t, and the tile on the right computes $T^{-1}$ along t (or T against t).

Figure 11 Wang tileset $\tau $ that computes T along a flow configuration for $BS(2,3)$ .

Proof. The proof follows the proof of [Reference Aubrun and Kari6, Lemma 9], since the function T we have chosen has immortal points. Fix a flow configuration $y\in Y_{\text {flow}}$ and choose x an immortal point for T. For every $g\in BS(2,3)$ , we put the tile $\tau (f,T^{h_y(g)}(x),g)$ in g, where $f=T$ if $y_g=t$ and $f=T^{-1}$ otherwise. This defines a configuration z in $\tau ^{BS(2,3)}$ . It remains to check that it is indeed in the SFT $Y_T$ . We need to check that the three matching rules conditions in §7.5 are satisfied.

(1) $z_{g}(r) = z_{g\cdot a^2}(\ell )$ ? We distinguish two cases, depending on whether $y_g=t$ or not. If this is the case, then $z_{g}(r)$ is $r(T,T^{h_y(g)}(x),g)$ :

$$ \begin{align*} z_{g}(r)=\tfrac{1}{2}T(\lfloor (2\unicode{x3bb}(g)+2)T^{h_y(g)}(x) \rfloor) - \tfrac{1}{3}\lfloor (3\unicode{x3bb}(g)+3)T^{h_y(g)+1}(x) \rfloor. \end{align*} $$

In this case, by the definition of $Y_{\text {flow}}$ (see Figure 7), we also have that $y_{g\cdot a^2}=t$ . Thus, $z_{g\cdot a^2}(\ell )$ is $\ell (T,T^{h_y(g\cdot a^2)}(x),g\cdot a^2)$ and since $h_y(g\cdot a^2)=h_y(g)$ , we get

$$ \begin{align*} z_{g\cdot a^2}(l)=\tfrac{1}{2}T(\lfloor 2\unicode{x3bb}(g\cdot a^2)T^{h_y(g)}(x) \rfloor) - \tfrac{1}{3}\lfloor 3\unicode{x3bb}(g\cdot a^2)T^{h_y(g)+1}(x) \rfloor. \end{align*} $$

It suffices to use the fact that $\unicode{x3bb} (g\cdot a^2)=\unicode{x3bb} (g)+1$ to conclude.

In the second case, $y_g\neq t$ , $z_{g}(r)$ is $r(T^{-1},T^{h_y(g)}(x),g)$ and the allowed patterns of Figure 7 impose that $y_{g\cdot a^2}\neq t$ . Thus, $z_{g\cdot a^2}(\ell )$ is equal to $\ell (T^{-1},T^{h_y(g)\cdot a^2}(x),g\cdot a^2)$ . The equalities $\unicode{x3bb} (g\cdot a^2)=\unicode{x3bb} (g)+1$ and $h_y(g\cdot a^2)=h_y(g)$ give that $z_{g}(r) = z_{g\cdot a^2}(\ell )$ .

(2) $z_{g}(b_{i+1}) = z_{g\cdot ta^{i}}(t_1)$ for $i=0,1,2$ ?

If $y_g=t$ , then

$$ \begin{align*} z_{g}(b_{i+1}) &= b_{i+1}(T,T^{h_y(g)}(x),g)\\ &= \lfloor(3\unicode{x3bb}(g)+i+1 )T^{h_y(g)+1}(x) \rfloor - \lfloor (3\unicode{x3bb}(g)+i )T^{h_y(g)+1}(x) \rfloor. \end{align*} $$

However,

$$ \begin{align*} z_{g\cdot ta^{i}}(t_1) &= t_1(T^{h_y(g\cdot ta^{i})}(x),g\cdot ta^{i})\\ &= \lfloor (2\unicode{x3bb}(g\cdot ta^{i})+1 )T^{h_y(g\cdot ta^{i})}(x) \rfloor - \lfloor (2\unicode{x3bb}(g\cdot ta^{i}) )T^{h_y(g\cdot ta^{i})}(x) \rfloor. \end{align*} $$

Using results from Proposition 7.5, we get that

$$ \begin{align*} z_{g\cdot ta^{i}}(t_1) &= \lfloor \bigg(2\bigg(\frac{3}{2}\unicode{x3bb}(g)+ \frac{i}{2}\bigg)+1 \bigg)T^{h_y(g)+1}(x) \rfloor - \lfloor \bigg(2\bigg(\frac{3}{2}\unicode{x3bb}(g)+ \frac{i}{2}\bigg) \bigg)T^{h_y(g)+1}(x) \rfloor\\ &= \lfloor (3\unicode{x3bb}(g)+i+1 )T^{h_y(g)+1}(x) \rfloor - \lfloor (3\unicode{x3bb}(g)+i )T^{h_y(g)+1}(x) \rfloor\\ &= z_{g}(b_{i+1}). \end{align*} $$

If $y_g\neq t$ , the calculations are quite similar, except that T is replaced by $T^{-1}$ in the expression of $z_g(b_{i+1})$ , which is compensated by the fact that, in that case, $h_y(g\cdot ta^{i})=h_y(g)-1$ .

$z_{g}(b_{i+1}) = z_{g\cdot ta^{i-1}}(t_2)$ for $i=0,1,2$ ?

This part is very similar to what precedes and left to the reader, since $t_2(x,g)$ is just a shift of $t_1(x,g)$ .

Let $(x_i)_{i\in \mathbb {Z}}$ be a bi-infinite sequence on the alphabet $\{k,k+1\}$ for some integer $k\in \mathbb {Z}$ . Then, $(x_i)_{i\in \mathbb {Z}}$ is a representation of a real number x if arbitrarily long sub-sequences have averages arbitrarily close to x. For instance, a given real number x has its Beatty sequence $(B_k(x))_{k\in \mathbb {Z}}$ , where $B_k(x)=\lfloor (k+1)x\rfloor - \lfloor kx\rfloor $ is a representation of x. A compactness argument shows that any bi-infinite sequence $(x_i)_{i\in \mathbb {Z}}$ represents at least one real number, but it may also represent different real numbers.

Proof. Let $z=(y,\tau )$ be a configuration in $Y_T$ and assume it possesses a period $g\in BS(2,3)$ . Thus for every $k\in \mathbb {Z}$ , one has that

$$ \begin{align*} z_{a^k}=z_{g^{-1}\cdot a^k} \end{align*} $$

so that the two $\langle a \rangle $ -cosets at $1$ and $g^{-1}$ are the same. If we denote by x one real number represented by $\tau $ on the $\langle a \rangle $ -coset of the identity, we get that

$$ \begin{align*} T^{h_y(g)}(x)=x. \end{align*} $$

However, the periodicity of z also constrains the flow configuration y. Necessarily by Proposition 7.3, if we decompose g into its normal form $g = wa^{p}$ , we have that y is characterized by either the infinite word $w^{\mathbb {N}}$ or $(w^{-1})^{\mathbb {N}}$ . Without loss of generality, we take $W(y) = w^{\mathbb {N}}$ , which in particular implies that $h_{y}(w) = h_y(g)=|g|_t+|g|_{t^{-1}}$ . Hence, we can rewrite $T^{h_y(g)}(x)=x$ as

$$ \begin{align*} T^{|g|_t+|g|_{t^{-1}}}(x)=x, \end{align*} $$

which in turn implies, by the aperiodicity of T, that $|g|_t+|g|_{t^{-1}}=0$ . Because the two terms are positive, they are necessarily zero. The period g is therefore a power of a that we denote as $a^{-N}$ for some $N\in \mathbb {Z}$ . We now know that for every group element $h\in BS(2,3)$ ,

$$ \begin{align*} z_{h}=z_{a^N\cdot h}, \end{align*} $$

so that each $\langle a \rangle $ -coset in the configuration y wears an N-periodic bi-infinite word. Since there are only finitely many possible words of length N, by following the flow component of y, there must exist two distinct integers $k,k'$ such that $T^k(x)=T^{k'}(x)$ . Again, the aperiodicity of T implies that $k=k'$ , which contradicts our initial assumption. We conclude that z has no period.

Combining Propositions 7.8 and 7.9 gives the existence of a strongly aperiodic SFT on $BS(2,3)$ . Since all non-residually finite Baumslag–Solitar groups are finitely presented, torsion free and quasi-isometric between them, [Reference Cohen17, [Theorem 4.1]] applies and we conclude that all the $BS(m,n)$ with $m,n>1$ and $m\neq n$ admit strongly aperiodic SFTs.

7.6 Consequences

Through the machinery provided by Theorem 4.1, we can push the result to a broader class of groups, namely those obtained as the fundamental group of a graph of virtual $\mathbb {Z}$ . This structure is the same as in Definition 3.2 but all vertex groups are virtually $\mathbb {Z}$ instead of just $\mathbb {Z}$ .

This way, Corollary 7.11 implies the following result.