A geometric simulation theorem on direct products of finitely generated groups

We show that every effectively closed action of a finitely generated group G on a closed subset of {0, 1}N can be obtained as a topological factor of the G-subaction of a (G×H1×H2)-subshift of finite type (SFT) for any choice of infinite and finitely generated groups H1, H2. As a consequence, we obtain that every group of the form G1 × G2 × G3 admits a non-empty strongly aperiodic SFT subject to the condition that each Gi is finitely generated and has decidable word problem. As a corollary of this last result we prove the existence of non-empty strongly aperiodic SFT in a large class of branch groups, notably including the Grigorchuk group.


Introduction
Dynamical systems can be often quite difficult to study and several tools have been developed in order to better understand them. A particularly interesting approach to do so is to restrict to subclasses of dynamical systems that can be defined using a finite amount of information. With this point of view, a reasonably large class of dynamical systems is that of effectively closed systems. Informally speaking, effectively closed systems are those where both the configurations in the system and the action can be described completely by a Turing machine. Even though these systems admit finite presentations, they remain quite complicated.
A natural question is whether an effectively closed system can be obtained as a subaction of another system which admits a simpler description. This question is motivated by the following: consider the class of subshifts of finite type (SFT), that is, the sets of colorings of a group which respect a finite number of local constrains -in the form of a finite list of forbidden patterns-and are equipped with the shift action. It can be easily shown that any system obtained as a restriction of the shift action to a subgroup is not necessarily an SFT, but under weak assumptions, such as the groups being recursively presented, we obtain that the subaction is still an effectively closed dynamical system, see [Hoc09] for the Z d case.
For the class of Z d -SFTs there is still no characterization of which dynamical systems can arise as their subactions, there are in fact some effective Zdynamical systems that cannot appear as a subaction of a Z d -SFT [Hoc09] or more generally, of any shift space. Nevertheless, in that same article, Hochman proves every effectively closed Z d -action over a Cantor set T : Z d X admits an almost trivial isometric extension which can be realized as the subaction of a Z d+2 -SFT. This result has subsequently been improved for the expansive case independently in [AS13] and [DRS10] showing that every effectively closed Z-subshift can be obtained as the projective subdynamics of a sofic Z 2 -subshift.
This type of results give powerful techniques to prove properties about the original systems. An example is the characterization of the set of entropies of Z 2 -SFTs as the set of right recursively enumerable numbers [HM10]. More recently, in an article of Sablik and the author [BS17], Hochman's theorem was extended to groups which are of the form G = Z 2 ⋊H. More specifically, it was shown that for every finitely generated group H, homomorphism ϕ : H → Aut(Z 2 ) and Heffectively closed dynamical system (X, T ) one can construct a (Z 2 ⋊ ϕ H)-SFT whose H-subaction is an extension of (X, T ). As a consequence of this result, a new class of groups admitting strongly aperiodic subshifts of finite type was found.
All of these previous results have a common denominator: they involve the use of a Z component in the group where the simulation is carried. In particular, a natural question would be to ask whether it is possible to obtain a realization result in a periodic group.
The purpose of this article is to prove a simulation theorem which does not involve a Z component. Specifically, Theorem 3.1. Let G be a finitely generated group and (X, T ) an effectively closed G-dynamical system. For every pair of infinite and finitely generated groups H 1 , H 2 there exists a (G × H 1 × H 2 )-SFT whose G-subaction is an extension of (X, T ).
This result is obtained through the combination of two different techniques already present in the literature. On the one hand, we use Toeplitz configurations to encode the dynamical system (X, T ) in an effectively closed Zsubshift and then extend this object to a Z 2 -sofic subshift through the theorems of [AS13,DRS10]. This has already been used in [BS17]. On the other hand, we employ a technique previously used by Jeandel [Jea15b] to force grid structures through local rules. Namely, a theorem by Seward [Sew14] shows that a geometric analogue of Burnside's conjecture holds, namely, that every infinite and finitely generated group admits a translation-like action by Z. From a graph theoretical perspective, this means that the group admits a set of generators such that its associated Cayley graph can be covered by disjoint bi-infinite paths. We use that theorem and Jeandel's technique to geometrically embed a two-dimensional grid into H 1 × H 2 and create the necessary structure to prove our main result.
In the case where the G-dynamical system is a subshift, we can give a stronger result.
Theorem 4.1. Let G be a recursively presented and finitely generated group and Y an effectively closed G-subshift. For every pair of infinite and finitely generated groups H 1 , H 2 there exists a sofic (G × H 1 × H 2 )-subshift X such that: • the G-subaction of X is conjugate to Y .
• the G-projective subdynamics of X is Y .
• The shift action σ restricted to H 1 × H 2 is trivial on X.
It is known that every Z-SFT contains a periodic configuration [LM95]. More generally, any SFT defined over a finitely generated free group also has a periodic configuration [Pia08]. However, it was shown by Berger [Ber66] that there are Z 2 -SFTs which are strongly aperiodic, that is, such that the shift acts freely on the set of configurations. This result has been proven several times with different techniques [Rob71,Kar96,JR15] giving a variety of constructions. However, the problem of determining which is the class of groups which admit strongly aperiodic SFTs remains open. Amongst the groups that do admit strongly aperiodic SFTs are: Z d for d > 1, hyperbolic surface groups [CGS15], Osin and Ivanov monster groups [Jea15a], the direct product G×Z for a particular class of groups G which includes Thompson's T group and PSL(Z, 2) [Jea15a] and groups of the form Z 2 ⋊ G where G is finitely generated and has decidable word problem [BS17]. It is also known that no group with two or more ends can contain strongly aperiodic SFTs [Coh17] and that recursively presented groups which admit strongly aperiodic SFTs must have decidable word problem [Jea15a].
As an application of Theorem 3.1 we present a new class of groups which admit strongly aperiodic SFTs, that is: Theorem 4.4. For any triple of infinite and finitely generated groups G 1 , G 2 and G 3 with decidable word problem, their direct product G 1 × G 2 × G 3 admits a non-empty strongly aperiodic subshift of finite type.
A result by Carroll and Penland [CP15] shows that having a non-empty strongly aperiodic subshift of finite type is a commensurability invariant of groups. Putting this together with Theorem 4.4 we deduce that any finitely generated group with decidable word problem which is commensurable to its square also has the property. In particular, as the Grigorchuk group is commensurable to its square, this yields the existence of non-empty strongly aperiodic subshifts of finite type in the Grigorchuk group.
Corollary 4.8. There exists a non-empty strongly aperiodic subshift of finite type defined over the Grigorchuk group.
This strengthens Jeandel's result from [Jea15b] where the Grigorchuk group was shown to admit a weakly aperiodic SFT, that is, a subshift such that the orbit of every configuration under the shift action is infinite.

Preliminaries
Consider a group G and a compact topological space X. The tuple (X, T ) where T : G X is a left G-action by homeomorphisms is called a G-dynamical system. Let (X, T ) and (Y, S) be two G-dynamical systems. We say φ : X → Y is a topological morphism if it is continuous and φ • T g = S g • φ for all g ∈ G.
A surjective topological morphism φ : X ։ Y is a topological factor and we say that (Y, S) is a factor of (X, T ) and that (X, T ) is an extension of (Y, S). When φ is a bijection and its inverse is continuous we say it is a topological conjugacy and that (X, T ) is conjugated to (Y, S).
In what follows, we consider the space X to be a Cantor set equipped with the product topology and G to be a finitely generated group. Without loss of generality, we consider X to be a closed subset of {0, 1} N and for a word w = w 0 , . . . , w n ∈ {0, 1} * we denote by [w] the set of all points x ∈ {0, 1} N such that x i = w i for i ≤ n. Let S ⊂ G a finite set of generators of G. An effectively closed G-dynamical system is a G-dynamical system (X, T ) where: , 1} * is a recursively enumerable language. That means that X is the complement of a union of cylinders which can be enumerated by a Turing machine.
2. T is an effectively closed action: there exists a Turing machine which on entry s ∈ S and w ∈ {0, 1} * enumerates a sequence of words (w j ) j∈J such that The idea behind the definition is the following: There is a Turing machine T which given a word g ∈ S * representing an element of G and n coordinates of x ∈ X ⊂ {0, 1} N returns a sequence of sets which yield successive approximations of the image of x by T s .
Let A be a finite alphabet and G a finitely generated group. The set A G = {x : G → A} equipped with the left group action σ : G × A G → A G given by: (σ h (x)) g = x h −1 g is the full G-shift. The elements a ∈ A and x ∈ A G are called symbols and configurations respectively. We endow A G with the product topology, therefore obtaining a compact metric space. The topology is generated by the metric d(x, y) = 2 − inf{|g| | g∈G: xg =yg} where |g| is the length of the smallest expression of g as the product of some fixed set of generators of G. This topology is also generated by the clopen subbase given by the cylinders Given a support F , a pattern with support F is an element p ∈ A F , i.e. a finite configuration and we write supp(p) = F . Analogously to words, we denote the cylinder generated by p centered in g as [p] g = h∈F [p h ] gh . If x ∈ [p] g for some g ∈ G we write p ⊏ x to say that p appears in x.
A subset X of A G is a G-subshift if and only if it is σ-invariant -for each g ∈ G, then σ g (X) ⊂ X-and closed for the product topology. Equivalently, X is a G-subshift if and only if there exists a set of forbidden patterns F that defines it: Said otherwise, the G-subshift X F is the set of all configurations x such that no p ∈ F appears in x.
If the context is clear enough, we will drop the group G from the notation and simply refer to a subshift. A subshift X ⊆ A G is of finite type -SFT for short -if there exists a finite set of forbidden patterns F such that X = X F . A subshift X ⊆ A G is sofic if it is the factor of an SFT. Finally, a subshift is effectively closed if there exists a recursively enumerable coding of a set of forbidden patterns F such that X = X F . More details can be found in [ABS17]. In the case of Z-subshifts, we say X is effectively closed if and only if there exists a recursively enumerable set of forbidden words F such that X = X F .
We say a Z 2 -subshift is nearest neighbor if there exists a set of forbidden patterns F defining it such that each p ∈ F has support {(0, 0), (1, 0)} or {(0, 0), (0, 1)}. While there are Z 2 -SFTs which are not nearest neighbor, each Z 2 -SFT is topologically conjugate to a nearest neighbor Z 2 -subshift through a higher block recoding, see for instance [LM95] for the 1-dimensional case. What is more, for every sofic Z 2 -subshift Y we can jointly extract a nearest neighbor Z 2 -SFT extension X and a 1-block topological factor φ : X ։ Y , that is, a topological factor such that there exists a local recoding of the alphabet Φ : Let H ≤ G be a subgroup and (X, T ) a G-dynamical system. The Hsubaction of (X, T ) is (X, T H ) where T H : H X is the restriction of T to H, that is, for each h ∈ H, then (T H ) h (x) = T h (x). In the case of a subshift X ⊂ A G there is also the different notion of projective subdynamics. The H-projective subdynamics of X is the set π H (X) = {y ∈ A H | ∃x ∈ X, ∀h ∈ H, y h = x h }. It is important to remark that subactions do not preserve expansivity, so in particular a subaction of a subshift is not necessarily a subshift. Nevertheless, the projective subdynamics of a subshift π H (X) is always an Hsubshift.

Simulation without Z
The purpose of this section is to prove the following result.
Theorem 3.1. Let G be a finitely generated group and (X, T ) an effectively closed G-dynamical system. For every pair of infinite and finitely generated groups H 1 , H 2 there exists a (G × H 1 × H 2 )-SFT whose G-subaction is an extension of (X, T ).
The general scheme of the proof is the following: First, in Section 3.1 we use Toeplitz sequences to encode the elements of X and the effectively closed action T : G X into an effectively closed Z-subshift Top 1D (X, T ). Subsequently, we extend Top 1D (X, T ) to a Z 2 -subshift by repeating its rows periodically in the vertical direction. Using a known simulation theorem [AS13,DRS10] we conclude that this object, which we call Top 2D (X, T ), is a sofic Z 2 -subshift from which we extract a nearest neighbor SFT extension Top(X, T ).
The next step is presented in Section 3.2 where we construct an SFT Ω in H 1 × H 2 with the property that each ω ∈ Ω induces a bounded Z 2 -action over H 1 × H 2 . We use a result by Seward [Sew14] to guarantee that for a specific choice of generators of H 1 and H 2 , then there is at least one ω ∈ Ω inducing a free action. We use these actions as replacements of two-dimensional grids, which we then proceed to use to embed configurations of a nearest neighbor Z 2 -subshift.
Finally, in Section 3.3 we use the simulated grids in H 1 × H 2 to encode Top(X, T ). This yields an (H 1 × H 2 )-SFT which factors onto a sofic (H 1 × H 2 )subshift where every grid codes an element x ∈ X and its image under T by the generators of G. We then proceed to extend this object to a G × H 1 × H 2 subshift of finite type by forcing every G-coset to have exactly the same grid structure and by linking the Top 1D (X, T ) layers through local rules.
We finish this section by defining the factor code and showing that it satisfies the required properties.

Encoding an effectively closed dynamical system using Toeplitz configurations
Let T : G X be an effectively closed action of a finitely generated group G over X ⊂ {0, 1} N . Here we show how to encode T into an effectively closed Zsubshift. This section presents the same ideas as that of [BS17], although here we will only treat a special simplified case which is enough for our purposes.
A configuration τ ∈ A Z is said to be Toeplitz if for every m ∈ Z there is p > 0 such that τ m = τ m+kp for each k ∈ Z. These configurations were initially defined by Jacobs and Keane [JK69] for one-sided dynamical systems and are quite useful to encode information in a recurrent way. Indeed, consider the function Ψ : {0, 1} N → {0, 1, $} Z given by: Technically speaking, Ψ(x) is not Toeplitz as m = 0 fails to satisfy the requirement, however, every other 1} N be the one-sided shift defined by σ(x) i = x i+1 and note that for every j ∈ Z we have that: The important property of Ψ(x) is that x can be recognized locally from any configuration y ∈ Orb σ (Ψ(x)). Indeed, each subword of length 3 in Ψ(x) is a cyclic permutation of a word of the form ax 0 $ where a ∈ {0, 1, $}, therefore x 0 can be recognized as it is the only non-$ symbol which is followed by a $. Similarly, any word of length 9 can be used to decode x 0 , x 1 and generally, a word of length 3 n is sufficient to decode x 0 , x 1 , . . . x n−1 . As any y ∈ Orb σ (Ψ(x)) must coincide in arbitrarily large blocks with a shift of Ψ(x) we have that this property holds for every configuration in Orb σ (Ψ(x)).
Let (X, T ) be a G-dynamical system. We use the encoding Ψ defined above to construct an effectively closed Z-subshift Top 1D (X, T ) which encodes the configurations of X and the action of T around a unit ball in G. Formally, let S ⊂ G be a finite and symmetric (S −1 ⊂ S) set of generators of G which contains the identity.
Given a word w ∈ ({0, 1, $} S ) * , we denote its s-th coordinate by w(s) ∈ {0, 1, $} * . We define Top 1D (X, T ) as the Z-subshift over the alphabet {0, 1, $} S given by the set of forbidden words F T , where F T = n∈N F n and F n is the set of words w of length 3 n+1 over the alphabet {0, 1, $} S which are accepted by the following algorithm.
For each s ∈ S do the following: fix j := 0 and check if there exists b ∈ {0, 1} such that each string of three contiguous symbols appearing in w(s) is a cyclic permutation of ab$ for some a ∈ {0, 1, $}. If no such b exists, accept the word as forbidden, otherwise, define u(s) j = b, set j := j + 1 and repeat with the string formed by the a's until either a problem is found (and thus the word is forbidden) or a valid symbol u(s) m is found for every s ∈ S and m ∈ {0, . . . , n}.
Finally, if this stage of the algorithm is reached without accepting the word, we can construct words u(s) := u(s) 0 u(s) 1 . . . u(s) n . In parallel, for each s ∈ S, run simultaneously the following two procedures: • Check whether [u(s)] ⊂ {0, 1} N \ X and accept if this is the case.
) and accept if this is the case.
These two last algorithms do exist in the case where (X, T ) is an effectively closed G-dynamical system. In particular, this shows that in this case F T is recursively enumerable and thus Top 1D (X, T ) is effectively closed. We state this formally in Proposition 3.2.
Proposition 3.2. If (X, T ) is an effectively closed G-dynamical system, then Top 1D (X, T ) is an effectively closed Z-subshift.
Consider y ∈ Top 1D (X, T ). By definition none of the words of length 3 n appearing in y belongs to F T and thus any w ⊏ y(s) of length 3 n defines a unique word u w ∈ {0, 1} n by the decoding process. Moreover, one can easily verify with the definition of F T by using a word of length 3 n+1 that u w does not depend on the specific choice w ⊏ y(s) but only on y(s). We can therefore define a family of functions γ n : Said otherwise, γ n recovers the n-th symbol from the s-th component of y ∈ Top 1D (X, T ). Furthermore, we can use the γ n to construct a function γ : S × Top 1D (X, T ) → {0, 1} N defined by γ(s, y) n := γ n (s, y). By definition of F T we have that for each n ∈ N then: From the fact that X is compact and T continuous, we deduce that γ(s, y) ∈ X and T s (γ(1 G , y)) = γ(s, y). Also, if we define Ψ(x) as the configuration Finally, from the fact that each γ n only depends on an arbitrary subword of length 3 n of y, we obtain that γ(s, y) = γ(s, σ m (y)) for each m ∈ Z and that γ is continuous.
Next we will make use of a known simulation theorem to lift our effectively closed Z-subshift up to a sofic Z 2 -subshift.
Using Theorem 3.3 we obtain a sofic Z 2 -subshift which we call Top 2D (X, T ). As every row in a configuration in Top 2D (X, T ) is the same, we can naturally extend the definition of γ to this subshift by restricting to Z × {0}. We resume the important points of all that has been constructed in this subsection in the following lemma.

Finding a grid in H 1 × H 2
The second ingredient of the proof uses the notion of translation-like action introduced by Whyte [Why99]. Instead of having the rigidity of a proper translation, translation-like actions only ask for the action to be free and that given an element of the acting group g then the image by g does not go arbitrarily far. Formally, Definition 3.1. A left action of a group G over a metric space (X, d) is translation-like if and only if it satisfies: This notion gives a proper setting to define geometric analogues of classical disproved conjectures in group theory concerning subgroup containments. For instance, the Burnside conjecture and the Von Neumann conjecture can be reinterpreted geometrically as the question of whether every infinite and finitely generated group admits a translation-like action by Z or by a non-abelian free group respectively.
In what concerns our study, we are only going to make use of the following result from Seward [Sew14] which is the geometric version of the Burnside conjecture: Theorem 1.4). Every finitely generated infinite group admits a translation-like action of Z.
Theorem 3.5 has already been used by Jeandel in [Jea15b] to show that the Domino problem -that is, the problem of deciding whether a finite set of forbidden patterns defines a non-empty subshift-is undecidable for groups of the form H 1 × H 2 where both H i are infinite and finitely generated. He also showed that groups containing such a product as a subgroup have undecidable Domino problem and admit weakly aperiodic SFTs, that is, an SFT such that every configuration has an infinite orbit. Here we make use of the same technique to prove our result.
Let S 1 , S 2 be finite and symmetric sets of generators for H 1 and H 2 respectively which contain the identity. Consider the alphabet B = S 1 × S 1 × S 2 × S 2 . For b = (s 1 , s 2 , s 3 , s 4 ) we write We can think of in i and out i as arrows pointing towards the left and right neighbor in a path.
In simpler words, the first and second rules just codify that when moving in a component H i , then the arrows associated to the other component do not change. For instance, each ω ∈ Ω satisfies that in 1 (ω (h1,h2) ) = in 1 (ω (h1,1H 2 ) ) for each h 2 ∈ H 2 .
The third and fourth rules code the fact that in any configuration ω ∈ Ω, if one looks at a particular position (h 1 , h 2 ), follows an arrow in a component and then in the new position follows the inverse arrow, one comes back to the starting position. In particular, we can interpret ω ∈ Ω as a product of two directed graphs with both in-degree and out-degree 1, that is, a collection of cycles and bi-infinite paths. Here in i (ω (h1,h2) ) codes the incoming arrow in the i-th component while out i (ω (h1,h2) ) codes the corresponding outgoing arrow. With this graph product interpretation, each connected component of some ω ∈ Ω can be either a product of two finite cycles, a product of a cycle and an infinite path, or a product of two infinite paths. Figure 1 shows how these grids look like in the case where both H 1 and H 2 are covered by bi-infinite paths. Clearly Ω is an (H 1 × H 2 )-SFT as S is finite. We can also give a group theoretical interpretation of Ω as a family of Z 2 actions over H 1 × H 2 , namely, given ω ∈ Ω we can define η(ω) : Z 2 H 1 × H 2 by: ,h2) )).
The first and second rules force η(ω) to be commuting, while the third and fourth rules make the inverses well defined.
In simpler words, C(ω, y, h 1 , h 2 ) is the configuration obtained by reading y along the two-dimensional grid defined by η(ω) where the pair (h 1 , h 2 ) is identified to (0, 0). By definition of F Ω(X) , we have that C(ω, y, h 1 , h 2 ) ∈ X as no forbidden patterns from F X can occur.
Proposition 3.6. There exist finite and symmetric sets of generators S 1 and S 2 for H 1 and H 2 respectively which contain the identity and such that Ω contains a configuration ω such that η(ω) is free.
Proof. By Theorem 3.5 there exist translation-like actions f 1 : Z H 1 and f 2 : Z H 2 . Consider generator metrics d 1 and d 2 on H 1 and H 2 respectively and define for i ∈ {1, 2} h)) .
As f i is a translation-like, we know that the distance from f i (1, h) to h is uniformly bounded, therefore S ′ i is finite. Let S i be a finite set of generators of H i with the desired properties and which contains S ′ i . We claim that S 1 , S 2 satisfy the requirements. Indeed, we can define ω ∈ Ω by: We verify directly that η(ω) (1,0) = f 1 and η(ω) (0,1) = f 2 . As both f i are free, we have that η(ω) is free.
Proof. We begin by fixing an arbitrary starting point in every grid, formally, define the equivalence relation over i∈I be a representing set of (H 1 × H 2 )/ ∼ which contains (1 H1 , 1 H2 ). We define y ∈ A H1×H2 X by y η(ω) z (h i 1 ,h i 2 ) := c z . By definition of ∼ and freeness of η(ω), y is well defined over all of H 1 × H 2 . Moreover, by definition of F Ω(Y ) , we have that (ω, y) ∈ Ω(Y ).
We collect everything we need from this subsection in the next lemma.

Proof of Theorem 3.1
Consider first the subshift Top 2D (X, T ) from Lemma 3.4. One can extract a nearest neighbor Z 2 -SFT extension Top(X, T ) and a 1-block map φ : Top(X, T ) ։ Top 2D (X, T ) defined by a local function Φ. We can thus construct the (H 1 ×H 2 )-SFT Ω( Top(X, T )) from Lemma 3.8. Let F Π be a finite set of forbidden patterns defining Ω( Top(X, T )). Recall that S is the finite set of generators of G with which Top 1D (X, T ) was defined. We construct a (G × H 1 × H 2 )-subshift Final using the alphabet of Ω( Top(X, T )) and the following forbidden patterns F := A 1 ∪ A 2 ∪ A 3 : 1. For each q ∈ F Π let p be the pattern with supp(p) = {1 G } × supp(q) such that for every (h 1 , h 2 ) ∈ supp(q) we have p(1 G , h 1 , h 2 ) = q(h 1 , h 2 ). We define A 1 as the set of all such p.
In simpler words: A 1 forces each (H 1 × H 2 )-coset to contain a configuration of Ω( Top(X, T )), A 2 constrains Final such that in every (H 1 × H 2 )-coset the first component (the ω ∈ Ω) is the same and A 3 links the different (H 1 × H 2 )cosets indexed by G forcing them to respect the action of T . This last rule is illustrated on Figure 2 where the grids are induced by ω and appear somewhere in In each of the two grids a configuration from Top(X, T ) is encoded. The forbidden patterns from A 3 code the fact that if we project these configurations to Top 2D (X, T ), then, if the configurations indexed by 1 G and s −1 1 code respectively x and y, we will forcefully have that . . .
. . . Figure 2: The set A 3 links the different cosets by forcing that in all configurations as above then Ψ(y) = Ψ(T s1 (x)).
Note that as A 1 ⊂ F , then (ω ′ , y ′ ) ∈ Ω( Top(X, T )) and thus the configuration C(ω ′ , y ′ , 1 H1 , 1 H2 ) ∈ Top(X, T ) as stated in Lemma 3.8. In turn, φ(C(ω ′ , y ′ , 1 H1 , 1 H2 )) ∈ Top 2D (X, T ) and thus γ and ϕ are also well defined. Moreover, in order to compute the first n coordinates of ϕ(ω, y) it suffices to know the values of C(ω ′ , y ′ , 1 H1 , 1 H2 ) restricted to a ball of diameter 3 n of Z 2 . And in turn, it suffices to know (ω ′ , y ′ ) restricted to the ball of diameter 3 n of H 1 × H 2 with respect to the generators S 1 × S 2 . This means that φ(ω, y) is continuous. In order to conclude we need to show that ϕ is onto and that it interchanges the subaction σ G with T .
Where the penultimate equality is from Lemma 3.4.
This implies that no patterns from A 3 appear. Therefore (ω, y) ∈ Final.
Adding up both claims and the previously proven properties of ϕ, we obtain that ϕ : (Final, σ G ) ։ (X, T ) is a topological factor. This proves Theorem 3.1.

Consequences and remarks
In this last section we explore some consequences of Theorem 3.1. In the case of expansive actions, we can give more detailed information about the factor. Specifically, we show that if G is a recursively presented group, then every effectively closed G-subshift can be realized as the projective subdynamics of a sofic (G × H 1 × H 2 )-subshift. Moreover, we prove that the sofic subshift can be picked in such a way that it is invariant under the shift action of This result is particularly helpful for the next part where we show that any group that can be written as the direct product of three infinite and finitely generated groups with decidable word problem admits a non-empty strongly aperiodic SFT.
Finally, we close this section by showing how the previous result can be used to prove the existence of non-empty strongly aperiodic subshifts in the Grigorchuk group.

The case of effectively closed expansive actions
The subshift Final constructed in the proof of Theorem 3.1 satisfies the required properties, however, it has an undesirable perk. Namely, it might happen that for (ω, y) ∈ Final then ϕ(ω, y) = ϕ(σ (1G ,h1,h2) ω, y) for some (h 1 , h 2 ) ∈ H 1 ×H 2 . The reason is that in Ω there might be many different grids and a priori there is no restriction forcing them to contain shifts of the same configuration.
The natural approach to get rid of this perk is to use the functions γ n to impose in every (H 1 × H 2 )-coset that the first n-coordinates of the coded configuration are the same everywhere. While we show that this works perfectly for an expansive system, it naturally fails in the case where (X, T ) is equicontinuous, see [Hoc09], Proposition 6.1. This makes expansive systems particularly interesting in this construction, specially in the proof of Theorem 4.4 where we show that every triple direct product of finitely generated groups with decidable word problem admit strongly aperiodic SFTs.
Given a group G generated by S and a finite alphabet A a pattern coding c is a finite set of tuples c = (w i , a i ) i∈I where w i ∈ S * and a i ∈ A. A set of pattern codings C is said to be recursively enumerable if there is a Turing machine which takes as input a pattern coding c and accepts it if and only if c ∈ C. A subshift X ⊂ A G is effectively closed if there is a recursively enumerable set of pattern codings C such that: Theorem 4.1. Let G be a recursively presented and finitely generated group and Y an effectively closed G-subshift. For every pair of infinite and finitely generated groups H 1 , H 2 there exists a sofic (G × H 1 × H 2 )-subshift X such that: • The G-subaction of X is conjugate to Y .
• The G-projective subdynamics of X is Y .
• The shift action σ restricted to H 1 × H 2 is trivial on X.
Proof. Let S ⊂ G be a finite generating set and consider a recursive bijection ϕ : N → S * where S * is the set of all words on S. As G is recursively presented, then its word problem WP(G) = {w ∈ S * | w = G 1 G } is recursively enumerable and there is a Turing machine M which accepts a pair (n, n ′ ) ∈ N 2 if and only if ϕ(n) = ϕ(n ′ ) as elements of G. For simplicity, fix ϕ(0) to be the empty word representing 1 G .
Here ϕ(⌊n/κ⌋) ∈ S * is identified as an element of G. Consider the set Z = ρ(Y ) and the G-action T : G Z defined by T g (ρ(y)) = ρ(σ g (y)). Clearly ρ is a topological conjugacy between (Y, σ) and (Z, T ). We claim that (Z, T ) is an effectively closed G-dynamical system. Indeed, let w ∈ {0, 1} * . A Turing machine which accepts w if and only if [w] ∈ {0, 1} N \ Z is given by the following scheme: First, if for some n < |w|/κ we have that w κn , . . . , w κn+κ−1 does not belong to υ(A) accept w. Then for each pair (κn, κn ′ ) in the support of w run M in parallel over the pair (n, n ′ ). If M accepts for a pair such that w κn , . . . , w κn+κ−1 = w κn ′ , . . . , w κn ′ +κ−1 then accept w (this means that w did not codify a configuration in A G as two different n, n ′ codifying the same group element have different symbols assigned to them). Also, in parallel, use the algorithm recognizing a maximal set of forbidden patterns for Y (this exists by [ABS17], Lemma 2.3) over the pattern coding c w = (ϕ(n), υ −1 (w κn , . . . , w κn+κ−1 )) n<|w|/κ .
This eliminates all w which codify configurations containing forbidden patterns in Y . For the analogue algorithm for T s ([w]) just note that as G is recursively presented, the set of pairs (n, m) such that ϕ(n) = G sϕ(m) also form a recursively enumerable set. Therefore T s ([w]) also admits the required algorithm. We use Theorem 3.1 to construct the (G × H 1 × H 2 )-SFT Final. We further restrict Final with an extra set of forbidden patterns A 4 . Let B n be the ball of size n in H 1 × H 2 with respect to the metric induced by the set of generators S 1 × S 2 used to construct Ω from Lemma 3.8. Let p be a pattern with support {1 G } × B 3 m +1 and let (ω, y) ∈ [p]. By definition of γ m , we have that for each (s 1 , s 2 ) ∈ S 1 × S 2 , γ m (1 G , φ(C(ω, y, s 1 , s 2 ))) only depends on the ball of size 3 m around (s 1 , s 2 ), therefore all of these functions depend only on p. Denote them by γ m (1 G , p, s 1 , s 2 ).
In other words, we force the first κ symbols coded in every simulated grid to coincide. As the size of the support of these patterns is bounded, A 4 is finite and Final defined by forbidding the patterns in A 4 is still an SFT. Moreover, the configuration constructed in Claim 3.2 clearly satisfies these constrains so ϕ is still onto.
Let X :=φ( Final). The function ϕ(σ (g,h1,h2) −1 (ω, y))| 0,...,κ−1 depends only on a finite support (a ball of size 3 κ around the identity for instance) and clearly commutes with the shift. Thereforeφ is indeed a topological factor and thus X is a sofic subshift. Also, by definition of A 4 and the fact that H 1 × H 2 is generated by S 1 × S 2 we obtain thatφ does not depend on (h 1 , h 2 ) and thus H 1 × H 2 acts trivially on X.

Strongly aperiodic SFTs in triple direct products
Next we show how Theorem 4.1 can be applied to produce strongly aperiodic subshifts of finite type. Recall that a G-subshift (X, σ) is strongly aperiodic if the shift action is free, that is, ∀x ∈ X, σ g (x) = x =⇒ g = 1 G .
Lemma 4.2. Let G i for i ∈ {1, 2, 3} be infinite and finitely generated groups such that there exists a non-empty effectively closed subshift Y i ⊂ A Gi which is strongly aperiodic. Then G 1 × G 2 × G 3 admits a non-empty strongly aperiodic SFT.
Proof. Recall the following general property of factor maps. Suppose there is a factor φ : (X, T ) ։ (Y, S) and let x ∈ X such that T g (x) = x. Then S g (φ(x)) = φ(T g (x)) = φ(x) ∈ Y . This means that if S is a free action then T is also a free action. In particular, it suffices to exhibit a non-empty strongly aperiodic sofic subshift to conclude.
By Theorem 4.1 we can construct for each i ∈ {1, 2, 3} a non-empty sofic (G 1 ×G 2 ×G 3 )-subshift X i whose G i -subaction is conjugate to Y i and is invariant under the action of G j × G k with i / ∈ {j, k}. Let X = X 1 × X 2 × X 3 . We claim that X is a non-empty strongly aperiodic sofic subshift.
On the other hand, as the G i -subaction is conjugate to Y i which is strongly aperiodic, we conclude that g i = 1 Gi . Therefore g = 1 G1×G2×G3 . As the choice of x was arbitrary, this shows that X is strongly aperiodic.
Lemma 4.2 requires the existence of a non-empty effectively closed and strongly aperiodic G i -subshift. Luckily, these objects always exist whenever the word problem of the group is decidable. Furthermore, in the class of recursively presented groups, non-empty effectively closed subshifts which are strongly aperiodic exist if and only if the word problem of the group is decidable. This is proven in [Jea15a] and [ABT15] and can be formally stated as follows. The only if part of this proof is a result by Jeandel [Jea15a] and is basically the fact that a strongly aperiodic SFT (or more generally, an effectively closed strongly aperiodic subshift) in a recursively presented group gives enough information to recursively enumerate the complement of the word problem of the group. Conversely, the existence part of the proof of Lemma 4.3 relies on a proof by Alon, Grytczuk, Haluszczak and Riordan [AGHR02] which uses Lovász local lemma to show that every finite regular graph of degree ∆ can be vertex-colored with at most (2e 16 + 1)∆ 2 colors in a way such that the sequence of colors in any non-intersecting path does not contain a square word. Using compactness arguments this result is extended to Cayley graphs Γ(G, S) of finitely generated groups where the bound takes the form 2 19 |S| 2 colors where |S| is the cardinality of a set of generators of G. One can also show that the set of square-free vertex-colorings of Γ(G, S) yields a strongly aperiodic subshift, which is thus non-empty if the alphabet has at least 2 19 |S| 2 symbols. In the case where G has decidable word problem, a Turing machine can construct a representation of the sequence of balls B(1 G , n) of the Cayley graph and enumerate a codification of all patterns containing a square colored path.
Adding up Lemma 4.2 and Lemma 4.3 gives us the following result.
Theorem 4.4. For any triple of infinite and finitely generated groups G 1 , G 2 and G 3 with decidable word problem, then G 1 × G 2 × G 3 admits a non-empty strongly aperiodic subshift of finite type.
Note that the hypothesis of having decidable word problem is necessary, if not, any finitely generated and recursively presented G i with undecidable word problem gives a counterexample by Lemma 4.3. On the other hand, to the best of the knowledge of the author, there are no known examples of a group of the form G 1 ×G 2 where both G i are infinite, finitely generated, have decidable word problem, and G 1 × G 2 does not admit a strongly aperiodic SFT. Therefore, it is possible that Theorem 4.4 can be improved in that direction.

A strongly aperiodic SFT in the Grigorchuk group
Here we exhibit a class of groups which admit strongly aperiodic SFTs. In particular, this class contains the Grigorchuk group [Gri85]. In order to present this result we need to recall the notion of commensurability. A result by Carroll and Penland [CP15] establishes that the group property of admitting a non-empty strongly aperiodic SFT is invariant under commensurability. In they article they say that a group G is weakly periodic if every non-empty SFT X ⊂ A G admits a periodic configuration, that is, there exists x ∈ X and g ∈ G \ {1 G } such that σ g (x) = x. In other words, a group G is weakly aperiodic if it does not admit a non-empty strongly aperiodic SFT. Finitely generated free groups are an example of weakly periodic groups [Pia08].
With this result in hand, we can show the following: Theorem 4.6. Let G be a finitely generated group with decidable word problem such that G is commensurable to G × G. Then G admits a non-empty strongly aperiodic subshift of finite type.
Proof. If G is finite, the result is immediate as X = {x ∈ {a, b} G | |x −1 (a)| = 1} is a non-empty strongly aperiodic subshift of finite type. We suppose from now on that G is infinite. If G is commensurable to G× G, then there exists H 1 ≤ G, H 2 ≤ G × G of finite index such that H 1 ∼ = H 2 . In particular, if we define H 1 = G × H 1 and H 2 = G × H 2 we get that H 1 ∼ = H 2 , H 1 is a finite index subgroup of G × G and H 2 is a finite index subgroup of G × G × G. Therefore G×G and G×G×G are commensurable. As commensurability is an equivalence relation in the class of groups we get that G is commensurable to G × G × G.
As G is infinite, finitely generated and has decidable word problem, Theorem 4.4 implies that G×G×G is not weakly periodic. It follows by Theorem 4.5 that G is not weakly periodic as well and thus admits a non empty strongly aperiodic SFT. These four actions can be represented in the Mealy automaton of Figure 3. Here an arrow of the form i → j means: "replace i by j and follow the arrow". To compute the image of x ∈ {0, 1} N under one of these involutions, start at the corresponding node and follow the arrow of the form x 0 → i. Replace x 0 by i and continue with x 1 and so on.
Besides the remarkable aforementioned properties, the Grigorchuk group is commensurable to its square.
Therefore, we can apply Theorem 4.6 to obtain: Corollary 4.8. There exists a non-empty strongly aperiodic subshift of finite type defined over the Grigorchuk group.