Distortion and the automorphism group of a shift

The set of automorphisms of a one-dimensional \shift $(X, \sigma)$ forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy \shiftno. In particular, we show that the Baumslag-Solitar groups ${\rm BS}(1,n)$ and all other groups that contain exponentially distorted elements cannot embed into ${\rm Aut}(X)$ when $h_{{\rm top}}(X)=0$. We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.


INTRODUCTION
If Σ is a finite alphabet and X ⊂ Σ Z is a closed set that is invariant under the left shift σ : Σ Z → Σ Z , then (X , σ) is called a subshift. The collection of homeomorphisms φ : X → X that commute with σ forms a group (under composition) called the automorphism group Aut(X ) of the shift (X , σ).
This group is always countable, but for many classical subshifts (including all mixing shifts of finite type) it has a complicated subgroup structure, containing isomorphic copies of all locally finite, residually finite groups, the fundamental group of any 2-manifold, the free group on two generators, and many other groups (see [10,2,13]).
On the other hand, for shifts with low complexity (see Section 2 for precise definitions), there are numerous restrictions that arise (see [5,6,7]). Theorems of this nature typically take the following form: suppose (X , σ) is a subshift with some dynamical assumption (such as minimality or transitivity) and suppose that the complexity function of (X , σ) grows more slowly than some explicitly chosen subexponential rate, then Aut(X ) has some particular algebraic property. Without these growth rate and dynamical assumptions, little is known about the algebraic structure of Aut(X ). It was asked in [7, Question 6.1] whether every countable group arises as the automorphism group of some minimal, zero entropy shift. We answer this question negatively, giving explicit countable groups that cannot embed. Moreover, we give an algebraic constraint on Aut(X ) that applies to any subshift with zero topological entropy (with no need for further assumptions on the dynamics).
To show how these constraints arise, we study the types of distortion that can arise (or not) in Aut(X ). For a finitely generated group G, an element g ∈ G is distorted with respect to a symmetric generating set S if the distance of its iterates to the identity grows sublinearly (with respect to iteration) in the word-length metric. A priori, this definition depends on the choice of a symmetric generating set S, but it is well-known that g is distorted with respect to one symmetric generating set if and only if it is with respect to every symmetric generating set. Thus we can refer to an element g as distorted without making explicit reference to the set S. Distortion can be quantified, depending on how slowly the distance of the iterates to the identity grows: we say that g is polynomially distorted if d S (e, g n ) = O(n 1/d ) (for some d ∈ N) and that g is exponentially distorted if d S (e, g n ) = O(log(n)).
A different notion of distortion is in terms of the range of the automorphism: we say that φ ∈ Aut(X ) is range distorted if the size of the shortest block-code defining φ n grows sublinearly in n. This idea is explored in [3], where it is shown that if φ is of infinite order and range distorted then the topological entropy of (X , φ), rather than that of shift (X , σ), is zero. The two notions of distortion are related: if G ⊂ Aut(X ) is a finitely generated subgroup and if φ ∈ G is distorted in G, then the automorphism φ is range distorted (see Proposition 3.4).
One of our main tools is the interplay between the level of distortion in Aut(X ) and the growth rate of the complexity function of (X , σ). We use this to study the algebraic structure of the group Aut(X ).
In [6], it is shown that for a minimal shift whose complexity grows at most polynomially, any finitely generated, torsion free subgroup of Aut(X , σ) is virtually nilpotent. For very low complexity systems, we improve on this result, showing that for any shift whose complexity function is o(n ((d +1)(d +2)/2)+2 ), any finitely generated, torsion free subgroup of the automorphism group is virtually d -step nilpotent (the precise statement is in Theorem 4.10). In particular, if the complexity is o(n 5 ), any finitely generated, torsion free subgroup of Aut(X , σ) is virtually abelian. In Theorem 4.4, we show that the growth rate of the complexity of (X , σ) provides further obstructions for an infinite nilpotent group G to embed into the automorphism group of a shift.
We would be remiss were we not to acknowledge that examples of non-abelian lattice actions as shift automorphisms are sorely lacking; in the setting of low complexity (zero entropy), we have none, and for positive entropy shifts, we can not rule out some of the simplest non-abelian groups. It is conceivable that few or no such actions exist. Even if this turns out to be the case, it is our hope that some of our results furnish the first steps towards non-existence proofs.
We conclude with several open questions, primarily on what sorts of restrictions can be placed on the automorphism group of a shift.

BACKGROUND ON SHIFTS
2.1. One-dimensional subshifts and automorphisms. We assume throughout that Σ is a finite set endowed with the discrete topology, and Σ Z is endowed with the product topology. For x ∈ Σ Z , we write x[n] ∈ Σ for the value of x at n ∈ Z.
The left shift σ : and is a homeomorphism from Σ Z to itself. The pair (X , σ) is a subshift, or just a shift when the context is clear, if X ⊂ Σ Z is a closed set that is invariant under the left shift σ : The system (X , σ) is said to be minimal if the orbit closure of any x ∈ X is all of X .
An automorphism of the shift (X , σ) is a homeomorphism φ : X → X such that φ•σ = σ•φ. The group of all automorphisms of (X , σ) is denoted Aut(X , σ), or simply Aut(X ) when σ is clear from the context.
A map φ : X → X is a sliding block code if there exists R ∈ N such that for any The least R such that this holds is called the range of φ.
By the Curtis-Hedlund-Lyndon Theorem [10], any automorphism φ : X → X of a shift (X , σ) is a sliding block code. In particular, Aut(X ) is always countable.

The language and complexity of a one-dimensional subshift. The words
L k (X ) of length k in X are defined to be the collection of all a 1 , . . . , a k ∈ Σ k such that there exist x ∈ X and m ∈ Z with x[m + i ] = a i for 1 ≤ i ≤ k. The length of a word w ∈ L(X ) is denoted by |w|. The language L(X ) = ∞ k=1 L k (X ) is defined to be the collection of all finite words.
A word w ∈ L(X ) is said to be right special (respectively, left special) if it can be extended in the language in at least two distinct ways to the right (respectively, to the left). Thus w is right special if |{x ∈ Σ : w x ∈ L(X )}| ≥ 2 and w is left special if |{x ∈ Σ : xw ∈ L(X )}| ≥ 2. A well-known consequence of the work of Morse and Hedlund [21] is that every infinite shift admits a right special word of length n for every n ≥ 1 (similarly for left special words). The complexity P X : N → N of the shift (X , σ) counts the number of words of length n in the language of X . Thus P X (n) = L n (X ) .
The exponential growth rate of the complexity is the topological entropy h top of the shift σ. Thus n .
This is equivalent to the usual definition of topological entropy using (n, ε)separated sets (see, for example [14]).

Two-dimensional subshifts.
With minor modifications, the previous notions may be extended to higher dimensions. For our needs dimension two suffices, and so we specialize to that case. The set of functions η : Z 2 → Σ is Σ Z 2 endowed with the product topology. The shift action of Z 2 on Σ Z 2 is given by (σ z (η))(·) := η(· − z) for every η ∈ Σ Z 2 , z ∈ Z 2 . Every σ z is a homeomorphism on Σ Z 2 . A two-dimensional shift is a closed subset X ⊂ Σ Z 2 invariant by the shift action. To avoid confusion with the one-dimensional case we denote by (X , σ| X , Z 2 ) the associated dynamical system. A function η ∈ Σ Z 2 is said to be vertically (resp. horizontally) periodic if it is a periodic point for σ (0,1) (resp. σ (1,0) ). We say that a subset A ⊂ Z 2 codes a subset B ⊂ Z 2 if for any η, θ ∈ X ⊂ Σ Z 2 coinciding on the set A (in other words, η| A = θ| A ) it follows that η and θ coincide on B (meaning that η| B = θ| B ).
We give a definition of the complexity function P X : {finite subsets of Z 2 } → N, which is analogous to that for one-dimensional shifts. Namely, for each finite set S ⊂ Z 2 the value P X (S) is defined to be the number of legal X colorings of the finite set S ⊂ Z 2 . If

SUBGROUPS OF THE AUTOMORPHISM GROUP
3.1. Group distortion.
where S (g ) denotes the length of the shortest presentation of g by elements of S (meaning the word length metric on the group 〈S 〉 generated by S with respect to the generating set S).
Note that since S (·) is subadditive, this limit exists by Fekete's Lemma. Furthermore, this definition also makes sense in a non-finitely generated group G. Also observe that any (positive or negative) power or root of a distorted element is still distorted.
An example of a distorted element is provided by the discrete Heisenberg group H, defined by One can check that for any n ∈ Z, we have that and so s is a distorted element of infinite order. In a similar way, for an automorphism φ the following limit, called the asymptotic range, exists (note that the sequence of ranges (range(φ n )) n∈N is subadditive). This can be interpreted as the average increase of the range along powers of φ. For instance, for the shift map σ on an infinite shift X , we trivially have that for n ≥ 1, range(σ n ) ≤ n. Since there always exists a right special word of every length, and so in particular of length 2n − 1, it follows that range(σ n ) = n and so range ∞ (σ) = 1.
It follows immediately from the definition that any power or root of a range distorted automorphism is still range distorted.
We check that if g ∈ G is group distorted, then it is also range distorted: If G is a finitely generated subgroup of Aut(X ) and g ∈ G is distorted, then g is also range distorted, and its topological entropy h top (g ) is null.
Proof. Let S denote a symmetric generating set for G. For all g 1 , g 2 ∈ Aut(X ), the range satisfies range(g 1 g 2 ) ≤ range(g 1 ) + range(g 2 ), and so it follows that for all m ∈ N range(g m ) ≤ S (g m )R S , since the element g is group distorted in G. Moreover, from the fact that g is range distorted it is not difficult to show that h top (g ) = 0. This is done, for example, in [3,Theorem 5.13].
However, we do not know if the converse holds, namely, if a range distorted element of Aut(X ) is a distortion element in the group Aut(X ).
A consequence of Proposition 3.4 is that for infinite X , the shift map σ is never distorted in Aut(X ). Of independent interest, since the center of the Heisenberg group is 〈s 〉, we have: Proof. Assume that (Z , T (s)) is expansive. Then it is conjugate to a subshift (X , σ) by [12]. Since s lies in the center of H, the conjugacy maps every element of T (H) into Aut(X ). Since T (s) is a distorted element, we have that σ is range distorted and hence X is finite. DEFINITION 3.6. For a finitely generated group G with generating set S, the element g ∈ G has exponential distortion if it has infinite order and there exists C > 0 such that for all sufficiently large m where S (·) denotes the word length of the element in the generating set S. The smallest such C satisfying this inequality (with the fixed generating set S) is Note that the property of an element having exponential distortion is independent of the generating set S, depending only on the algebraic properties of the group. However, the constant C depends on the choice of generators S.
We also say an element g of infinite order has polynomial distortion when- EXAMPLES 3.7. The following groups have elements with exponential distortion: • SL(k, Z) for any k ≥ 3 (see [15]).
To see this for the Baumslag-Solitar group BS(1, n) = 〈a, b : bab −1 = a n 〉 with n > 1, take the generators We show (Corollary 3.10) that BS(1, n) does not embed in the automorphism group of any shift of zero entropy.
An example of Hochman [11] gives a subshift of polynomial complexity with an automorphism of infinite order that is (polynomially) range distorted but the full automorphism group of the shift constructed is not explicit and so it is unknown (to us) if this automorphism is group distorted.
3.2. Entropy obstructions to embedding. For a subgroup G of Aut(X ) containing an element φ with exponential distortion, the two quantities R S and C S (φ) determine a lower bound on the possible entropy of the shift (X , σ). Recalling the definition of coding, this means that if η, θ ∈ U and η| H = θ| H , then Since range(φ m ) ≤ Rlog(m) for all m > 0, it follows that if 1 ≤ m ≤ 2 n and C 0 = Rlog(2), we have that Thus the horizontal segment of length 2 C 0 n + 1 centered at (0, 0) codes the vertical segment V . We deduce that the number of distinct vertical words of height 2 n that occur in U is at most P X (2 C 0 n + 1).
Suppose for contradiction that P X (2 C 0 n + 1) ≤ 2 n for some n ∈ N. Then, by the Morse-Hedlund Theorem [22] each vertical column is periodic with period at most 2 n . This in turn implies that φ has finite order, a contradiction of the hypothesis.
Thus we have P X (2 C 0 n + 1) > 2 n and hence

REMARK 3.9.
Recall that for a finite set S of generators for a subgroup G ⊂ Aut(X ), the range, R S , of S is defined to be R S = max g ∈S range(g ). Also we defined C S (φ) to be the smallest C such that S (φ m ) ≤ C log(m), for all m > 0.
For all g 1 , g 2 ∈ Aut(X ), the range satisfies range(g 1 g 2 ) ≤ range(g 1 ) + range(g 2 ). It follows that for all m ∈ N, Hence the number R := R S C S (φ) satisfies the hypothesis of Theorem 3.8 and we conclude that .
The quantity C S (φ) depends only on the algebraic properties of the abstract group G and not on the realizations of these automorphisms as sliding block codes, whereas R S depends only on the range of the sliding block code generators of S.
In a private communication, Hochman indicated how to modify the construction in [11] to obtain an infinite order, exponentially range distorted automorphism.
Recall that a group G is almost simple if every normal subgroup is either finite or has finite index. The Margulis normal subgroups theorem (see [19]) implies that many Lie group lattices are almost simple (including for example SL(n, Z) for n ≥ 3). COROLLARY 3.10. Let (X , σ) be a shift with zero entropy. Suppose G is group and some element g ∈ G has exponential distortion. Then if Φ : G → Aut(X ) is a homomorphism, the element Φ(g ) ∈ Aut(X ) has finite order. Moreover, if G is almost simple, then Φ(G) is a finite group.
Proof. If φ = Φ(g ) is not of finite order, then it is an element with exponential distortion in the subgroup Φ(G) of Aut(X ). Moreover, since the range is subadditive, there are a finite set S ⊂ Φ(G) and positive constants R S , C S , such that range(φ k ) ≤ R S C S log(k) for each k ≥ 1. (See Remark 3.9.) This assumption would contradict Theorem 3.8.
Suppose now that G is an almost simple group and Φ : G → Aut(X ) is a homomorphism. If g ∈ G has exponential distortion then, as above, φ = Φ(g ) has finite order. So the kernel K of Φ contains infinitely many distinct powers of g and, in particular, K is infinite. But since G is almost simple, this implies K has finite index and we conclude that Φ(G) ∼ = G/K is finite.
Since SL(k, Z), k ≥ 3, and the Baumslag-Solitar group BS(1, n) have elements which are exponentially distorted, Corollary 3.10 implies they are examples of finitely generated groups that do not embed into the automorphism group of any shift with zero entropy. In particular, this provides an answer to Question 6.1 of [7]. However, we are unable to give a positive entropy shift for which SL(k, Z), k ≥ 3, or BS(1, n) do not embed.
On a related note, if m > 1 and n > m, then BS(m, n) is not residually finite [20]. Thus if X is a mixing shift of finite type, then BS(m, n) does not embed in Aut(X ).

Periodicity in two dimensions.
We recall some results about two-dimensional shifts which we then use to describe properties of the automorphism group of a one-dimensional shift. [4]). Let η : Z 2 → Σ and suppose there exist n, k ∈ N such that P η (n, k) ≤ nk 2 . Then there exists (i , j ) ∈ Z 2 {(0, 0)} such that η(x + i , y + j ) = η(x, y) for all (x, y) ∈ Z 2 . LEMMA 4.2. Let (X , σ, Z 2 ) be a two-dimensional subshift such that each element is vertically periodic. Then there exists a constant T > 0 such that each element of X is fixed by σ (0,T ) .

THEOREM 4.1 (Cyr & Kra
Proof. Let Z be the collection of all the sequences along the vertical columns of elements in X . The set Z defines a one-dimensional subshift where each sequence is periodic.
If the subshift Z is infinite, its language contains arbitrarily long right special words. Taking an accumulation point, there exist two different sequences x, y ∈ Z sharing the same past. This is impossible because x and y are both periodic. Hence the set Z is finite. So a power T of the shift map is the identity on Z . This shows the lemma.

Complexity obstructions to embedding.
In this section, we show a subshift with an infinite order polynomially range distorted automorphism cannot have a sub-polynomial complexity. Then we deduce a restriction on the complexity of a shift which contains a nilpotent group in its automorphism group.
We start with a sufficient condition for an automorphism to be non-distorted: . This implies that range(σ j m ) ≥ | j |m, proving the lemma. n d +1 > 0. Recall that [11] provides an example of a subshift with polynomial complexity and an infinite order automorphism polynomially range distorted. Furthermore, the exponent may be arbitrairly small.
Proof. Let C 0 be a constant such that for any n ∈ N and all integers k ≤ n d , range(φ k ) ≤ C 0 n.
Consider the φ-spacetime U and let V be a rectangle of height n d and width 2n + 1 in Z 2 , with the horizontal base of V centered at (0, 0). Recall that an horizontal segment of length 2 range(φ k ) + 1 centered at the origin codes the point {(0, k)}.
Since range(φ k ) ≤ C 0 n, we have that r (n) ≤ 2C 0 n + 2n + 1 ≤ C n, where C = 2C 0 + 2 + 2. We conclude that there are at most P X (C n) possible colorings of the rectangle V .
Again letting P U (k, n) denote the complexity of the k × n rectangle in U, this remark implies that P U (n, n d ) ≤ P X (C n). We proceed by contradiction and assume that lim inf n P X (n)/n d +1 = 0. Since for each n, P X (C n/C ) ≤ P X (n), we also have lim inf n P U (n, n d )/n d +1 = 0. It follows that P U (n, n d ) < n d +1 /2 for infinitely many n ∈ N. By Theorem 4.1, we conclude that if x 0 ∈ X is a fixed aperiodic element of X , then φ i (x 0 ) = σ j (x 0 ) for some i > 0 and j ∈ Z. By Lemma 4.3, range(φ i m ) ≥ | j |·m for all m ∈ N. On the other hand, since φ is distorted we also have that lim range(φ k )/k = 0. These two properties can only be simultaneously true if j = 0. Therefore, for any aperiodic x 0 ∈ X , there exists i x 0 ∈ N such that φ i x 0 (x 0 ) = x 0 . Hence, the map φ is periodic on each aperiodic sequence of X .
Since the set of periodic sequences of a given period is finite and the automorphism φ has to preserve this set, the map φ is also periodic on each periodic sequence. By Lemma 4.2 applied to the φ-spacetime U, the automorphism φ has a finite order. But this contradicts the hypothesis that φ has infinite order.
Let us recall some basics on nilpotent groups. If G is a group and A, B ⊂ G, let [A, B ] denote the commutator subgroup, meaning the subgroup generated by {a −1 b −1 ab : a ∈ A, b ∈ B }. Given a group G, we inductively define the lower central series by setting G 1 = G and G k+1 = [G,G k ] for k > 0. If d is the least integer such that G d +1 is the trivial group {e}, then we say that G is d -step nilpotent, and we say that G is nilpotent if it is d -step nilpotent for some d ≥ 1.
We use a few standard facts about nilpotent groups: (1) Any subgroup of a finitely generated nilpotent group G is finitely generated. (2) The set of elements of finite order in a nilpotent group form a normal subgroup T , called the torsion subgroup. (3) A finitely generated torsion subgroup of a nilpotent group is finite.
We also use the following standard fact about commutators in any group (see [24,Equation 2.3b] for a more general statement and further references):

LEMMA 4.6. Suppose G is a finitely generated nilpotent group with torsion subgroup T and assume that the quotient G/T is d -step nilpotent with d ≥ 2.
Then there exists an element z ∈ G d of infinite order that is polynomially distorted. More precisely, there exists a finite set S ⊂ G such that Proof. We first claim that it suffices to prove the result when T is trivial. Namely, since T is normal and finite, for any z ∈ G, where S 0 is a set of generators for G/T , K is the order of T , and S is a set of generators of G containing T and a representative of each coset in S 0 . Hence it suffices to show that in the torsion free group G/T , there is an element zT ∈ (G/T ) d such that Moreover the element z ∈ G has infinite order as soon as zT is not T .
Thus we now assume that H = G/T is torsion free and d -step nilpotent. In particular, the group H d is nontrivial. Since it is generated by the elements  [a 1 , [a 2 , . . . , [a d −1 , a d  In particular, for any integers 1 ≤ q, 0 ≤ α ≤ q and 0 ≤ i < d , For an integer n ≥ 1, let q be the smallest integer such that q > n 1 d . Write n in base q as where 0 ≤ α i < q. Since S 0 (ab) ≤ S 0 (a)+ S 0 (b) for every a, b ∈ H , the inequality in (4.1) leads to We deduce the following corollary Proof. Let z ∈ G be the element guaranteed to exist by Lemma 4.6. If S ⊂ G is a finite set, for all n ∈ N we have The result follows from Theorem 4.4.

4.3.
The automorphism group for subshifts whose complexity is subpolynomial. For minimal shifts of polynomial growth, there are strong constraints on the automorphism group: THEOREM 4.8 (Cyr & Kra [6]). Suppose (X , σ) is a minimal shift and there exists ∈ N such that P X (n) = o(n +1 ). Then any finitely generated, torsion-free subgroup of Aut(X ) is a group of polynomial growth of degree at most .
For instance, if the Heisenberg group is embedded into the automorphism group of a minimal shift (X , σ), we must have at least lim sup n P X (n)/n 4 > 0. Using distortion, we obtain a better bound, and we start with an algebraic lemma on the growth rate of the nilpotent group: LEMMA 4.9. If G is a finitely generated, torsion free d -step nilpotent group for some d ≥ 2, then G has polynomial growth rate of degree at least d (d + 1)/2 + 1.
Proof. Letting Z (H ) denote the center of the group H , we inductively define a sequence of normal subgroups. Set Z 0 (G) = {1}. Given Z i (G), let π i : G → G/Z i (G) denote the quotient map and define By induction on i , it is easy to check that G d +1−i is a subgroup of Z i (G). By a result of Mal'cev [16,17], each quotient Z i +1 (G)/Z i (G) is torsion free. Hence, each group G d −i /G d −i +1 is torsion free, as it embeds into Z i +1 (G)/Z i (G). In particular, the rank of each G d −i /G d −i +1 is at least 1.
We next check that the rank of G/G 2 is at least 2. LetḠ denote the group G/G 3 . It is a nilpotent group of step at most 2 and the groupḠ/Ḡ 2 is abelian. We claim thatḠ/Ḡ 2 is not cyclic. If not, thenḠ/Ḡ 2 is generated by the coset xḠ 2 and soḠ is generated byḠ 2 and x. Since the generators commute (recall that G 2 lies in the center ofḠ), it follows thatḠ is abelian. However, this contradicts the assumption that G is d -step for some d ≥ 2. Therefore, G/G 2 has at least two independent generators, and so its rank is at least 2.
By the Bass-Guivarc'h formula [1,9], G has polynomial growth rate of degree k≥1 k rank(G k /G k+1 ), (4.2) where rank(G k /G k+1 ) is the torsion free rank of the abelian group G k /G k+1 . Since the rank of each G k /G k+1 , 1 ≤ k ≤ d is positive and rank(G 1 /G 2 ) ≥ 2, the lemma follows.
Recall that a group G is virtually nilpotent (of degree d ) if it contains a finite index (d -step) nilpotent subgroup. THEOREM 4.10. Let (X , σ) be an infinite minimal shift such that for some d ≥ 1 we have P X (n) = o(n (d +1)(d +2)/2+2 ). Then any finitely generated, torsion-free subgroup of Aut(X ) is virtually nilpotent of step at most d .
In particular, for an aperiodic minimal shift such that P X (n) = o(n 5 ), any finitely generated, torsion-free subgroup of Aut(X ) is virtually abelian.
Proof. Let G < Aut(X ) be a finitely generated, torsion-free subgroup of Aut(X ). Theorem 4.8 ensures that G has a polynomial growth with degree at most (d +1) (d + 2)/2 + 1. By Gromov's Theorem [8], G contains a nilpotent subgroup H with finite index. We proceed by contradiction and assume that H is a k-step nilpotent group for some k > d .
Assume first that 〈σ〉 ∩ H = {1}. Then the group Aut(X ) contains 〈σ〉 ⊕ H , and by Lemma 4.9 this group has polynomial growth of degree at least k(k + 1)/2 + 2. But this is a contradiction of Theorem 4.8.
Otherwise, we assume that 〈σ〉 ∩ H is not trivial. Then the group H /(〈σ〉 ∩ H ) is nilpotent. Let z ∈ H k be the element given by Lemma 4.6. Thus z is distorted and z ∉ 〈σ〉 ∩ H , since any element in 〈σ〉 is not distorted (see the computations in Section 3). It follows that H /(〈σ〉 ∩ H ) is k -step nilpotent for some k ≥ k. By Lemma 4.9, this group has polynomial growth of degree at least k(k + 1)/2 + 1. Since 〈σ〉 ∩ H is an infinite, finitely generated group, H has polynomial growth rate of degree at least k(k + 1)/2 + 2 (see [18,Proposition 2.5(d)] for instance). Again, this contradicts Theorem 4.8.
In fact one can extract from the proof a more general, but more technical, statement, relating the homogeneous dimension given by (4.2) to the step of any finitely generated, torsion-free subgroup of the automorphism group for an infinite minimal shift. Interest in the Heisenberg group in particular arises from Theorem 4.8. Consequently, Question 5.1 becomes most interesting if X is assumed to be minimal and have P X (n) = O(n d ) as we then have a dichotomy in the possible behaviors. If there exists a subshift such that the Heisenberg group embeds in its automorphism group, then Question 5.1 is resolved affirmatively. If no such system (X , σ) exists, then by Theorem 4.8 any finitely generated, torsion-free subgroup of Aut(X ) is virtually abelian, as the Heisenberg group is a subgroup of any finitely generated, torsion-free, nonabelian nilpotent group, resolving Question 5.1 negatively.

OPEN QUESTIONS
More generally we have the same question for higher dimensions: By Corollary 3.10, these groups do not embed into Aut(X ) for any shift X with entropy zero. We note that if BS(1, p) embeds in Aut(X , σ) for some subshift of finite type σ and some prime p, this would answer both Questions 3.4 and 3.5 of [2] which ask if some some automorphism of infinite order has an infinite chain of p th roots. If G ∼ = BS(1, p) is a subgroup of Aut(X ) and has generators a, b with relation b −1 ab = a p , then it is straightforward to show that c k := b k ab −k , k ≥ 0, satisfies c p k = c k−1 and c 0 = a, and so a has an infinite chain of p th roots.