2 Preliminaries

All necessary material concerning subshifts of finite type can be found in [Reference Lind and MarcusLM21]. This section mainly serves to introduce notation. Our natural numbers $\mathbb {N}$ start at $1$ and we denote the non-negative integers by $\mathbb {Z}_+$ . In this paper, a topological dynamical system $(X,f)$ consists of a compact metrizable state space X and a continuous self map $f:X \to X$ . The full shift over the alphabet $\mathcal {S}$ is the topological dynamical system $(\mathcal {S}^{\mathbb {Z}},\sigma )$ , where $\sigma : \mathcal {S}^{\mathbb {Z}} \to \mathcal {S}^{\mathbb {Z}}, \sigma (x)_i = x_{i+1}$ is the left shift, $\mathcal {S}$ is endowed with the discrete topology, and $\mathcal {S}^{\mathbb {Z}}$ with the product topology. The elements of $\mathcal {S}^{\mathbb {Z}}$ are called configurations. A subshift is a subsystem of a full shift. A subshift X is of finite type if there is a finite set of forbidden words such that X consists of all configurations not containing any of these words. An isomorphism between two topological dynamical systems $(X,f)$ and $(Y,g)$ is a homeomorphism $\varphi : X \to Y$ such that $\varphi \circ f = g \circ \varphi $ . We denote the group of automorphisms of the topological dynamical system $(X,f)$ by $\operatorname {\mathrm {Aut}}(X,f)$ . Following [Reference Hartman, Kra and SchmiedingHKS22], we call $\operatorname {\mathrm {Aut}}^{\infty }(X,f) = \bigcup _{k \in \mathbb {N}} \operatorname {\mathrm {Aut}}(X,f^k)$ the stabilized automorphism group of $(X,f)$ .

If $(X,\sigma )$ is a subshift over the alphabet $\mathcal {S}$ , then we can consider $(X,\sigma ^k)$ as a subshift over the alphabet $\mathcal {S}^k$ . We say that two subshifts $(X,\sigma )$ and $(Y,\sigma )$ are eventually conjugate if and only if $(X,\sigma ^k)$ and $(Y,\sigma ^k)$ are topologically conjugate for sufficiently large k.

Our graphs are tuples $\Gamma =(V,E,\mathrm {i}_\Gamma ,\mathrm {t}_\Gamma )$ consisting of a vertex set $V=V(\Gamma )$ , an edge set $E=E(\Gamma )$ , together with two maps $\mathrm {i}_\Gamma ,\mathrm {t}_\Gamma : E \to V$ mapping edges to their initial and terminal vertices, respectively. In this context, it will be notationally convenient to consider matrices whose rows and columns are indexed by an arbitrary finite set M not necessarily of the form $\{1,\ldots ,m\}$ . If $\Gamma =(V,E,\mathrm {i}_\Gamma ,\mathrm {t}_\Gamma )$ is a graph, its adjacency matrix is the $V \times V$ matrix A with

$$ \begin{align*}A_{i,j}=|\{e \in E \mid \mathrm{i}_\Gamma(e)=i, \mathrm{t}_\Gamma(e)=j\}|.\end{align*} $$

We call a matrix essential if it has no zero rows or columns and we call a graph essential if its adjacency matrix is essential. A non-negative square matrix A is called irreducible if for each pair of indices $i,j$ , there is $k \in \mathbb {N}$ such that $A^k_{i,j}>0$ . If k can be chosen independently of $i,j$ , we call A primitive. Every essential graph $\Gamma $ defines an edge shift $X_\Gamma $ . This SFT over the alphabet $E(\Gamma )$ consists of all biinfinite sequences of edges $(e_k)_{k \in \mathbb {Z}}$ with $\mathrm {i}_\Gamma (e_{k+1})=\mathrm {t}_\Gamma (e_{k})$ for all $k \in \mathbb {Z}$ .

Up to renaming the edges, the graph, and hence the edge shift, is uniquely determined by the adjacency matrix. Hence, for a graph $\Gamma $ with adjacency matrix A, we denote by $X_A$ the edge shift of $\Gamma $ . For $A \in \mathbb {Z}_+^{V \times V}$ , we can canonically label the edges in the graph defined by A from i to j by $(i,j,1),(i,j,2),\ldots ,(i,j,A_{i,j}) \in V \times V \times \mathbb {N}$ .

An automorphism of a graph is a pair of bijections $\theta _V: V \to V$ and $\theta _E: E \to E$ such that $\theta _V \circ \mathrm {i}_\Gamma = \mathrm {i}_\Gamma \circ \theta _E$ and $\theta _V \circ \mathrm {t}_\Gamma = \mathrm {t}_\Gamma \circ \theta _E$ . For essential graphs, the map on the edges uniquely determines the map on the vertices. Every graph automorphism induces an automorphism of the corresponding edge shift. If a group G is acting by graph automorphisms on a graph $\Gamma $ , we sometimes do not write the action explicitly and simply write $gi$ or $ge$ for the image of the vertex i or the edge e, respectively, under the action of $g \in G$ .

We are interested in various objects associated to a subshift $(X,\sigma )$ . We denote by

$$ \begin{align*} \operatorname{\mathrm{Per}}_k(X,\sigma):= \{x \in X \mid \sigma^k(x)=x\} \end{align*} $$

the set of k-periodic points of X. The cardinalities of these sets can be conveniently encoded in the zeta function

$$ \begin{align*} \zeta_X(t) := \exp\bigg(\sum_{k=1}^\infty \frac{|\!\operatorname{\mathrm{Per}}_k(X,\sigma)|}{k} t^k\bigg). \end{align*} $$

Now, let $A \in \mathbb {Z}_+^{V \times V}$ be essential. The dimension group $D_A$ of $X_A$ is defined as the direct limit of the diagram

$$ \begin{align*} \mathbb{Z}^V \xrightarrow{A} \mathbb{Z}^V \xrightarrow{A} \mathbb{Z}^V \xrightarrow{A}\cdots. \end{align*} $$

Formally, we define $D_A$ as $\mathbb {Z}^V \times \mathbb {Z} / \sim $ , where $\sim $ is the equivalence relation defined by $(x,n) \sim (xA,n+1)$ . One can further endow this group with an order and the automorphism induced by the shift map to obtain the so-called dimension triple, but we do not need this additional structure here.

If A is an $M \times M$ matrix and B is a $K \times K$ matrix, the Kronecker product $A \otimes B$ of A and B is the $(M \times K) \times (M \times K)$ matrix with entries $(A \otimes B)_{(i,k),(j,\ell )} = A_{i,j}B_{k,\ell }$ . If M and K are totally ordered and $M \times K$ is endowed with the lexicographic ordering, then this corresponds to the usual Kronecker product. For a set M, we denote by the vector in $\mathbb {Z}^M$ whose entries are all equal to $1$ and we denote by the $M \times M$ matrix which equals $1$ in every entry. The $M \times M$ identity matrix is denoted by $I_{M \times M}$ .

3 Finite groups acting freely on SFTs

In the following, let G be a finite group whose identity element we denote by $1_G$ . We start by introducing our central objects of interest.

The following proposition appears in [Reference Adler, Kitchens and MarcusAKM85, Observation 1-3] and is credited without reference to Franks in [Reference Boyle, Lind and RudolphBLR88], proofs can be found, e.g., in [Reference Boyle, Lind and RudolphBLR88, Proposition 2.9] or [Reference SaloSal19, Proof of Theorem 7.2].

Every action of G on an SFT X induces a G-action on the dimension group of this SFT. This is called the dimension group representation of the action of G on X. If $X=X_\Gamma $ is an edge shift and the action on $X_\Gamma $ is induced by an action of G on the graph $\Gamma $ , the dimension group representation can be constructed as follows. Let V be the vertex set and let A be the adjacency matrix of $\Gamma $ . Since G acts by graph automorphisms, we have $A_{gi,gj}=A_{i,j}$ for all $g \in G$ and $i,j \in V$ . Our group G acts on $\mathbb {Z}^V$ by $w \mapsto gw$ with $(gw)_{i} = w_{g^{-1}i}$ . This action extends to an action of G on $D_A$ simply by

$$ \begin{align*} [(w,n)] \mapsto [(gw,n)]. \end{align*} $$

This action is well defined since $g(wA)_i = \sum _{j \in V} w_j A_{j,g^{-1}i} = \sum _{j \in V} w_{g^{-1}j}A_{j,i} = ((gw)A)_i$ . Alternatively, one can use the fact that the dimension group construction is functorial in the sense that every isomorphism between SFTs induces a unique isomorphism of the corresponding dimension groups. This isomorphism can be either defined directly using Krieger’s internal construction of the dimension group from [Reference KriegerKri80] using equivalence classes of rays as explained, e.g., in [Reference Boyle, Lind and RudolphBLR88, §6]. Otherwise, one uses the fact that each isomorphism can be decomposed into a series of state splittings and amalgamations. Between edge shifts, these can in turn be represented as strong shift equivalence matrix pairs. To show well-definedness of the resulting isomorphism, that is, independence from the decomposition into splittings and amalgamations, one then uses Wagoner’s complex. This approach can be found in [Reference WagonerWag92, §2] and [Reference Lind and MarcusLM21, Theorem 7.5.7]. More concretely, if $A=RS$ and $B=SR$ is an elementary strong shift equivalence between matrices A and B, then an element $[(w,n)]$ in the dimension group of A is simply mapped to $[(wR,n)]$ , an element in the dimension group of B. Again, this is well defined since $wAR=wRB$ . In the case of a graph automorphism as above, we simply have $A=R_g R_{g^{-1}}A=R_{g^{-1}}A R_g$ , where $R_g$ is the $V \times V$ permutation matrix with $(R_g)_{i,j}=1$ if and only if $j=gi$ , so our action is given by $[(w,n)] \mapsto [(wR_g,n)]$ , which agrees with the first definition since $(wR_g)_{i}=w_{g^{-1}i}=(gw)_i$ .

Applying the explicit construction of the dimension group representation for edge shifts, we obtain the following characterization of inertness, which appeared as a remark in [Reference FiebigFie93, p. 497].

Proposition 3.6. (Inertness of free graph automorphisms)

Let $A \in \mathbb {Z}_+^{V \times V}$ be essential and let $(X_A,\sigma ,\alpha )$ be a free G-SFT induced by an action of G on the graph defined by A. Then, $(X_A,\sigma ,\alpha )$ is inert if and only if there is $\ell \in \mathbb {N}$ such that for all $g \in G,i,j \in V$ , we have

$$ \begin{align*} (A^\ell)_{i,j}=(A^{\ell})_{i,gj}. \end{align*} $$

Proof. Let A be a $V \times V$ matrix. Let $[(w,n)]$ be an element in the dimension group of $(X_A,\sigma )$ . By the explicit construction of the dimension group representation, a group element g fixes $[(w,n)]$ if and only if there is $\ell \in \mathbb {N}$ such that $w A^\ell = (gw) A^\ell =g(w A^\ell )$ . However, this is equivalent to

$$\begin{align*}(w A^\ell )_{gj} = (g(wA^\ell))_{gj}=(wA^\ell)_{j} \end{align*}$$

for all $j \in V$ . This holds for all $w \in \mathbb {Z}^V$ if it holds for all standard basis vectors, which in turn is equivalent to

$$\begin{align*}(A^\ell)_{i,j}=(A^\ell)_{i,g j} \end{align*}$$

for every $i,j \in V$ .

4 Matrices over the integral group ring and G-extensions of SFTs

In this section, we look at G-SFTs from the opposite direction. Instead of considering an SFT together with a free G-action and obtaining the orbit space as a quotient, we start with an orbit space, which is itself an SFT, and construct an SFT as a G-extension. These extensions can be defined by square matrices over the integer group ring. Our goal is then to show that every free G-SFT can be represented in that way. We use a very concrete definition of G-extensions. For their connection to cocycles and the general theory of extensions of dynamical systems, see [Reference Boyle and SchmiedingBS17].

By $\mathbb {Z}[G]$ , we denote the integer group ring over G consisting of all formal linear combinations of elements in G with integer coefficients. The set $\mathbb {Z}_+[G]$ consists of all elements in $\mathbb {Z}[G]$ whose coefficients are non-negative.

For $h \in G$ , let $\pi _h$ be the map from $\mathbb {Z}[G] \to \mathbb {Z}$ defined by $\pi _h(\sum _{g \in G} a_g g):=a_h$ . When we apply this map entry wise, we also get a map $\pi _h: \mathbb {Z}[G]^{V \times V} \to \mathbb {Z}^{V \times V}$ and we can write every matrix $B \in \mathbb {Z}[G]^{V \times V}$ as $B = \sum _{h \in G} \pi _h(B) h$ .

Let $B \in \mathbb {Z}_+[G]^{V \times V}$ . We define two matrices $\mathcal {A}(B) \in \mathbb {Z}_+^{V \times V}$ and $\mathcal {E}(B) \in \mathbb {Z}_+^{(V \times G)\times (V \times G)}$ as follows:

$$ \begin{align*} \mathcal{A}(B) &:=\sum_{g \in G} \pi_g(B), \\ \mathcal{E}(B)_{(i,g),(j,h)} &:=\pi_{g^{-1}h}(B)_{i,j}. \end{align*} $$

We call $\mathcal {A}(B)$ the augmentation of B and $\mathcal {E}(B)$ the extension defined by B.

Using the Kronecker product, we can write this more concisely. Let $P_g$ be the permutation matrix defined by

$$ \begin{align*} (P_g)_{h,k} = \begin{cases} 1 &\text{ if } k=hg, \\ 0 &\text{ otherwise}. \end{cases} \end{align*} $$

Then,

$$\begin{align*}\mathcal{E}(B) = \sum_{g \in G} \pi_g(B) \otimes P_g. \end{align*}$$

We get an action $\theta _B$ of G on the graph defined by $\mathcal {E}(B)$ as follows. The automorphism $\theta _B(g)$ acts on the vertex set $V \times G$ of this graph simply by $\theta _B(g)(i,h)=(i,gh)$ . We can extend this map to the edges, and thus define a graph automorphism, since

$$\begin{align*}\mathcal{E}(B)_{(i,rg),(j,rh)} =\pi_{g^{-1}h}(B)_{i,j}=\mathcal{E}(B)_{(i,g),(j,h)} \quad \text{for every } r \in G. \end{align*}$$

Therefore, we can define that $\theta _B(r)$ maps the kth edge between $(i,g)$ and $(j,h)$ to the kth edge between $(i,rg)$ and $(j,rh)$ . This action by graph automorphisms immediately extends to an action $\beta _B$ of G on the edge shift $X_{\mathcal {E}(B)}$ .

The graphs defined by B, $\mathcal {A}(B)$ , and $\mathcal {E}(B)$ are related as follows: B defines a graph $\Gamma $ with edge labels in G; the graph defined by $\mathcal {A}(B)$ simply forgets about the edge labels; and $\mathcal {E}(B)$ defines a covering graph or more specifically a $|G|$ -lift of $\Gamma $ .

Example 4.1. (Augmentation and extension)

Consider $\mathbb {Z}/2\mathbb {Z}=\{e,g\}$ , where e is the identity element, and

$$\begin{align*}B=\begin{pmatrix}g & e \\ g &0\end{pmatrix}. \end{align*}$$

The labeled graph corresponding to B is depicted on the left in Figure 2. We have

$$ \begin{align*} \mathcal{A}(B)&=\begin{pmatrix} 1 & 1 \\ 1 &0 \end{pmatrix},\\ \mathcal{E}(B)&=\begin{pmatrix} 0 & 1 \\ 0 &0 \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 &1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 1 &0 \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 1 &0 \end{pmatrix} \\ &= \begin{pmatrix}\begin{array}{@{}c|c@{}}\begin{matrix} 0 & 1 \\ 1 &0 \end{matrix} &\begin{matrix} 1 & 0 \\ 0 &1 \end{matrix} \\ \hline \begin{matrix} 0 & 1 \\ 1 &0 \end{matrix} &\begin{matrix} 0 & 0 \\ 0 &0 \end{matrix}\end{array}\end{pmatrix}. \end{align*} $$

Figure 2

A $\mathbb {Z}/2\mathbb {Z}$ extension of the golden mean shift.

There is a canonical factor map from $X_{\mathcal {E}(B)}$ to $X_{\mathcal {A}(B)}$ induced by the graph homomorphism from $\Gamma _{\mathcal {E}(B)}$ to $\Gamma _{\mathcal {A}(B)}$ that maps the $\ell $ th edge from $(i,g)$ to $(j,h)$ onto the $\ell $ th edge from i to j labeled by $g^{-1} h$ . We now want to describe the preimages of $(e_k)_{k \in \mathbb {Z}} \in X_{\mathcal {A}(B)}$ under this factor map. To do so, we have to introduce slightly more notation. We may relabel the edges from i to j in the graph defined by $\mathcal {A}(B)$ by $(i,j,t,m)$ with $t \in G$ and $m \in \{1,\ldots ,\pi _t(B_{i,j})\}$ . The preimages of $(i_k,j_k,t_k,m_k)_{k \in \mathbb {Z}}$ under the factor map described above are of the form

$$ \begin{align*}((i_k,g_k),(j_k,h_k),m_k)_{k \in \mathbb{Z}},\end{align*} $$

where $g_0$ is any element in G and where $g_{k+1}=h_k=g_k t_k$ . These preimages are precisely the G-orbits in $X_{\mathcal {E}(B)}$ of the action $\beta _B$ . Therefore, $(X_{\mathcal {A}(B)},\sigma )$ is conjugate to the orbit space $(X_{\mathcal {E}(B)}/\beta _B,\sigma )$ .

Parry showed in [Reference ParryPar91, Appendix II] and further unpublished work, see [Reference Boyle and SullivanBS05, §2.7], that every free G-SFT is isomorphic to a G-extension defined by a matrix over $\mathbb {Z}_+[G]$ . This construction is sketched in [Reference Boyle and SullivanBS05, §2]. For completeness, we give a short explicit proof using the notation introduced above in the next proposition.

Proposition 4.2. (Representing free G-SFTs by matrices over $\mathbb {Z}_+[G]$ )

For every free G-SFT $(X,\sigma ,\alpha )$ , we can find $B \in \mathbb {Z}_+[G]$ such that

$$\begin{align*}(X,\sigma,\alpha) \cong (X_{\mathcal{E}(B)},\sigma,\beta_B). \end{align*}$$

The operations of forming $\mathcal {A}(B)$ and $\mathcal {E}(B)$ behave nicely with respect to matrix powers as the following lemma shows.

Lemma 4.3. (Compatibility of matrix multiplication with augmentation and extension)

For $B \in \mathbb {Z}_+[G]^{V \times V}$ and $k \in \mathbb {N}$ , we have

$$ \begin{align*} \mathcal{A}(B^k)&=\mathcal{A}(B)^k,\\ \mathcal{E}(B^k)&=\mathcal{E}(B)^k. \end{align*} $$

Proof. This can be shown by induction since

$$ \begin{align*} \mathcal{A}(B^{k+1})_{i,j} &= \sum_{g \in G} \sum_{\ell \in V} \pi_g( (B^k)_{i,\ell}B_{\ell,j})\\ &= \sum_{\ell\in V} \sum_{g \in G} \sum_{h \in G} \pi_h( (B^k)_{i,\ell}) \pi_{h^{-1}g}(B_{\ell,j}) \\ &= \sum_{\ell \in V} \sum_{h \in G} \pi_h( (B^k)_{i,\ell}) \sum_{g \in G} \pi_{g}(B_{\ell,j})\\ &= \sum_{\ell \in V} \mathcal{A}(B)^k_{i,\ell} \mathcal{A}(B)_{\ell,j}= \mathcal{A}(B)^{k+1}_{i,j}. \end{align*} $$

Similarly,

$$ \begin{align*} \begin{aligned} \mathcal{E}(B^{k+1})_{(i,s),(j,t)} &= \pi_{s^{-1}t} (B^{k+1}_{i,j}) \\ &=\sum_{\ell \in V} \pi_{s^{-1}t} (B^{k}_{i,\ell} B_{\ell,j})\\ &=\sum_{\ell \in V} \sum_{g \in G} \pi_{s^{-1}g}(B_{i,\ell}^k) \pi_{g^{-1}t}(B_{\ell,j})\\ &=\sum_{\ell \in V} \sum_{g \in G}\mathcal{E}(B)^k_{(i,s),(\ell,g)} \mathcal{E}(B)_{(\ell,g),(j,t)} \\ &= \mathcal{E}(B)^{k+1}_{(i,s),(j,t)}. \end{aligned}\\[-38pt] \end{align*} $$

The usual definition of shift equivalence due to Williams [Reference WilliamsWil73] for integer matrices carries over to the case of matrices over rings or subsets of rings, see [Reference Boyle and SchmiedingBS17].

In his seminal paper [Reference WilliamsWil73], Williams showed that strong shift equivalence of non-negative integer matrices is equivalent to conjugacy of the corresponding edge shifts. From this, it immediately follows that shift equivalence implies eventual conjugacy. This converse direction, that eventual conjugacy of edge shifts implies shift equivalence of the defining matrices, was shown by Kim and Roush in [Reference Kim and RoushKR79, Theorem 3.3].

The same theorems hold for matrices over $\mathbb {Z}_+[G]$ and free G-SFTs. This was shown by Boyle and Sullivan [Reference Boyle and SullivanBS05, Proposition 2.7.1], who attribute the result to a personal communication with Parry, and respectively Boyle and Schmieding [Reference Boyle and SchmiedingBS17, Proposition B.11].

5 A characterization of free inert G-SFTs

In this section, we collect the results about inert free G-SFTs which we need in the proof of our main result. An important role is played by the element $u_G : = \sum _{g \in G} g \in \mathbb {Z}_+[G]$ .

Proof. (i) $\Leftrightarrow $ (ii): this is a direct consequence of Lemma 4.3 and Proposition 3.6.

(ii) $\Leftrightarrow $ (iii): this equivalence follows directly from $\mathcal {E}(B^\ell )_{(i,s),(j,t)} = \pi _{s^{-1}t}(B^\ell )_{i,j}$ as follows. Using this equation, we can reformulate (1) as: for every $g,h_1,h_2 \in G$ , we have $\pi _{h_1^{-1}h_2}(B^\ell )=\pi _{h_1^{-1}gh_2}(B^\ell )$ , which in turn is equivalent to $\pi _g(B^\ell )=\pi _{1_G}(B^\ell )$ for all $g \in G$ . However, this is nothing else than $B^\ell \in u_G \mathbb {Z}_+^{V \times V}$ .

(iii) $\Rightarrow $ (iv): let $\ell \in \mathbb {N}$ such that $B^\ell \in u_G \mathbb {Z}_+^{V \times V}$ . Then,

Furthermore,

$$ \begin{align*} \sum_{g \in G}\pi_{g}(B) \pi_{1_G}(B^{\ell}) &= \sum_{g \in G} \pi_{g}(B)\pi_{g^{-1}}(B^{\ell}) = \pi_{1_G}(B^{\ell+1})\\ &= \sum_{g \in G} \pi_{g^{-1}}(B^{\ell})\pi_{g}(B) \\ &= \pi_{1_G}(B^{\ell})\sum_{g \in G}\pi_{g}(B). \end{align*} $$

Setting and , we therefore have

Therefore, the pair $(R,S)$ defines a shift equivalence with lag $\ell $ between $\mathcal {A}(B)$ and $\mathcal {E}(B)$ .

(iv) $\Rightarrow $ (v): the zeta function is an invariant of shift equivalence, see, e.g., [Reference Lind and MarcusLM21, Corollary 7.4.12].

(v) $\Rightarrow $ (ii): this result as well as the converse direction are due to Fiebig [Reference FiebigFie93, Theorem B]. We give a simplified proof, since this allows us to also get a quantitative bound on the lag of the shift equivalence in condition (iii). Let $(w_g)_{g \in G}$ be an orthogonal basis of $\mathbb {C}^G$ such that . Let F be the $G \times G$ matrix with columns $(w_g)_{g \in G}$ . For every $g \in G$ , we have $P_g w_{1_G} = w_{1_G}$ and $w_{1_G}^\top P_g = w_{1_G}^\top $ , and hence,

$$ \begin{align*} F^{-1} P_g F = \begin{pmatrix} \begin{array}{@{}c|c@{}}1 & 0 \\ \hline 0 & Q_g\end{array} \end{pmatrix} \end{align*} $$

for some matrix $Q_g \in \mathbb {C}^{\tilde {G} \times \tilde {G}}$ with $\tilde {G} = G \setminus \{1_G\}$ . Therefore,

$$ \begin{align*} (I_{V \times V} \otimes F)^{-1} \mathcal{E}(B) (I_{V \times V} \otimes F) &= \sum_{g \in G} \pi_g(B) \otimes (F^{-1} P_g F) \\ &= \sum_{g \in G} \pi_g(B) \otimes \begin{pmatrix} \begin{array}{@{}c|c@{}}1 & 0 \\ \hline 0 & Q_g\end{array} \end{pmatrix}. \end{align*} $$

Calculating the characteristic polynomial of the resulting block diagonal matrix gives $\chi _{\mathcal {E}(B)}(t) = \chi _{\mathcal {A}(B)}(t) \chi _{Q}(t)$ , where $Q:= \sum _{g \in G} \pi _g(B) \otimes Q_g$ . Since the characteristic polynomials of $\mathcal {E}(B)$ and $\mathcal {A}(B)$ agree with the inverse of the zeta function up to a power of t, we get $\chi _{Q}(t)=t^k$ , where k is the size of Q. Hence, Q is nilpotent and we get

$$ \begin{align*} (I_{V \times V} \otimes F)^{-1} \mathcal{E}(B^k) (I \otimes F) &= \mathcal{A}(B^k) \otimes \begin{pmatrix} \begin{array}{@{}c|c@{}}1 & 0 \\ \hline 0 & 0\end{array} \end{pmatrix} \end{align*} $$

and, hence,

In other words, condition (ii) is satisfied.

As a last ingredient, we also need the fact that for square matrices $B,C$ over the integer group ring whose entries eventually lie in $u_G\mathbb {Z}_+$ , we can lift an SE between $\mathcal {A}(B)$ and $\mathcal {A}(C)$ to an SE between $\mathcal {E}(B)$ and $\mathcal {E}(C)$ . This was shown by Boyle and Schmieding in [Reference Boyle and SchmiedingBS17]. The key observation is that for a matrix B with entries in $\mathbb {Z}_+[G]$ , we have $u_G B = u_G \mathcal {A}(B)$ .

6 Proof of the main theorem

With the preparation from the previous sections, we are now ready to prove our main theorems.

7 A theorem of Kim and Roush

Theorem 5.2 gives a new way to interpret a theorem by Kim and Roush from [Reference Kim and RoushKR97]. Namely, in our language, their theorem characterizes the existence of an inert $\mathbb {Z}/p\mathbb {Z}$ extension of a given mixing SFT.

Recall that for a subshift X, we denote by $\operatorname {\mathrm {Per}}_k(X,\sigma ) = \{x \in X \mid \sigma ^k(x)=x\}$ the $\sigma $ -periodic points in X with period k. Denote by $\widetilde {\operatorname {\mathrm {Per}}}_k(X,\sigma )$ the set of $\sigma $ -periodic points in X with minimal period k, that is, $\widetilde {\operatorname {\mathrm {Per}}}_k(X,\sigma ) := \operatorname {\mathrm {Per}}_k(X,\sigma ) \setminus \bigcup _{\ell <k} \operatorname {\mathrm {Per}}_\ell (X,\sigma )$ . Endow $\operatorname {\mathrm {Per}}(X) := \bigcup _{k \in \mathbb {N}} \operatorname {\mathrm {Per}}_k(X,\sigma )$ with the discrete topology.

The Boyle–Handelmann condition first appeared in a special case in [Reference Boyle and HandelmanBH91, 6.2 ‘Degrees example’]. Its necessity follows directly from the invariance of the zeta function under shift equivalence. In [Reference Kim and RoushKR97, Lemma 2.2], Kim and Roush showed that it can be formulated arithmetically as in condition (iii). The main work in the proof of Kim and Roush goes into (iii) $\Rightarrow $ (i), which is an impressive technical tour de force using the polynomial representation of SFTs.

Surprisingly, this theorem is basically everything that is known about the existence of inert extensions and actions. A few other results on inert actions can be found in [Reference Boyle and ChuysurichayBC18, Reference LongLon09].

8 Shift equivalence over $\mathbb {Z}_+[G]$ and equivariant flow equivalence

In [Reference Boyle, Carlsen and EilersBCE20], Boyle, Carlsen, and Eilers showed the following theorem as an application of their characterization of G-equivariant flow equivalence. For the somewhat lengthy definition of G-equivariant flow equivalence, see, e.g., [Reference Boyle and SullivanBS05, §2]. We only need here that it is an equivalence relation between subshifts, which is weaker than topological conjugacy and can algebraically be characterized as in Theorem 8.2.

We will show in Theorem 8.6 that at least for cyclic groups and irreducible free G-SFTs, equivariant shift equivalence already implies equivariant flow equivalence. As a consequence, we obtain Corollary 8.8 as a generalization of Theorem 8.1. This echos the recent result by Boyle [Reference BoyleBoy25] that shift equivalence implies flow equivalence between SFTs in the ordinary non-equivariant setting even without the irreducibility assumption.

A complete set of invariants for G-invariant flow equivalence in the irreducible case was given by Boyle and Sullivan already in [Reference Boyle and SullivanBS05].

Their main result is the following theorem. All relevant notions are briefly defined afterwards. For a slightly simplified version treating the case of irreducible extensions, see [Reference Boyle and ChuysurichayBC18, Theorem 8.5].

Here are the necessary definitions to make sense of this theorem. For a matrix $A \in \mathbb {Z}[G]^{V \times V}$ and $i \in V$ , define $W_i(A)$ as the subgroup of G consisting of elements of the form $\{g \in G \mid \text { there exists } n \in \mathbb {N}: \pi _g(A^n_{i,i})>0\}$ . These are all elements of G which appear as products over the labels along a cycle based at i in the graph defined by A. We call such a product the weight of the cycle. The weight class of A is then defined as the conjugacy class of $W_i(A)$ and this conjugacy class is independent of the choice of i. Recall (see, e.g., [Reference Boyle and SullivanBS05, Definition 3.1]) that a square matrix $A \in \mathbb {Z}_+[G]^{V \times V}$ is irreducible if for every $i,j \in V$ , there is $k \in \mathbb {N}$ such that $A^k_{i,j} \neq 0$ . For a matrix $A \in \mathbb {Z}[G]^{n \times n}$ , we define $A_\infty $ as the $\mathbb {N} \times \mathbb {N}$ matrix with

$$ \begin{align*} (A_\infty)_{i,j} =\begin{cases} A_{i,j} &\text{ if } i,j \leq n,\\ 0 &\text{ if } \max(i,j)>n,\; i\neq j, \\ 1 &\text{ if } \max(i,j)>n,\; i=j. \end{cases} \end{align*} $$

In other words, we take the $\mathbb {N} \times \mathbb {N}$ identity matrix and replace the upper left $n \times n$ block by A. For $z \in \mathbb {Z}[G], i,j \in \mathbb {N}, i \neq j$ , we denote by $E_{i,j}(z)$ the $\mathbb {N} \times \mathbb {N}$ matrix which agrees with the identity matrix everywhere besides the entry at position $i,j$ , where it equals z. We say that two square matrices A and B over $\mathbb {Z}_+[G]$ are elementarily positively equivalent if there are $g \in G$ and $i,j \in \mathbb {N}$ with $\pi _g(A_{i,j})>0$ and $(I-B)_\infty =E_{i,j}(g)(I-A)_\infty $ or $(I-B)_\infty =(I-A)_\infty E_{i,j}(g)$ . We call the equivalence relation on square matrices over $\mathbb {Z}_+[G]$ generated by this notion positive equivalence. The stabilized general linear group over $\mathbb {Z}[G]$ is defined as $\operatorname {\mathrm {GL}}_\infty (\mathbb {Z}[G]):= \{G_\infty \mid G \in \operatorname {\mathrm {GL}}_n(\mathbb {Z}[G])$ , $n \in \mathbb {N}\}$ . The subgroup generated by the elementary matrices $\{E_{i,j}(z) \mid z \in \mathbb {Z}[G],\; i,j \in \mathbb {N}\}$ is denoted by $\operatorname {\mathrm {EL}}_\infty (\mathbb {Z}[G])$ .

In addition to the results appearing in [Reference Boyle, Carlsen and EilersBCE20], we also need the following two lemmas concerning the weight class.

We now show that the conditions of Theorem 8.2 are met when A and B satisfy the conditions of Theorem 8.1, thus showing that in this case, equivariant shift equivalence implies equivariant flow equivalence. To do so, we need another characterization of shift equivalence coming from the ‘positive K-theory’ framework.

Theorem 8.5. [Reference Boyle and SchmiedingBS19, Theorem 6.2]

Let $A,B$ be square matrices over a ring $\mathcal {R}$ . Then, A and B are SE over $\mathcal {R}$ if and only if $(I-tA)_{\infty }$ and $(I-tB)_{\infty }$ are $\operatorname {\mathrm {GL}}_\infty (\mathcal {R}[t])$ equivalent, that is, there are $U,V \in \operatorname {\mathrm {GL}}_\infty (\mathcal {R}[t])$ such that

$$ \begin{align*} U(t) (I-tA)_\infty V(t) = (I-tB)_\infty. \end{align*} $$

Proof. Let A and B be SE matrices as above. By Lemma 8.3, they have the same weight class. Let H be a group in this weight class. Since G is Abelian, H is normal. Then, by [Reference Boyle and SullivanBS05, Proposition 4.4], there are diagonal matrices $C,D$ with diagonal entries in G such that $A' := DAD^{-1}$ and $B' :=CBC^{-1}$ are matrices over $\mathbb {Z}_+[H]$ . Here, A and $A'$ are SSE over $\mathbb {Z}_+[G]$ . The same holds for $B'$ and B. Additionally, $A, A', B$ , and $B'$ are all SE to each other over $\mathbb {Z}_+[G]$ .

Therefore, $A'$ and $B'$ are also SE over $\mathbb {Z}_+[H]$ by Lemma 8.4.

By Theorem 8.5, we know that there are matrices $U,V \in \operatorname {\mathrm {GL}}_\infty (\mathbb {Z}[H][t])$ such that

$$ \begin{align*} U(t) (I-tA')_\infty V(t) = (I-tB')_\infty. \end{align*} $$

We now have to upgrade this $\operatorname {\mathrm {GL}}_\infty (\mathbb {Z}[H][t])$ -equivalence to an $\operatorname {\mathrm {SL}}_\infty (\mathbb {Z}[H][t])$ -equivalence. Since G and hence also H are Abelian by assumption, both $\mathbb {Z}[H]$ and $\mathbb {Z}[H][t]$ are commutative rings. Therefore, we can take the determinant and, setting $u(t) := \det U(t)$ , $v(t) := \det V(t)$ , we see that $u(t)$ and $v(t)$ are units in $\mathbb {Z}[H][t]$ which satisfy

$$ \begin{align*} u(0)v(0)=1. \end{align*} $$

Now, $\mathbb {Z}[H]$ contains no nilpotent elements, which can be seen, e.g., by applying Wedderburn’s theorem to the commutative group algebra $\mathbb {C}[H]$ . Therefore, the units in $\mathbb {Z}[H][t]$ are precisely the units in $\mathbb {Z}[H]$ , as the coefficients of order at least one in a unit in a commutative polynomial ring have to be nilpotent, see, e.g., [Reference Dummit and FooteDF04, Exercise 7.3.33]. Hence, $u(t)=u(0)$ , $v(t)=v(0)$ , and therefore, $u(t)v(t)=1$ . Conjugating our equation above by $u(t)$ and setting $t=1$ gives

$$ \begin{align*} u(1)^{-1} U(1) (I-A')_\infty V(1)v(1)^{-1} = (I-B')_\infty. \end{align*} $$

Both matrices $u(1)^{-1}U(1)$ and $V(1)v(1)^{-1}$ are contained in

$$ \begin{align*} \operatorname{\mathrm{SL}}_\infty(\mathbb{Z}[H]) = \{W \in \operatorname{\mathrm{GL}}_\infty(\mathbb{Z}[H])\:| \det(W)=1\}. \end{align*} $$

By a K-theoretic result of Oliver, see [Reference OliverOli88, Theorem 14.2(iii)], we have $\operatorname {\mathrm {SL}}_\infty (\mathbb {Z}[H])=\operatorname {\mathrm {EL}}_\infty (\mathbb {Z}[H])$ for the cyclic group H. Hence, $(I-A')_\infty $ and $(I-B')_\infty $ are $\operatorname {\mathrm {EL}}_\infty (\mathbb {Z}[H])$ equivalent. Therefore, Theorem 8.2(ii) is satisfied, and $(X_{\mathcal {E}(A)},\sigma ,\alpha _A)$ and $(X_{\mathcal {E}(B)},\sigma ,\alpha _B)$ are G-flow equivalent.