Flow Equivalence of G-Sfts

FLOW EQUIVALENCE OF G-SFTS

MIKE BOYLE, TOKE MEIER CARLSEN, AND SØREN EILERS

Abstract. In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G = Z2, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata.

Contents 1. Introduction 1 2. Applications of G-SFTs 3 3. Background 7 4. (G, P) coset structures 14 5. The main results 21 6. From G-ﬂow equivalence to positive ElP (H) equivalence 23 7. The Factorization Theorem: setting 35 8. The Factorization Theorem: proof 42 9. Conclusion of proofs 50 Appendix A. Cohomology as positive equivalence 52 Appendix B. Permutation similarity as positive equivalence 55 Appendix C. Resolving extensions 56 Appendix D. G-SFTs following Adler-Kitchens-Marcus 59 Appendix E. A special case 67 References 79

1. Introduction The shifts of ﬁnite type (SFTs) are the fundamental building blocks of symbolic dynamics. Elaborations of these include the class of G- SFTs: SFTs equipped with a continuous action by a group G which

Date: August 16, 2019. 1 FLOW EQUIVALENCE OF G-SFTS 2 commutes with the shift. Apart from a few remarks, in this paper G- SFT means G-SFT with G finite and acting freely (gx = x only when g = e). We will give an algebraic classification of these G-SFTs up to equivariant flow equivalence (G-flow equivalence). This generalizes the G- flow equivalence classification for irreducible G-SFTs in [12] and the Huang flow equivalence classification for general SFTs without group action [4, 9]. Square matrices over Z+G (the positive cone in the integral group ring of G) present G-SFTs. When such matrices A and B present nontrivial1 mixing G-SFTs, these G-SFTs are G-flow equivalent if and only if there exist n in N and identity matrices Ij, Ik such that there are matrices U, V in the elementary group El(n, ZG) such that U((I −A)⊕ Ij)V = (I − B) ⊕ Ik. This reduces the dynamical classification of G- SFTs up to G-flow equivalence to the algebraic classification of square matrices over ZG up to the stabilized elementary equivalence described above. The algebraic classification is in general highly nontrivial, but far more manageable than the dynamical problem. For nonmixing G-SFTs, this stabilized elementary ZG equivalence no longer implies G-flow equivalence. To get an analogous result (Theorem 5.1) for general G-SFTs, we consider G-SFTs presented by matrices in a special block triangular form, with entries in an ij block lying in ZHij for some union Hij of double cosets of G, and their equivalence by elementary matrices from a restricted class subordinate to this blocked coset structure. Using this, we classify G-SFTs up to G-flow equivalence (Theorems 5.1 and 5.3). The paper is organized as follows. In Section 2, we discuss uses of G-SFTs, and application of the results of this paper. This includes a new use of the G-SFT structure, for cellular automata. Especially, the main theorem of the current paper is a key result for the classification up to flow equivalence of a large class of irreducible sofic shifts in [7]. In Section 3, we give a bare- bones review of the necessary background. In Section 4, we introduce the notion of coset structure, crucial for defining our restricted class of elementary matrices, and define various classes of related matrices. We prove Proposition 4.23 which tells us that, in order to classify G-SFTs up to G-flow equivalence, it is enough to work with square matrices over Z+G having a special block form. In Section 5, we present our classification, with comments. The full classification statement is Theorem 5.3; the essence is the simpler statement of Theorem 5.1 for the case the G-flow equivalence respects the ordering of irreducible components. In each statement, for matrices in a suitable class chosen to present the G-SFTs, a matrix condition

1By definition, an irreducible SFT is trivial iff it contains only one orbit. The only trivial mixing SFT is a single point; so, a trivial mixing G-SFT has G = {e}. FLOW EQUIVALENCE OF G-SFTS 3 is necessary and sufficient for the G-flow equivalence. In Section 6, we prove a strengthened version of necessity of the matrix condition, Theorem 6.2. In Section 7, we present the functorial Factorization Theorem 7.2, which we later use to prove sufficiency of the matrix condition in Theorem 5.3, and develop the setting for its proof. In Section 8, we give that proof. Section 9 contains the proof of Theorem 5.3 (a short argument appealing to Theorems 6.2 and 7.2), a result on range of invariants and a finiteness result. In Appendices A, B, and C, we establish three types of positive ElP (H)-equivalences which we use in the paper. In Appendix D, we relate the group actions viewpoint on G-SFTs developed in the Adler-Kitchens-Marcus paper [2] to our matrix-based setup. (This is analogous to the relation of matrices and linear trans- formations in elementary linear algebra.) Adler, Kitchens and Marcus were concerned only with nonwandering SFTs. We explain in Remark 4.25 how our coset structures give a kind of algebraic calculus to describe the transitions among irreducible components of a general G- SFT. In Appendix E, we work out algebraic invariants of G-flow equivalence for a special class of systems with exactly two irreducible components. For the subclass with G = Z2, we give a complete algebraic classification, with algorithms to answer all questions (at least, the natural ones we thought of). This supplies a tractable collection of examples and points to some of the issues involved in a general algebraic classification.

2. Applications of G-SFTs G-SFTs arise in several contexts, including the following. (Below, we occasionally assume background reviewed in Section 3.) (i) In [2], Adler, Kitchens and Marcus introduced invariants of nonwandering G-SFTs (and a more general class), and used these in [1] to classify factor maps between irreducible SFTs up to almost topological conjugacy. Here, a construction replaces a given factor map with a map ϕ: S0 → S, equivalent up to almost conjugacy, which is constant n-to-1. The map ϕ gives rises to a continuous function τ : S → Sn (with Sn the group of permutations on n symbols). This τ is used as a skewing function to generate a G- SFT (with G = Sn) extension T of S, with maps π : T → S and 0 α : T → S such that π = ϕ◦α. Group invariants of the Sn-action on T are then related to ϕ and used for the classiﬁcation, which requires further constructions. (ii) Field, Golubitsky and Nicol used G-SFTs in studies of “symmetry in chaos” and related equivariance [15, 16, 17]. FLOW EQUIVALENCE OF G-SFTS 4

(iii) The Livˇsictheorem, restricted to dimension zero, states that two real-valued Hölder functions on an irreducible SFT are cohomologous if and only if on each periodic orbit the sum of their output values is the same. G-SFTs arise as a case of the study of analogous rigidity possibilities for skew products in terms of group- valued skewing functions (see [24, 28, 10]). (iv) In [31, 30, 32], Michael Sullivan introduced “twistwise flow equivalence”. Here G = Z2. For an SFT basic set of the return map to a cross-section under a flow on a 3-manifold, the return map to the cross-section induces a map on the local stable set which is orientation preserving or reversing. This additional data is given by a function X → Z2, which can be encoded as a matrix A over ZG. Sullivan found G-FE invariants of A to produce new invariants of the template (branched two-manifold fitted with an expansive semiflow) for the flow. The complete algebraic G-FE classification (for G = Z2) for the mixing case gave further invariants (see [12, Sec. 7] and [33] for more). (v) The mapping class group of a nontrivial irreducible SFT (see [8]) is the countable group of homeomorphisms of the mapping torus of the SFT which respect orientation of the suspension flow, up to isotopy. This is a challenging group to understand. The classification of irreducible G-SFTs up to G-flow equivalence has provided at least a little information: for example (see [8, Theorem 8.6]), if the irreducible SFT is defined by a matrix A such that det(I − A) is odd, then the free orientation preserving involutions of the mapping torus are contained in a finite set of conjugacy classes of this mapping class group. (For a speculative application related to mapping class groups, see Remark 7.3.) (vi) The main result of the current paper has already been applied to the classification of sofic shifts up to flow equivalence in [7]. We will give more detail on this below. The automorphism group Aut(S) of an SFT S is the group of homeomorphisms commuting with S. The G-SFTs offer a different tool set to the study of finite subgroups of Aut(S). For simplicity, we consider just the case of G = Z2. If U is a free involution in Aut(S), then the pair (S,U) presents a G-SFT. When S is σn, the full shift on n symbols, this can also be described as a free involution of an invertible one-dimensional cellular automata. A longstanding question (recalled in [14, p.492]) asks for n = 2 whether two such involutions must be conjugate in Aut(σn). This conjugacy in the group Aut(σn) is equivalent to topological conjugacy of the corresponding G-SFTs. By Proposition 3.4.1, conjugacy in the group is equivalence to strong shift equivalence over Z+G of presenting matrices. A necessary condition for this conjugacy (a test) is that the G-SFTs be G-flow equivalent. (Equivalently, the induced involutions of the FLOW EQUIVALENCE OF G-SFTS 5 mapping torus are conjugate in the mapping class group of σn.) We prove next that this necessary condition is satisfied. For the proof, we assume the background and notation of Section 3. n For a matrix A over a commutative ring, the sequence (trace(A ))n∈N determines and is determined by the polynomial det(I − tA). We have n n trace(A ) = |Fix(σA)| when σA is an SFT defined by a matrix A over Z+. We assume in the proof some familiarity with the Bowen-Franks group invariant of flow equivalence. Finally, note that a full shift on an odd number of symbols has an odd number of fixed points, and therefore admits no free involution.

Theorem 2.1. Suppose G = Z2, k is a positive integer and σ2k is the full shift on 2k symbols. Then there is a free involution commuting with 0 0 σ2k. For any two such involutions U, U the G-SFTs (σ2k,U), (σ2k,U ) are G-flow equivalent. Proof. Let G = {e, g}. For the existence claim, choose a free order two permutation of the 2k symbols; this defines a one-block code which defines a free involution U of σ2k. Now let A be an m × m matrix over Z+G presenting a skew product G-SFT isomorphic to one defined by (U, σ2k). This A presents a skewing function on an SFT T which is a factor of σ2k under the map π which collapses G-orbits to points. Therefore (e.g. by [14, Theorem A]), det(I − tA) = 1 − 2kt. So, for every n, n |Fix(T n)| = |Fix(T )| = (2k)n .

n Let trace(A ) = αne + βng. From the structure of the skew product n construction, one can check that αn is the number of fixed points of T whose preimages are contained in Fix(T n). As π maps Fix(T n) 2-to-1 n n n into Fix(T ), we have αn = (1/2)(2k) , and therefore βn = (1/2)(2k) , for each n. This forces det(I − tA) = det(I − tB), for the 1 × 1 matrix B = (k(e + g)), because Bn = (1/2)(2k)ne + (1/2)(2k)ng. Therefore det(I − tA) = e − tk(e + g). Because G = Z2 and 1 − 2k is odd, by [12, Theorem 8.1] the matrix I−A is El(m, ZG)-equivalent to a diagonal matrix, D, which must have determinant e − k(e + g). It can happen, for some k, that e − k(e + g) factors in ZG. Nevertheless, D must be (e − k(e + g)) ⊕ Im−1. This is because the Bowen-Franks group for T is isomorphic to Z2k−1 (because det(I − tA) = 1 − 2kt) and is also isomorphic to cok(I − A). So, the El(ZG) class of I − A is the same for any choice of free involution U. It follows from [12, Theorem 6.4] (the case P = 1 of Theorem 5.1) that all these G-SFTs fall in the same G-flow equivalence class. By definition, the involutions U, U 0 are conjugate on periodic points if there is a bijection Per(T ) → Per(T 0) which intertwines the actions of (T,U) and T,U 0. A consequence of Ulf Fiebig’s work [14] (or the FLOW EQUIVALENCE OF G-SFTS 6 arguments above) is that free involutions of σ2k are conjugate on periodic points (for k = 1 this is explicitly contained in [14, Corollary 1.12]). Fiebig did much more, in particular for G actions which need not be free. In contrast to the free case, we do not have a Z+G matrix framework which handles the nonfree actions. Problem 2.2. Develop a useful matrix framework of matrices over Z+G for G-SFTs for which the G action need not be free. The framework should in particular capture topological conjugacy and flow equivalence of the G-SFT. The papers [14, 29] are relevant to Problem 2.2. A solution to Prob- lem 2.2 would give a ZG matrix framework for the entire vast collection of finite subgroups of Aut(σn). It would also naturally involve general, reducible G-SFTs. The discussions above involved G-SFTs which are irreducible as SFTs. The advance of the current paper is in addressing G-flow equivalence of G-SFTs which are reducible. We expect the general reducible case to be meaningful to related applications (especially, given a solution to Problem 2.2), but for the most part we have not developed reducible applications related to the items above. However, our main result, Theorem 5.3, is an essential tool for our classification in [7] of a large collection of irreducible sofic shifts up to flow equivalence (those which are “point extension type”, or PET). We emphasize that our results on G-FE for reducible G-SFTs is used for the flow equivalence classification of sofic shifts which are irreducible (or, equivalently for flow equivalence, mixing). For simplicity, we will describe only a subclass C of the PET sofic shifts. Let C be the class of nontrivial irreducible strictly sofic shifts such that for the right Fischer cover π : X → Y , the set M = {x : |π−1(πx)| > 1} is a closed proper subset and satisfies the following condition for some finite group G: there is a shift-commuting embedding of T to M which takes G-orbits to the fibers of π. Given the easily computed flow equivalence class of the irreducible SFT X, we show in [7] that the G-flow equivalence class of T is a complete classification invariant for the flow equivalence class of the sofic shift Y . For every X, every G-SFT G-flow equivalence class arises in this construction. This theorem appeals to the full strength of our classification result Theorem 5.3 (as noted in [7, Remark 7.14]). We give next an example to indicate how this works. a b c d b a d c Example 2.3. Let A = 0 0 k ` be a matrix in which the nonzero 0 0 ` k letters represent positive integers. There is a free involution γ on the edge SFT σA coming from a graph automorphism γ corresponding to the involution of vertices 1 ↔ 2, 3 ↔ 4. (E.g., there are d edges from vertex 1 to vertex 4, and these are mapped bijectively to the d edges FLOW EQUIVALENCE OF G-SFTS 7 from vertex 2 to vertex 3.) The edge SFT σA, together with γ, is a G-SFT, with G = Z2. Now take any 4 × 4 matrix B which is entrywise greater than A. Define a one block code ϕ from XB which maps two edges (the symbols of XB) to the same symbol if and only if they are edges of XA paired by γ. The image shift is a mixing strictly sofic shift. Now suppose a b c0 d0 0 b a d0 c0 A = 0 0 k ` , again with letters representing positive integers, and 0 0 ` k 0 0 again with B entrywise greater than A . Construct ϕ from XB just as ϕ was constructed. Are the two image sofic shifts flow equivalent? The 0 result of [7] tell us they are if and only if the G-SFTs on XA and XA are G-flow equivalent. Here, for any choices of the letters, we can translate to the completely worked classification of Appendix E and look up the answer. For the ae+bg ce+dg translation, first A becomes 0 ke+`g . There is an isomorphism from ZG to the subring R of Z2 consisting of the (α, β) such that α ≡ β mod 2, given by ae + bg 7→ (a + b, a − b) := (α, β). We apply this map entrywise to A to arrive at a matrix (αp,βp)(α,β) , where (α , β ) = 0 (αq,βq) p p 0 (a+b, a−b), (αq, βq) = (k+`, k−`) and (α, β) = (c+d, c−d). For A we do the same, arriving at the same matrix except that (α0, β0) replaces (α, β). From Proposition E.12 or Proposition E.13, we determine the ideal J to which the classifying Theorem E.9 applies; and then we can determine from Theorem E.9 whether the G-flow equivalence holds. Example E.17 gives a complete discussion of the classification for the case ae + bg = 54 − 42g, ke + `g = 16e − 8g.

3. Background This section provides a bare-bones review of the background mate- rial, assuming some familiarity with the subject. For basic background on shifts of finite type, see [19, 20]. For a detailed presentation with proofs of the basic theory of G-SFTs and G-flow equivalence for finite G, see [12]. The basic ideas of skew product constructions are of fundamental importance in various branches of dynamics; the exposition in [12] is tailored to our topic and also includes facts specific to it. See [10] for further developments, and a correction [10, Appendix A] to [12].

3.1. Shifts of finite type and matrices over Z+. Given an n × n square matrix A over Z+ = {0, 1,... }, let GA be a graph (in this paper, graph means directed graph) with vertex set {1, . . . , n}, edge set E = EA and adjacency matrix A. Define XA to be the subset of Z E realized by bi-infinite paths in GA. With the natural topology, XA is a zero-dimensional compact metrizable space. The homeomorphism σA : XA → XA given by the shift map σA, defined by (σA(s))i = si+1, FLOW EQUIVALENCE OF G-SFTS 8 is the edge SFT defined by A. Every SFT is topologically conjugate to some edge SFT.

3.2. Matrices over Z+G. Let G be a finite group, let ZG be the integral group ring of G, and let Z+G be the subset containing the P elements g∈G ngg with ng ≥ 0 for all g. Suppose A is a square matrix over Z+G. Let A denote the standard augmentation of A: the matrix over Z+ obtained by applying entrywise the standard augmentation P P map, g∈G ngg 7→ g ng. By an irreducible matrix A over ZG we mean a square matrix over Z+G whose augmentation A is an irreducible matrix. An irreducible component of A is a maximal irreducible principal submatrix of A. A matrix A is said to be essentially irreducible if it has a unique irreducible component. If A is essentially irreducible, then its unique irreducible component is called the irreducible core of A. P 2 An element g ngg of ZG is G-positive when ng > 0 for all g ∈ G. A matrix A over ZG is G-positive if every entry is G-positive. 3.3. G-SFTs. In this paper, by a G-SFT we mean an SFT together with a free continuous action on its domain by a finite group G which commutes with the shift. (In general, a “G-SFT” is not restricted to free actions or finite groups.) Two G-SFTs are G-conjugate (isomorphic as G-SFTs) if there is a topological conjugacy between them which intertwines their G actions. For a left G-SFT, the G action is from the left: gh : y 7→ g(hy)(h acts first). For a right G-SFT, the G action is from the right: gh : y 7→ (yg)h (g acts first). Standing Convention 3.3.1. Unless mentioned otherwise, in this paper a G-SFT is a left G-SFT (although we might sometimes repeat the declaration for clarity). This is the choice which aligns with matrix invariants (see [10, Appendix A]). (The G-SFTs of [2] are implicitly left G-SFTs; the G-SFTs of [1, p.493] are right G-SFTs.)

Suppose A is a square matrix over Z+G. Then A can be interpreted as the adjacency matrix of a labeled graph GA, where the underlying graph is G , and the label of an edge of G is the corresponding element A P A of G (so if the (s, t) entry of A is g∈G ngg, then there is for each g ∈ G, ng is the number of edges from s to t with label g). The labeled graph defines a skewing function τA : XA → G which sends x to the label of x0. The skew product construction then gives a homeomorphism TA : XA × G → XA × G defined by (x, g) 7→ (σA(x), gτA(x)), and TA is an SFT. (We consider every map topologically conjugate to an edge SFT to be SFT.) The continuous free left G action g :(x, g0) 7→ (x, gg0) commutes with TA. Together with this action, TA is a G-SFT. The map collapsing G-orbits to points is given by (x, g) 7→ x; it defines a

2“G-positive” replaces the term “very positive” used in [12]. FLOW EQUIVALENCE OF G-SFTS 9 factor map from the SFT TA to the edge SFT deﬁned by A. Every G-SFT is isomorphic to one presented as a group extension in this way by some A over Z+G. 3.4. Cohomology. Continuous functions τ and ρ from an SFT (X, σ) into G are cohomologous (written τ ∼ ρ) if there is another continuous function ψ from X into G such that for all x in X, τ(x) = [ψ(x)]−1ρ(x)ψ(σx). In this equation, the product on the right is a product in the group G. This is the form appropriate for our consid- eration of left G-SFTs (for which τ skews from the right). For right G-SFTs we would use instead the equation τ(x) = [ψ(σx)]ρ(x)ψ(x)−1 for all x. For nonabelian G, these coboundary equations are not equivalent. The following result is fundamental for us. Proposition 3.4.1. [12, Proposition 2.7.1] Suppose G is a ﬁnite group and A, B are square matrices over Z+G. Then the following are equivalent.

(1) A and B are strong shift equivalent over Z+G. (2) There is a topological conjugacy ϕ: XA → XB such that τB ∼ τA ◦ ϕ. (3) The G-SFTs TA and TB are G-conjugate. As seen in the Section 2, G-SFTs may arise in some setting directly, or in terms of a function from an SFT into G, corresponding to (2) above. Both possibilities are addressed by the matrix invariant (1). 3.5. Flow equivalence. Let Y be a compact metrizable space. In this paper, a flow on Y is a continuous R-action on Y with no fixed point. Two flows are topologically conjugate, or conjugate, if there is a homeomorphism intertwining their R-actions. Two flows are equivalent if there is a homeomorphism between their domains taking R-orbits to R-orbits and preserving orientation (i.e., respecting the direction of the flow). A cross-section to a flow γ : Y × R → Y is a closed subset C of Y such that the restriction of γ to C × R is a surjective local homeomorphism onto Y . In that case, the return time function τC : C → R given by τC (x) = min{t > 0 : γ(x, t) ∈ C} is well defined and continuous. The map rC : C → C given by rC (x) = γ(x, τC (x)) is call the return map of C.A section of a flow is the return map of a cross-section of the flow. For i = 1, 2 suppose Si : Xi → Xi is a homeomorphism of a compact metrizable space, and Yi is its mapping torus with the induced suspension flow. The homeomorphisms S1,S2 are flow equivalent if they are topologically conjugate to sections of a common flow; equivalently, after a continuous time change, the flows on Y1 and Y2 become topologically conjugate; equivalently, there is a homeomorphism Y1 → Y2 which on each Y1 flow orbit is an orientation preserving homeomorphism to a Y2 flow orbit. A flow equivalence S1 → S2 is such a homeomorphism. FLOW EQUIVALENCE OF G-SFTS 10

By a G-flow we mean a flow together with a continuous free left G-action which commutes with the flow. A free G action commuting with a section lifts to a free G action commuting with the flow. Two G-flows are G-conjugate if the flows are topologically conjugate by a map which intertwines the G-actions. Two G-flows are G-equivalent if the flows are equivalent by a map which intertwines the G-actions (i.e., by a G-flow equivalence). The standard theory carries over to the G setting. We call two G- homeomorphisms G-flow equivalent if they are conjugate to G-sections of the same G-flow. G-sections of two G-flows are G-flow equivalent if and only if the flows are G-equivalent. If A and B are square matrices over Z+G, then a G-flow equivalence TA → TB of their G-SFTs induces a flow equivalence of the SFTs defined by their standard augmentations A, B.

3.6. Positive equivalence. Suppose B,B0, U, V are n × n matrices over ZG with U, V in GL(n, ZG). We say (U, V ): B → B0 is an equivalence if UBV = B0. If {U, V } is contained in a subset M of GL(n, Z), then it is an M-equivalence, and the matrices B,B0 are M- equivalent. A basic elementary matrix is a matrix Est(x), which denotes a square matrix equal to the identity except for perhaps the off-diagonal st entry (so, s 6= t), which is equal to an element x of ZG. Suppose E = Est(g) and A is a square matrix over Z+G such that g is a summand of A(i, j) (i.e., g ∈ G and the coefficient of g in A(i, j) is positive). Then we say that each of the equivalences (E,I):(I − A) → E(I − A) , (E−1,I): E(I − A) → (I − A) , (I,E):(I − A) → (I − A)E, (I,E−1):(I − A)E → (I − A) is a basic positive ZG-equivalence. Here the equivalences (E,I) and (I,E) are forward and the other two are backward. An equivalence (U, V ):(I − A) → (I − B) is a positive ZG-equivalence if it is a composition of basic positive equivalences. A basic positive equivalence (I − A) → (I − B) induces a G-flow equivalence TA → TB. Every G-flow equivalence TA → TB is induced (up to isotopy, see [4, Section 6]) by a positive ZG-equivalence. For a justification of this claim, we refer to [12]; for more on its place in the positive K-theory classifications for symbolic dynamics, see [5]. The elementary group El(n, ZG) is the group of n×n matrices which are products of basic elementary matrices. A positive equivalence (I − A) → (I − B) through n × n matrices is an El(n, ZG) equivalence, but in general, an El(n, ZG) equivalence need not be a positive ZG equivalence, even if A is primitive (see for instance [12, Example 4.3]). Therefore, we do not in general have that an equivalence (I−A) → (I− B) induces a G-flow equivalence TA → TB. Still, we will in Theorem 7.2 FLOW EQUIVALENCE OF G-SFTS 11 show that if A and B satisfy specified conditions, and the equivalence (U, V ):(I − A) → (I − B) preserves specified structures (the poset structure, the cycle components (see later in this section) and the coset structure (see Section 4)), then it must be a positive ZG-equivalence and thus induce a G-flow equivalence TA → TB (see Theorem 5.3). For the proofs in Appendices A and B, we will use the graphical viewpoint described next (this description can also be found in [12]). 3.7. A row cut basic positive equivalence. Suppose (E,I):(I − A) → (I −B) is a basic forward positive equivalence, E = Est(g). Then A and B agree except perhaps in row s, where B(s, r) = A(s, r) + gA(t, r) if r 6= t , and B(s, t) = A(s, t) + gA(t, t) − g .

Consequently the labeled graph GB associated to B is constructed from the labeled graph GA as follows. An edge e from s to t with label g is deleted from GA. Then, for each GA-edge f beginning at t, an additional edge (called [ef]) from s to r with label gh (where h is the G-label of f and r is the terminal vertex of f) is added in to form GB. We refer to this type of positive equivalence as a (g, s, t) row cut (of the matrix A, or of an edge e labeled g), or just a row cut. When E(s, t) = p ≤ A(s, t), we may likewise refer to the positive equivalence implemented by (E,I) as a (p, s, t) row cut. See Figure 1 for an example of a (g, s, t) row cut of an edge from s to t labeled g. For a matrix example corresponding to Figure 1, with (s, t) = (1, 2), and r in Figure 1 set to r = 3, we use matrices E = Est(g) and    0 00  p11 g + p12 p13 p11 gh + p12 gh + p13 A =  0 h0 h00  and B =  0 h0 h00  p31 p32 p33 p31 p32 p33 in which the pij are arbitrary elements of Z+G, suppressed from the ﬁgure, and row 2 has just two entries for simplicity. The change from GA to GB is the replacement of the dashed edge of the left graph with the dashed edges of the right graph. On the left, g, h0, h00 are labels of edges e, f 0, f 00; on the right gh0, gh00 label edges named [ef 0], [ef 00].

@ r 5 @ r gh00 h00 h00 / s / t s / t g R gh0 R h0 h0

Figure 1. A row cut of an edge from s to t. FLOW EQUIVALENCE OF G-SFTS 12

@ t 5@ t h00g g h0g / r / s r / s h00 Q h00 Q h0 h0

Figure 2. A column-cut of an edge from s to t.

The correspondence of the graphs GA, GB induces a bijection of σA- orbits and σB-orbits, e.g. . . . b e f 00 c e f 0 f 0 d . . . ↔ . . . b [ef 00] c [ef 0] f 0 d . . . .

This bijection of orbits does not arise from a bijection of points for the SFTs, but it does correspond to a G-equivariant homeomorphism of their mapping tori (after changing time by a factor of 2 over the clopen sets {x : x0 = [ef]}, the new ﬂow is conjugate to the old one), which lifts to a G-equivariant homeomorphism of the respective mapping tori.

3.8. A column cut basic positive equivalence. The other type of basic forward positive equivalence is (I,E):(I − A) → (I − B), with E = Est(g). Then A and B agree except perhaps in column t, where B(r, t) = A(r, t) + A(r, s)g if r 6= t , and B(s, t) = A(s, t) + A(t, t)g − g .

The labeled graph GB associated to B is constructed from the labeled graph GA as follows. An edge e from s to t with label g is deleted from GA. Then, for each GA-edge f ending at s, an additional edge (called [fe]) from r to t with label hg (where h is the G-label of f and r is the initial vertex of f) is added in to form GB. We refer to this type of positive equivalence as a (g, s, t) column cut (of the matrix A, or of an edge e labeled g), or just a column cut. Figure 2 gives the column-cut analogue of Figure 1.

Example 3.8.1. When A is a square matrix over Z+G with some diagonal entry Att = 0, row cuts may be applied to zero out each entry of column t, and then column cuts to zero out each entry of row t, giving a permutation matrix P and a smaller matrix M such that P −1AP = M ⊕ 0. We call this type of operation a trim move. Note A and M define G-SFTs which are G-flow equivalent. For instance, with G the symmetric group S3, using A22 = 0 and the standard notation recalled in Definition 4.7, the matrix (23)(12) A = (13) 0 FLOW EQUIVALENCE OF G-SFTS 13 can be row cut, using (E12((12)),I):(I − A) → (I − B), to the matrix (23) + (132) 0 B = , (13) 0 which can be column cut, using (I,E21((13)) : (I − B) → (I − C), to the matrix (23) + (132) 0 C = = ((23) + (132)) ⊕ 0 . 0 0 3.9. Poset-blocked matrices. In order to handle general G-SFTs (having more than one irreducible component), as for the case G = {e} addressed in [4, 9] we need to consider matrices with block structures corresponding to irreducible components and transitions between them. Throughout this paper, P = {1,...,N} is a poset (partially ordered set) with a partial order relation chosen such that i j =⇒ i ≤ j. We will write i ≺ j if i j and i 6= j. For a PN vector of positive integers n = (n1, . . . , nN ), let n = j=1 nj, and Pi−1 Pi−1 Pi let Ii = {( j=1 nj) + 1, ( j=1 nj) + 2,..., j=1 nj} for each i ∈ P. If s ∈ {1, 2, . . . , n}, then we let i(s) be the unique integer such that s ∈ Ii(s). For an n × n matrix A and i, j ∈ P, we let A{i, j} denote the submatrix of A obtained by deleting the rows corresponding to indices not belonging to Ii and columns corresponding to indices not belonging to Ij. The matrix A is called an (n, P)-blocked matrix if A{i, j}= 6 0 =⇒ i j. For S a subset of ZG, we let MP (n,S) denote the set of (n, P)-blocked matrices with entries in S, and we let MP (S) be the union over n of the sets MP (n,S). o The set MP (n, Z+G) is the set of matrices A in MP (n, Z+G) satisfying the following conditions: (1) Each diagonal block A{i, i} is essentially irreducible. (2) If i ≺ j, then there are r > 0, an index s corresponding to a row in the irreducible core of A{i, i}, and an index t corresponding to a column in the irreducible core of A{j, j} such that Ar(s, t) 6= 0. o For A ∈ MP (n, Z+G), i in P corresponds explicitly to an irreducible component of the SFT defined by XA, with i ≺ j if and only there exists an orbit in XA backwardly asymptotic to component i and forwardly o 0 o asymptotic to component j. If A ∈ MP (Z+G) and A ∈ MP0 (Z+G), then a flow equivalence TA → TB induces a poset isomorphism P → P0. We say the flow equivalence respects the component order if this isomorphism is k 7→ k, 1 ≤ k ≤ N. o o We let MP (Z+G) be the union over n of the sets MP (n, Z+G).

o 3.10. Cycle components. For a matrix A in MP (n, Z+G), a cycle component is a component i in P such that the irreducible core of FLOW EQUIVALENCE OF G-SFTS 14

A{i, i} is a cyclic permutation matrix. The cycle components con- tribute significantly to technical difficulties in the classification of G- o SFTs up to G-flow equivalence. For A in MP (n, Z+G), C(A) denotes the set of its cycle components. For a subset C of P,

o o MP (C, n, Z+G) := {A ∈ MP (n, Z+G): C(A) = C} . o o We let MP (C, Z+G) be the union over n of the sets MP (C, n, Z+G).

3.11. Stabilizations. An unblocked matrix A0 is a stabilization (or 0- stabilization) of an m × n matrix A if A equals an upper left corner of A0, and A0 is zero in all remaining entries. (I.e., A0(s, t) = A(s, t) if 1 ≤ s ≤ m and 1 ≤ t ≤ n, and otherwise A0(s, t) = 0.) A matrix 0 0 A in MP (n , ZG) is a stabilization (or 0-stabilization) of a matrix A 0 0 in MP (n, ZG) if n ≥ n (i.e., ni ≥ ni for 1 ≤ i ≤ N) and for 1 ≤ i, j ≤ n, the i, j block submatrix of A0 is a stabilization of the i, j block submatrix of A. 0 0 A matrix M in MP (n , ZG) is a 1-stabilization of a matrix M in 0 MP (n, ZG) if M − I is a 0-stabilization of M − I. When its entries come from Z+G, a matrix A and its 0-stabilizations deﬁne the same G-SFT, but it is I − A and its 1-stabilizations which will share the algebraic invariants for G-ﬂow equivalence.

4. (G, P) coset structures

The classification up to G-flow equivalence of G-SFTs TA defined by irreducible matrices A over Z+G required a reduction to the case that the “weights group” of TA is all of G. (This essentially amounts to re- ducing to the case that TA is mixing as an SFT, as recalled in Appendix D.) For general A over Z+G, we will need an analogous reduction on the irreducible components of A, and then we will capture invariants of transitions between components using double coset conditions. In this section we prepare the formal structure for this. We begin with the double coset conditions. Below, G is the given finite group and P = {1,...,N} is the given finite poset, with a partial order relation satisfying i j =⇒ i ≤ j. Let Hi and Hj be subgroups of G. An (Hi,Hj) double coset is a nonempty set equal to HigHj for some g in G. Definition 4.1. A(G, P) coset structure H is a function which assigns to each pair (i, j) in P × P such that i j a nonempty subset Hij of G such that

(4.2) i j k =⇒ HijHjk ⊂ Hik .

Consequently, Hii (also denoted Hi) is a subgroup of G and for i ≺ j Hij is a nonempty union of (Hi,Hj) double cosets. FLOW EQUIVALENCE OF G-SFTS 15

If Hi,Hj are subgroups of an abelian group G, then HiHj is a group, and a double coset HigHj is a coset gHiHj. For general G, H is actually a double coset structure; we use “coset structure” for brevity.

Deﬁnition 4.3. (Notation) Pn denotes the poset {1, . . . , n} with the linear order: i ≺ j iﬀ i < j.

In the next two examples, we consider H a coset structure for (G, P3), with G abelian, using additive notation. Here the possibilities for H are as follows.

• H1,H2,H3 are arbitrary subgroups of G, • H12 is an arbitrary nonempty union of cosets of H1 + H2, • H23 is an arbitrary nonempty union of cosets of H2 + H3, • H13 is an arbitrary union of cosets of H1 + H3 containing H12 + H23; because G is abelian, H12 + H23 is a union of cosets of H1 + H2 + H3.

Example 4.4. Let G = Z/27, H1 = H3 = 9G = {0, 9, 18}, H2 = H12 = H23 = 3G = {0, 3,..., 24}, H13 = {1, 10, 19} ∪ 3G. So, H1 + H2 + H3 = 3G. Now H13 contains a coset of H1 + H2 + H3, but H13 is not a union of cosets of H1 + H2 + H3.

Example 4.5. Let G = Z/30, H1 = 15G, H2 = 6G and H3 = 10G. Then H1 +H2 = 3G, H2 +H3 = 2G, H1 +H3 = 5G and H1 +H2 +H3 = G. Because H13 contains a coset of H1 + H2 + H3, H13 must be G.

Remark 4.6. For G not necessarily abelian, with subgroups Hi,Hj, let us recall some elementary facts about the double cosets HigHj, g ∈ G. As in Example 4.8) HiHj need not be a group; HigHj need not be a coset of a subgroup of G; HigHj is a union of right cosets of Hi (or left cosets of Hj), but the union is not arbitrary. A double coset HigHj is the orbit of g under the (right) action on G by Hi ⊕ Hj given by −1 (h, k): g 7→ h gk. Consequently, two (Hi,Hj) double cosets are equal or disjoint. Thus, if there are exactly r (Hi,Hj) double cosets in G, r there are exactly 2 − 1 sets which are nonempty unions of (Hi,Hj) double cosets. For g ∈ G, deﬁne the subgroup −1 Mij(g) = {h ∈ Hi : ∃k ∈ Hj, h gk = g} −1 −1 = {h ∈ Hi : g hg ∈ Hj} = Hi ∩ gHjg .

Then the isotropy group of g for this action of Hi ⊕ Hj on G is −1 {(h, k) ∈ Hi ⊕ Hj : h ∈ Mij(g), k = g hg} , with cardinality |Mij(g)|, and therefore |HigHj| = |Hi||Hj|/|Mij(g)|. In particular, as the cardinality |Mij(g)| of the isotropy group might vary with g, so might the cardinality of the double coset |HigHj|.

Deﬁnition 4.7. (Notation) Sn is the group of permutations of {1,..., n} (to avoid confusion, we reserve sans serif numbers to indicate the elements on which Sn acts). An is the alternating group in Sn. We use FLOW EQUIVALENCE OF G-SFTS 16 cycle notation to denote elements of Sn. For example, (123) is the cyclic permutation 1 → 2 → 3;(12)(13) = (132). We let e denote the identity. For a subset K of a group, hKi denotes the subgroup generated by K; for example, in Sn, h(12)i = {(12), e}.

Example 4.8. Suppose (G, P) = (S3, P2) with coset structure H. If H11 = S3 or H22 = S3, then H12 must be S3. At the opposite extreme, if H11 = H22 = {e}, then H12 can be any nonempty subset of S3. Now suppose H1 = h(12)i and H2 = h(13)i. For g in S3, −1 −1 |H1gH2| = |H1||H2|/|H1 ∩ gH2g | = 4/|H1 ∩ gH2g | . −1 So, |H1gH2| = 2 if gH2g = H1, and otherwise |H1gH2| = 4 . There are exactly two double cosets H1gH2: D1 = {(23), (132)} and its complement D2 in S3. Neither double coset is a subgroup of S3. D1 is the only coset of H1 which is a double coset. D2 is neither a left coset nor a right coset of a subgroup of S3. D1 is a right coset of H1 and a left coset of H2, but it is neither a left coset of H1 nor a right coset of H2.

Example 4.9. Suppose (G, P) = (S3, P3) with coset structure H. Let H1 = H12 = h(12)i, H2 = {e}, H3 = (13) with H12 = H1 and H23 = H3 ∪ (23)H3 . As seen in Example 4.8, there are two (H1,H3) double cosets: {(23), (132)}, and its complement in S3. By (4.2), H13 contains H12H23, which here contains {e, (23)}. Therefore H13 intersects both (H1,H3) double cosets, and must equal S3. Deﬁnition 4.10. Two (G, P) coset structures H, H0 are G-cohomologous if there exist elements γ1, . . . , γN in G such that −1 0 i j =⇒ Hij = γi Hijγj . The “G” in “G-cohomologous” matters (see Example 4.11). Still, because (G, P) is ﬁxed, we sometimes write just “coset structure” in place of “(G, P) coset structure”.

Example 4.11. Let P be the trivial poset P1. Then (G, P) coset 0 0 structures H, H are cohomologous iﬀ the groups H1,H1 are conjugate subgroups of G. Deﬁne order two subgroups of S4, H1 = h(12)(34)i and 0 H1 = h(13)(14)i. Let H (isomorphic to Z/2 ⊕ Z/2) be the subgroup 0 0 generated by H1 and H1. Then H1,H1 are conjugate as subgroups of S4, but not as subgroups of H. 0 Example 4.12. Suppose G is abelian and H, H are (G, P2) coset 0 0 0 structures such that H1 = H1 and H2 = H2. If |H12| = 1 = |H12|, 0 0 Then H and H are G-cohomologous if and only if H12 = g + H12 for 0 some g in G. This always holds if |H12| = 1 = |H12|, but need not hold 0 if e.g. |H12| = 2 = |H12|.

Example 4.13. Let H be a (S3, P2) coset structure, with H11 = H22 = {e}. We noted in Example 4.8 that any nonempty subset of S3 may 0 serve as H12. When H12 is another such subset, of course it is necessary FLOW EQUIVALENCE OF G-SFTS 17

0 that |H12| = |H12| for the structures to be cohomologous. But this is not a suﬃcient condition; for instance H12 = {e, (12)} gives a system 0 which is not cohomologous to that from H12 = {e, (123)}, as is seen by applying the sign function to the deﬁning relation.

Definition 4.14. For a matrix A over Z+G, with τA the associated labeling of edges of GA, the weight of a path of edges p = p1p2 ··· pk in GA is defined to be τA(p) = τA(p1)τA(p2) ··· τA(pk). o Definition 4.15. Suppose A ∈ MP (Z+G). Then a (G, P) coset structure H for A is defined as follows. (1) For each i ∈ P, choose a vertex v(i) from the irreducible core of the block A{i, i}. (2) For i j, Hij is the set of weights of paths from v(i) to v(j).

The group Hii (also denoted Hi) was called a weights group for Aii in [12].

Example 4.16. With G = S3, consider the matrices  0 (12) 0  (12)(132) (12)(13) A = ,A = ,A = (12) e (12) . 1 (123) e 2 0 (12) 3   0 0 e

For A1, P = P1. If v(1) = 1, then H1 = h(12)i; if v(1) = 2, then from −1 −1 (132) = (123) we have H1 = (123) h(12)i (123) = h(13)i. For A2 and A3, P = P2. For A2, H1 = H2 = h(12)i and H12 = S3 \ h(12)i. For A3, H1 = H2 = {e}; if v(1) = 1 then H12 = {e}, and if v(1) = 2 then H12 = {(12)}. Even though the example above shows that they may be different, all coset structures for A will be G-cohomologous. To see this, suppose v1, v2 are vertices in the irreducible core of A{i, i} and γi is the weight of a path from v1 to v2. Replacing a choice v(i) = v2 with the 0 −1 choice v(i) = v1 has the effect of replacing Hi with Hi := γiHiγi , and 0 0 replacing Hij with Hij := γiHij when i ≺ j. Therefore H and H are G-cohomologous. Definition 4.17. The (G, P) coset structure class of A is the G-cohomology class of a (G, P) coset structure for A.

Example 4.18. Let G = Z/4Z with generator g. By Example 4.12, 2 2 2 g 2 1+g ( 0 2 ) and 0 2 have the same coset structure class, but 0 2 and 2 1+g2 0 2 do not have the same coset structure class. 1 1 1 Example 4.19. Suppose G is a nontrivial group. Define A = 0 1 1 0 1 1 0 1 1+g 1 0 0 and A = 0 1 1 . Coset structures H, H for A, A are given by 0 1 1 0 0 0 H1 = H2 = H12 = H1 = H2 = {1} and H12 = {1, g}. Because 0 0 |H12|= 6 |H12|, the coset structures H, H are not G-cohomologous (and FLOW EQUIVALENCE OF G-SFTS 18 therefore, by our structure theorem, the matrices A, A0 define G-SFTs which are not G-flow equivalent). However, (I − A) and (I − A0) are El(n, P2, Z+G)-equivalent, with n = (1, 2), because 1 0 g 0 −1 −1 0 −1−g −1 0 E13(g)(I − A) = 0 1 0 0 0 −1 = 0 0 −1 = I − A . 0 0 1 0 −1 0 0 −1 0 1 2 0 1 2+g A smaller example is given by C = ( 0 2 ) and C = 0 2 , which respectively define G-SFTs which respectively are G-flow equivalent to those defined by A and A0.

Example 4.19 shows that El(P, n, ZG) equivalence, even by matrices with ith diagonal block entries in Z+Hi for all i, does not give an algebraic relation capturing G-ﬂow equivalence. This is why we are led to introduce ElP (n, H) equivalence later in this section. Example 4.20. Notice that it can happen that not every coset structure in the (G, P) coset structure class of a matrix A is a (G, P) coset structure for A. If for example (G, P) = (S3, P2) and e e A = , 0 e then H11 = H12 = H22 = {e} is the only (S3, P2) coset structure for 0 0 0 A; but H11 = H22 = {e}, H12 = {(12)} is also in the (S3, P2) coset structure class of A.

o The classes MP (n, ZG) and MP (C, n, Z+G) used below were deﬁned in Subsections 3.9 and 3.10. Deﬁnition 4.21. Let H be a (G, P) coset structure.

•M P (n, H) is the set of matrices A ∈ MP (n, ZG) such that for i j, the entries of A{i, j} belong to ZHij. o o •M P (C, n, H) is the set of A in MP (n, H)∩MP (C, n, Z+G) such that H is a (G, P) coset structure for A. o o •M P (C, H) = ∪n MP (C, n, H). o o •M P (H) = ∪C MP (C, H). o Deﬁnition 4.22. A matrix in MP (C, Z+G) satisﬁes Condition C1 (or, o is C1) if it belongs to MP (C, n, Z+G) for n = (n1, n2, . . . , nN ) such that the following condition holds for all i ∈ P: ni = 1 if and only if i ∈ C.

A square matrix A over Z+G is nondegenerate if it has no zero row and no zero column. Notice that this is equivalent to the graph GA being nondegenerate (that is, every vertex of GA belongs to a bi-inﬁnite path). The next proposition applies in particular to a nondegenerate matrix A (which cannot be nilpotent).

Proposition 4.23. Let A be a nonnilpotent square matrix over Z+G. Then there is an N, a partial order on P := {1,...,N} satisfying i j =⇒ i ≤ j, a subset C of P, a (G, P) coset structure H, and a FLOW EQUIVALENCE OF G-SFTS 19

o C1 matrix B ∈ MP (C, H) with each diagonal block irreducible such that TA and TB are G-flow equivalent. B can be produced algorithmically from A. Example 4.24. The first two of the matrices e (12) 0 0 (12) , e + (12) , 0 0 e (13) 0   0 0 e are not C1, and the third does not satisfy the diagonal block irreducibility condition of Proposition 4.23. The algorithm outlined in the proof below will give, respectively, e (12) e (12) (123) , , e (12) 0 e which satisfy both conditions. See Example 3.8.1 for details of the first example. For an example of the step B = DAD−1 in the proof of Prop. 4.23, we use P = P2, G = S3, H1 = H2 = H12 = A3, ν(1) = 1 and ν(2) = 4. Then e e e (123)(123) ! −1 (123)(213)(123) e (123) B = DAD = e e e e e 0 0 0 e e 0 0 0 (132)(123) e 0 0 0 0 !  e (12) e (123)(13)  e 0 0 0 0 ! 0 (12) 0 0 0 (23)(123)(23)(12)(123) 0 (12) 0 0 0 = 0 0 e 0 0  e (12) e e e  0 0 e 0 0 0 0 0 e 0 0 0 0 e (23) 0 0 0 e 0 0 0 0 0 (23) 0 0 0 0 (23) 0 0 0 (12)(132) Proof of Prop. 4.23 . If a diagonal entry of A is zero, then we may apply the trimming move of Example 3.8.1 to produce a smaller matrix over Z+G, which defines a flow equivalent G-SFT, and which is also not nilpotent. By iteration of this move, we may assume without loss of generality that every diagonal entry of A is nonzero. Then there is a poset P = {1,...,N}, a vector n = (n1, n2, . . . , nN ) and a permutation −1 o matrix P and such that P AP ∈ MP (n, Z+G). We may assume P AP −1 = A. Let C be the set of cycle components for A in P. For 1 ≤ i ≤ N, P set mi = h

∗ because Ii = Ii we may choose ds the weight of some path from v(i) to s and bs the weight of some path from s to v(i). Let D be the diagonal −1 matrix with D(s, s) = ds. Define B = DAD . There is a positive ZG-equivalence from I − A to I − B (Proposition A.1), so TB is G-flow equivalent to TA. Suppose s ∈ Ii and t ∈ Ij. We claim that B(s, t) ∈ ZHij. To prove this claim, note that btdt is the weight of a path from t to t. Pick k > 0 k k−1 −1 −1 such that (btdt) = e. Then (btdt) bt = (dt) , and therefore (dt) is −1 the weight of a path from t to v(j). Therefore B(s, t) = dsA(s, t)(dt) is the weight of a path from v(i) to v(j), and therefore is in Hij. Because I −A and I −B are positive ZG equivalent, a coset structure for B must be G-cohomologous to the coset structure H of A. By construction, a coset structure H0 for B defined from the vertex choices 0 v(i) has Hij ⊂ Hij if i j. By the G-cohomology, this containment o must be equality, so H is a coset structure for B, and B ∈ MP (C, H).

++ Remark 4.25. Suppose A ∈ MP (n, H) and TA is the G-SFT defined by A in Section 3.3. Let π : XA × G → XA be the map collapsing G- orbits. For i ∈ P, let T i : Xi → Xi be the mixing Z-SFT defined by A{i, i}. Let Xi = Xi × Hi, and let Ti be the restriction of TA to −1 Xi. Then Hi is the isotropy group of Xi, and π (Xi) is the disjoint union of |G/Hi| mixing Z-SFTs; the mixing SFTs g1Xi, g2Xi are equal if and only if g1Hi = g2Hi. Let us write g1Xi → g2Xj if there exists a point backwardly asymptotic to g1Xi and forwardly asymptotic to −1 g2Xj. Then g1Xi → g2Xj if and only if Xi → (g1) g2Xj. From the definition of TA, we see Xi → gXj iff g ∈ Hij, and this holds for every element or no element of a double coset HigHj.

We now give terminology for the equivalences fundamental to our results. For a positive vector n = (n1, . . . , nN ), let ElP (n, H) be the group of matrices generated by the basic elementary matrices in MP (n, H). We define an ElP (n, H)-equivalence to be an equivalence (U, V ):(I − A) → (I − B) with U, V in ElP (n, H) and A, B in MP (n, H). A basic positive ElP (n, H)-equivalence is a basic positive ZG-equivalence which is also an ElP (n, H)-equivalence. A positive ElP (n, H)-equivalence is defined to be a composition of basic positive ElP (n, H)-equivalences. An (positive) ElP (H)-equivalence from I − A to I − B is defined to 0 0 be any (positive) ElP (n, H)-equivalence (U, V ):(I − A ) → (I − B ) such that A0,B0 are stabilizations of A, B. It is easy to check that if there is a (positive) ElP (H)-equivalence from I − A to I − B, and a (positive) ElP (H)-equivalence from I − B to I − C, then there is a (positive) ElP (H)-equivalence from I − A to I − C. FLOW EQUIVALENCE OF G-SFTS 21

5. The main results

We can now state the primary result of the paper. The C1 condition in the statement was given in Definition 4.22. Given γ = (γ1, . . . , γN ) ∈ N n G , we define Dγ ∈ MP (n, Z+G) to be the diagonal matrix D such that D(s, s) = γi(s) (i.e., for 1 ≤ i ≤ N, the diagonal block D{i, i} is γiI). Theorem 5.1. Suppose G is a finite group; P = {1,...,N} is a poset; 0 H and H are (G, P) coset structures; A and B are nondegenerate C1 o o 0 0 matrices in MP (C, n, H) and MP (C, n , H ), respectively. Then the following are equivalent.

(1) There is a G-flow equivalence of the G-SFTs TA and TB which respects the component ordering. N n −1 n (2) For some γ ∈ G , for C = (Dγ ) BDγ there exist m and C1 <0> <0> o stabilizations A ,C of A, C in MP (C, m, H) such that the <0> <0> matrices I − A and I − C are ElP (m, H)-equivalent. By condition (2) of Theorem 5.1, there is an immediate corollary. Corollary 5.2. The (G, P) coset structure class is an invariant of component-order-respecting G-flow equivalence. We will give a more complicated statement next for a flow equivalence which need not respect component order. By Proposition 4.23, G- SFTs can be presented by matrices in the form addressed by Theorem 5.1. Therefore, Theorem 5.1 (as elaborated in Theorem 5.3) gives a classification of G-SFTs up to G-flow equivalence. If P = {1,...,N} and P0 = {1,...,N} are finite posets given by partial order relations satisfying i j =⇒ i ≤ j, α : P → P0 is a poset isomorphism, and n = (n1, . . . , nN ) is a positive vector, then ∗ we denote by α (n) the vector (m1, . . . , mN ) with mi = nα−1(i), and n PN PN we let Qα be the n × n matrix (where n = j=1 nj = i=1 mi) with Pα(i)−1 Pi−1 entries qrs = 1 if r = l + j=1 mj and s = l + k=1 nk for some n −1 n i ∈ P and some l ∈ {1, . . . , ni}, and 0 otherwise. Then (Qα) AQα ∈ ∗ MP (n, Z+G) whenever A ∈ MP0 (α (n), Z+G). Theorem 5.3. Let G be a finite group, let P = {1,...,N} and P0 = {1,...,N 0} be finite posets given by partial order relations satisfying i j =⇒ i ≤ j, let C and C0 be subsets of P and P0 respectively, and 0 0 0 let H = {Hij}i,j∈P and H = {Hij}i,j∈P0 be (G, P) and (G, P ) coset structures. o o 0 0 0 Suppose A ∈ MP (C, n, H) and B ∈ MP0 (C , n , H ) are C1 stabilizations of nondegenerate matrices. Then the following are equivalent.

(1) The G-SFTs TA and TB are G-ﬂow equivalent. (2) There exists a poset isomorphism α : P → P0 such that α(C) = 0 N −1 0 C , and there exists γ ∈ G such that Hij = γi Hα(i)α(j)γj for FLOW EQUIVALENCE OF G-SFTS 22

i, j ∈ P with i j, and such that for the matrix

n −1 n −1 n n C = (Dγ ) (Qα) BQαDγ <0> the following holds: there exist m and C1 stabilizations A , <0> o C in MP (C, m, H) of A, C such that the following holds: <0> <0> the matrices I − A and I − C are ElP (m, H)-equivalent. In Theorem 5.3, the implication (1) =⇒ (2) is a part of the more general result Theorem 6.2. The implication (2) =⇒ (1) is a consequence of a much stronger constructive statement, the Factorization Theorem 7.2. We therefore postpone the proof of Theorem 5.3 to Sec- tion 9. After finite reductions, it is now clear that there is a procedure for determining G-flow equivalence of TA and TB (hence of G-SFTs) if there is a procedure for answering the following. Question 5.4. Let G be a finite group. Given a (G, P) coset structure o H, C ⊂ P and C1 matrices A, C in MP (C, n, H), does there exist a procedure to decide whether the following holds: there exists m, and <0> <0> o C1 stabilizations A ,C in MP (C, m, H) of A, C, such that the <0> <0> matrices I − A and I − C are ElP (m, H)-equivalent. We will give a more algebraically phrased question next for future reference. An answer yes to Question 5.5 gives an answer yes to Ques- tion 5.4 (using L = I − A, M = I − B). Question 5.5. Let G be a finite group. Given a (G, P) coset structure H, C ⊂ P, and n such that ni = 1 for i ∈ C, and matrices L, M in MP (n, H), is there a procedure to decide whether the following holds: <1> <1> there exist m, with mi = 1 for i ∈ C, and 1-stabilizations L ,M <0> <0> of L, M in MP (m, H), such that the matrices L and M are ElP (m, H)-equivalent. There is an affirmative answer to Question 5.5 for G = {e} (see [11]), and perhaps for all G. In some cases the invariants for SFTs (which we can regard as G-SFTs with G = {e}) have allowed practical computation of examples and subclasses (see e.g. [18, 9]). We note that if G is abelian, then there are only finitely many G-flow equivalence classes of G-SFTs defined by matrices A for which the diagonal block determinants of I − A are prescribed regular elements (i.e., are not zero-divisors) in ZG (Theorem 9.3). For the case G = {e}, there is a complicated diagram of homomorphisms of finitely generated abelian groups (the reduced K-web of [9], useful for applications to Cuntz-Krieger algebras as explained in [3] which (with regard to an appropriate notion of diagram isomorphism) is a complete invariant for the ElP (H)-equivalence. For general G, one can define a K-web invariant in the same way, using ZG-modules FLOW EQUIVALENCE OF G-SFTS 23 and module homomorphisms in place of abelian groups and homomorphisms of abelian groups. However, this invariant is no longer complete, because new obstructions arise to passing from diagram isomorphism to the elementary matrix equivalence. (We thank Takeshi Katsura for showing us examples of this.) Developing a complete invariant from the ZGK-web by characterizing the allowed diagram isomorphisms is a nontrivial but perhaps accessible problem. We use the C1 condition in Theorem 5.3 to get a precise characteriza- tion in terms of matrix equivalence. To see that Theorem 5.3 would be false if the C1 condition were dropped, consider the following example: (5.6) (I − A) U = I − B 1 − g −e −e 0  1 0 0 0 1 − g −3e −2e 0   0 0 0 −e  0 2 1 0  0 0 0 −e      =   .  0 0 0 −e  0 1 1 0  0 0 0 −e  0 0 0 1 − h 0 0 0 1 0 0 0 1 − h

Here e is the identity in G (e = 1 in ZG) and U ∈ ElP (H), for a coset structure H for A. But A and B present skewing functions on SFTs with 2 and 5 infinite orbits, respectively, so TA and TB are certainly not G-flow equivalent. In the case G = {e} we understand by [4, Theorem 3.3] exactly when an ElP (H)-equivalence is a positive ElP (H)-equivalence (i.e., arises from a G-flow equivalence). We do not know how to do the same for nontrivial G, if a block A{i, i} defining a cycle component is allowed to be larger than 1 × 1. We need to require the C1 condition to be able to prove in the Factorization Theorem 7.2 below, that every ElP (H)-equivalence is a positive ElP (H)-equivalence. Partly because of complications arising from cycle components, we avoid the infinite matrices used in [4] and [12] to describe stabilizations.

6. From G-flow equivalence to positive ElP (H) equivalence We will now present and prove Theorem 6.2, from which we directly get the implication (1) =⇒ (2) in Theorem 5.3. First we introduce Condition C1+, which we shall use in the proof of Theorem 6.2. Definition 6.1. Let G be a finite group, let P = {1,...,N} be a finite poset given by a partial order relation satisfying i j =⇒ i ≤ j, and let n = (n1, . . . , nN ) be a vector of positive integers. Given C ⊂ P, a matrix M in MP (n, ZG) satisfies Condition C1+ if for every i in C there exists si in Ii such that the following hold.

(1) If {s, t} ⊂ Ii and M(s, t) 6= 0, then (s, t) = (si, si). (2) If s ∈ Ii, with M(s, t) 6= 0 or M(t, s) 6= 0, then s = si. FLOW EQUIVALENCE OF G-SFTS 24

Notice that if M is a stabilization of a C1 matrix, then M satisfies condition C1+. Theorem 6.2. Let G be a finite group, let P = {1,...,N} and P0 = {1,...,N 0} be finite posets given by partial order relations satisfying i j =⇒ i ≤ j, let C and C0 be subsets of P and P0 respectively, and 0 0 0 let H = {Hij}i,j∈P and H = {Hij}i,j∈P0 be (G, P) and (G, P ) coset structures. o o 0 0 0 Suppose A ∈ MP (C, n, H) and B ∈ MP0 (C , n , H ) are stabilizations of nondegenerate matrices, and TA and TB are G-flow equivalent. Then there exist a poset isomorphism α : P → P0 such that α(C) = C0 and N γ = (γ1, . . . , γN ) ∈ G (where N is the number of elements of P) such −1 0 that Hij = γi Hα(i)α(j)γj for i, j ∈ P with i j, and such that for the matrix n −1 n −1 n n C = (Dγ ) (Qα) BQαDγ the following holds: there exist m and stabilizations A<0>,C<0> in o MP (C, m, H) of A, C with a positive ElP (m, H)-equivalence (U, V ): (I − A<0>) → (I − C<0>). <0> Moreover, if A and B satisfy Condition C1, then the matrices A , <0> C can be chosen to satisfy Condition C1. Before we give the proof, which is a nontrivial elaboration of the proof for the case that A, B are essentially irreducible [12, Proposition 4.7], we outline the argument. A discrete cross-section for a homeomorphism T : X → X of a compact zero-dimensional metrizable space X is a clopen subset K ⊂ X such that every point of X is mapped into K by some positive power of T . In this case, for x ∈ K there is a smallest positive integer ρK (x) ρK (x) such that T is in K and the return map RK : K → K is then the map x → T ρK (x)(x) (see for example [6] for details). If T is an SFT, then RK is again SFT. If K is a G-invariant discrete cross-section for TA, then there is a (unique) discrete cross-section C for σA such that K = C × G and R ((x, g)) = (R (x), gτ (x)τ (σ (x)) . . . τ (σρC (x)−1(x))) K C A A A A A for x ∈ C and g ∈ G. The Parry-Sullivan argument [26] shows that any flow equivalence of mapping tori of SFTs is isotopic to one which is induced by a conjugacy of return maps to discrete cross-sections (again, see for example [6] for details). It follows that since TA and TB are G-flow equivalent, there exist G-invariant discrete cross-sections KA and KB for TA and TB such that the return maps RKA and RKB are G-conjugate. Let CA and CB be discrete cross-sections for σA and σB such that KA = CA × G and KB = CB × G. <1> <2> o Our strategy is to first construct matrices A ,A ∈ MP (C, H) <2> such that A presents the G-SFT RKA , and such that there are FLOW EQUIVALENCE OF G-SFTS 25

<1> positive ElP (H)-equivalences from I − A to I − A and from I − A<1> to I − A<2> (this is done in Step 1 and Step 2). Similarly, we <1> <2> o 0 0 get matrices B ,B ∈ MP0 (C , H ) such that there is a positive 0 <1> 0 ElP0 (H )-equivalence (I − B) → (I − B ) and a positive ElP0 (H )- <1> <2> equivalence (I − B ) → (I − B ), and such that TB<2> is G- conjugate to RKB . Since RKA and RKB are G-conjugate, it follows that TA<2> and TB<2> are G-conjugate. We use this in Step 3 to construct <3> o <3> o 0 0 0 matrices A ∈ MP (C, r, H), B ∈ MP0 (C , r , H ) and a poset isomorphism α : P → P0 such that α∗(r) = r0 and α(C) = C0, and <2> such that there is a positive ElP (H)-equivalence from I − A to <3> 0 <2> <3> I − A , a positive ElP0 (H )-equivalence from I − B to I − B , <3> r −1 <3> r and such that A = (Qα) B Qα and on this common domain <3> r −1 <3> r the matrices A and (Qα) B Qα define skewing functions which are cohomologous. In Step 4 we then find a vector γ ∈ GN such that −1 0 Hij = γi Hα(i)α(j)γj for i, j ∈ P with i j, and such that there are positive ElP (H)-equivalences <3> r −1 r −1 <3> r r (I − A ) → (I − (Dγ) (Qα) B QαDγ) and r −1 r −1 <3> r r (I − (Dγ) (Qα) B QαDγ) → (I − C) , n −1 n −1 n n where C = (Dγ ) (Qα) BQαDγ . This completes the proof of the first half of the theorem. To show that the matrices A<0>,C<0> can be chosen to satisfy Con- dition C1 if A and B satisfy Condition C1, we refine the construction of <0> <0> o Steps 1–4 in order to obtain stabilizations Ae , Ce ∈ MP (C, H) of A and C and a positive ElP (H)-equivalence (U,e Ve):(I − Ae<0>) → (I − Ce<0>) such that for every i ∈ C the matrices Ue{i, i}, Ve{i, i} are the identity <0> <0> o matrix. We then get C1 stabilizations A ,C in MP (C, m, H) of A, C and with a positive ElP (m, H)-equivalence (U, V ):(I − A<0>) → (I − C<0>) as wanted by letting A<0>, C<0>, U, W be principal submatrices of Ae<0>, Ce<0>, Ue, Ve. This is done in Steps 5–8. Proof. Step 1: Higher block presentation. We begin by construct- <1> ing a higher block presentation A of TA such that the discrete cross- section CA corresponds to a union of vertices in the graph GA<1> , and <1> such that there is a positive ElP (H)-equivalence from I −A to I −A . There is a k and a subset S of the 2k + 1-blocks of XA such that CA = {x ∈ XA : x[−k, k] ∈ S}. Let P2k+1 be the set of paths in GA of length 2k +1. For p ∈ P2k+1, let s(p) be the initial vertex of the middle edge of p. Index the elements of P2k+1 = {p1, p2, . . . , pn} such that if <1> s(pi) < s(pj), then i < j. We will now construct an n × n matrix A over Z+G (actually it will be a matrix over G). Let 1 ≤ s, t ≤ n. If FLOW EQUIVALENCE OF G-SFTS 26 there exist edges e, f in GA such that pse = fpt, then the (s, t) entry <1> of A is the label of the middle edge of ps. If there are no edges <1> e, f in GA such that pse = fpt, then the (s, t) entry of A is 0. It <1> o follows from Proposition C.1 that A ∈ MP (C, H) and that there <1> is a positive ElP (H)-equivalence from I − A to I − A . Since A is a stabilization of a nondegenerate matrix, it follows that A<1> is nondegenerate. Step 2: Discrete cross-section. <2> o In this step we produce a nondegenerate matrix A ∈ MP (C, H) which presents the G-SFT RKA , and explain that there is a positive <1> <2> ElP (H)-equivalence from I − A to I − A . The matrix A<2> is the adjacency matrix of the labelled graph which has vertex set {s ∈ {1, 2, . . . , n} : ps ∈ S} and where there for each path p in GA<1> which starts and ends in vertices s and t for which ps, pt ∈ S, but which otherwise go through vertices v for which pv ∈/ S, is an edge from s to t with label equal to the weight τA<1> (p) of p (so in particular, if e is an edge in GA<1> which starts and ends in vertices s and t for which ps, pt ∈ S, then there is an edge in GA<2> from s to t with the same label as e). We then have that TA<2> is G-conjugate to

RKA . <1> We will now construct a positive ElP (H)-equivalence from I − A to I − A<2>. This is accomplished by iterating a matrix move. Given o a matrix M in MP (C, m, H) and a vertex s such that M(s, s) = 0, the move produces a positive ElP (m, H)-equivalence (I − M) → (I − Ms) o for a related matrix Ms in MP (C, m, H), where Ms has row s and column s zero. For a description of this move, let M(r, s) = p 6= 0, let Er be the basic elementary matrix Er,s(p). Let U be the product of these Er (so, U(r, s) = M(r, s) if r 6= s, and in other entries U = I) and let M 0 be the matrix such that U(I−M) = I−M 0. Then (U, I):(I−M) → (I−M 0), as a composition of the equivalences (Er,I), is a positive ElP (m, H) equivalence. Column s of M 0 is zero. Row s of M 0 equals row s of M. Next, we zero out row s of M 0. Deﬁne V (s, t) = M(s, t) if s 6= t 0 and V = I otherwise. Let M(s) be the matrix such that (I − M )V = 0 I − M(s). Then (I,V ):(I − M ) → (I − M(s)) is a positive ElP (m, H)- equivalence. We have ( M(r, t) + M(r, s)M(s, t) if r 6= s 6= t M (r, t) = (s) 0 if r = s or s = t.

The matrix M(s) presents a skewing function into G induced by the return map to the clopen set of points x for which the initial and terminal vertices of x0 do not equal s. <1> Altogether, for {1, . . . , n}\ S = {s(1), . . . , s(k)}, set M0 = A , and set Mi = (Mi−1)(s(i)) for 1 ≤ i ≤ k. Then we have a positive <1> <2> ElP (H)-equivalence from I − A = 1 − M0 to I − Mk = I − A . FLOW EQUIVALENCE OF G-SFTS 27

Step 3: The resolving tower and matrix cohomology. Similar to how we constructed A<1> and A<2>, we can construct <1> <2> o 0 0 nondegenerate matrices B ,B ∈ MP0 (C , H ) such that there is 0 <1> a positive ElP0 (H )-equivalence (I − B) → (I − B ) and a positive 0 <1> <2> ElP0 (H )-equivalence (I − B ) → (I − B ), and such that TB<2> is G-conjugate to RKB . Since RKA and RKB are G-conjugate, it follows that TA<2> and TB<2> are G-conjugate. It therefore follows from Proposition 3.4.1 that there is a topological conjugacy

ϕ : XA<2> → XB<2> which takes the skewing function τA<2> to a function cohomologous to <3> o τB<2> . In this step we will construct matrices A ∈ MP (C, r, H), <3> o 0 0 0 0 B ∈ MP0 (C , r , H ) and a poset isomorphism α : P → P such that ∗ 0 0 0 α (r) = r and α(C) = C , and such that there is a positive ElP0 (H )- <2> <3> 0 equivalence from I −A to I −A , a positive ElP0 (H )-equivalence <2> <3> <3> r −1 <3> r from I − B to I − B , and such that A = (Qα) B Qα and <3> r −1 <3> r on this common domain the matrices A and (Qα) B Qα deﬁne skewing functions which are cohomologous. There is a standard decomposition for the topological conjugacy ϕ [23] (see also [20, Theorem 7.1.2]). It follows from this that there is a matrix M over Z+, with one-block conjugacies (given by graph homomorphisms) ϕ1 : σM → σA<2> and ϕ2 : σM → σB<2> such that −1 (1) ϕ is ϕ1 followed by ϕ2, −1 (2) ϕ1 is left resolving, with ϕ1 a composition of conjugacies given by row splittings, −1 (3) ϕ2 is right resolving, with ϕ2 a composition of conjugacies given by column splittings. −1 Since ϕ1 a composition of conjugacies given by row splittings, it fol- <3> −1 lows that there is a ZG-matrix A such that ϕ1 can be lifted to a G-conjugacy ψA : TA<2> → TA<3> , also given by row splittings, and a −1 <3> permutation matrix PA such that PA A PA = M and ψA((x, g)) = −1 (ηPA (ϕ1 (x)), g) for (x, g) ∈ XA<2> × G, where ηPA : σM → σA<3> is the conjugacy given by PA. It follows from Proposition C.1 that <3> o A ∈ MP (C, H) and that there is a positive ElP (H)-equivalence from I − A<2> to I − A<3>. Since A<2> is nondegenerate, so is A<3>. <3> o Choose r such that A ∈ MP (C, r, H). Similarly, there is a vector r0, a nondegenerate ZG-matrix B<2.5> ∈ o 0 0 0 MP0 (C , r , H ), a G-conjugacy ψB : TB<2> → TB<2.5> , a permutation −1 <2.5> −1 matrix PB such that PB B PB = M and ψB((x, g)) = (ηPB (ϕ2 (x)), g) for (x, g) ∈ XB<2> × G, where ηPB : σM → σB<2.5> is the conjugacy 0 <2> given by PB, and a positive ElP0 (H )-equivalence from I − B to <2.5> −1 <3> −1 <2.5> I − B . Let P = PBPA . Then A = P B P . It follows that there is a poset isomorphism α : P → P0 such that α∗(r) = r0 and 0 0 0 α(C) = C , and a permutation matrix P ∈ MP0 (r , Z+) such that P = FLOW EQUIVALENCE OF G-SFTS 28

0 r0 <3> 0 −1 <2.5> 0 o 0 0 0 P Qα . Let B = (P ) B P ∈ MP0 (C , r , H ). It then follows 0 0 from Proposition B.1 that there is a positive ElP0 (r , H )-equivalence <2.5> <3> r −1 <3> r (I−B ) → (I−B ). We furthermore have that (Qα) B Qα ∈ o r −1 r <3> <3> r −1 <3> r <3> MP (C, r, H), (Qα) B Qα = A , and τ(Qα) B Qα and τA are cohomologous. Step 4: ElP (H)-equivalence. In this step we complete the proof apart from the (nontrivial) “moreover” statement. We continue with the notation of the last step.

Let ψ be the continuous function from XA<3> into G such that −1 <3> r −1 <3> r τA (x) = (ψ(x)) τ(Qα) B Qα (x)ψ(σA<3> (x)) for all x ∈ XA<3> . Then proof of Parry for [24, Lemma 9.1] (translated from his vertex SFTs to our edge SFTs) shows that if x ∈ XA<3> , then ψ(x) is determined by the initial vertex of the edge x0. Because <3> r −1 <3> r A and (Qα) B Qα are nondegenerate, this implies that there is a diagonal matrix D, with each diagonal element an element of G, such that −1 <3> r −1 <3> r (6.3) D A D = (Qα) B Qα.

We let Mi denote a diagonal block M{i, i} of a P-blocked matrix. <3> r −1 <3> r The matrices Ai and ((Qα) B Qα)i are essentially irreducible <3> r −1 <3> r and the group Hi is a weights group for Ai and ((Qα) B Qα)i. It then follows from the proof of Theorem 4.7 of [12] that there exists N r γ ∈ G such that for each i the diagonal matrix (DDγ)i has every entry in Hi. Then r −1 m0 −1 <3> m0 r r −1 −1 <3> r (Dγ) (Qα ) B Qα Dγ = (Dγ) (D A D)Dγ r −1 <3> r = (DDγ) A (DDγ). r −1 r −1 <3> r r o Applying Proposition A.1, we have (Dγ) (Qα) B QαDγ ∈ MP (H), with a positive ElP (H)-equivalence <3> r −1 r −1 <3> r r (I − A ) → (I − (Dγ) (Qα) B QαDγ). −1 0 So, Hij = γi Hα(i)α(j)γj for i, j ∈ P with i j, and we have positive ElP (H)-equivalences <3> r −1 r −1 <3> r r (I − A) → (I − A ) → (I − (Dγ) (Qα) B QαDγ). n −1 n −1 n n o Let C = (Dγ ) (Qα) BQαDγ . Then C ∈ MP (H) because B ∈ o 0 0 0 0 MP0 (C , H ), α is a poset isomorphism from P to P such that α(C) = C −1 0 and Hij = γi Hα(i)α(j)γj for i, j ∈ P with i j. It remains to show that there is a positive ElP (H)-equivalence from r −1 r −1 <3> r r I − C to I − (Dγ) (Qα) B QαDγ. We have proved that there is a 0 <3> positive ElP0 (H )-equivalence from I − B to I − B given by a path

(E ,F ) (E ,F ) (E ,F ) (I − B0) −−−−→·1 1 −−−−→·2 2 · · −−−−−→T T (I − (B<3>)0) FLOW EQUIVALENCE OF G-SFTS 29

0 o 0 0 <3> 0 o 0 0 in which B ∈ MP0 (m , H ) is a stabilization of B,(B ) ∈ MP0 (m , H ) <3> is a stabilization of B , and the Et and Ft are basic elementary 0 0 0 p0 −1 p0 −1 p0 p0 matrices in ElP0 (m , H ). But then Et := (Dγ ) (Qα ) EtQα Dγ 0 p0 −1 p0 −1 p0 p0 and Ft := (Dγ ) (Qα ) FtQα Dγ are basic elementary matrices in ElP (H), and we have a positive ElP (H)-equivalence

0 0 0 0 p0 −1 p0 −1 0 p0 p0 (E1,F1) (E2,F2) (I − (Dγ ) (Qα ) B Qα Dγ ) −−−−→· −−−−→ 0 0 (ET ,FT ) p0 −1 p0 −1 <3> 0 p0 p0 ··· −−−−−→ (I − (Dγ ) (Qα ) (B ) Qα Dγ ). p0 −1 p0 −1 0 p0 p0 Since (Dγ ) (Qα ) B Qα Dγ is a stabilization of C and p0 −1 p0 −1 <3> 0 p0 p0 (Dγ ) (Qα ) (B ) Qα Dγ r −1 r −1 <3> r r is a stabilization of (Dγ) (Qα) B QαDγ, this shows that there is a r −1 r −1 <3> r r positive ElP (H)-equivalence from I−C to I−(Dγ) (Qα) B QαDγ and thus that there is a positive ElP (H)-equivalence from I−A to I−C. “Moreover”. For the rest of the proof, we assume that A and B satisfy Condition C1 . It remains to show that we can ﬁnd stabiliza- <0> <0> o tions A ,C ∈ MP (C, m, H) of A, C with a positive ElP (m, H)- equivalence (U, V ):(I − A<0>) → (I − C<0>). For this we reﬁne the construction of Steps 1-4 in order to obtain <0> <0> o stabilizations Ae , Ce ∈ MP (C, t, H) of A and C and a positive <0> <0> ElP (t, H)-equivalence (U,e Ve):(I − Ae ) → (I − Ce ) such that for every i ∈ C the matrices Ue{i, i}, Ve{i, i} are the identity matrix. We <0> <0> o then get C1 stabilizations A ,C in MP (C, m, H) of A, C and with <0> <0> a positive ElP (m, H)-equivalence (U, V ):(I − A ) → (I − C ) as wanted by letting A<0>, C<0>, U, W be principal submatrices of Ae<0>, Ce<0>, Ue, Ve. <0> <3> Step 5: Getting a Cu-equivalence (I − A ) → (I − A ). In this step we will show that the positive equivalence I − A → I − A<3> of Steps 1-3 can be chosen such that there are stabilizations 000 <3>000 o <3> A ,A ∈ MP (C, s, H) of A and A and a positive ElP (s, H)- 0000 <3>0000 equivalence (UA,VA):(I − A ) → (I − A ) such that for every i ∈ C the matrices UA{i, i}, VA{i, i} are unipotent upper triangular. <1> Recall that the positive ElP (H)-equivalence from I −A to I −A is the composition of positive ElP (H)-equivalences (I − At) → (I − At+1) obtained by applying Proposition C.1. At each stage the P blocking of At+1 is the lift of the P blocking of At. At step t, there is an index st such that either row st of At is split into two rows, or column st is split into two columns. We will choose st, 1 + st to be the indices associated to the splitting (so, if an index j of At is greater than s, then it corresponds to index j + 1 of At+1). In the case that st is an index in a cycle component we place additional conditions as follows. If At(st, st) = g 6= 0, then we require that

At+1(st + 1, st + 1) = g and At+1(st, st) = 0 FLOW EQUIVALENCE OF G-SFTS 30 in the case At → At+1 is a column splitting, and we require

At+1(st + 1, st + 1) = 0 and At+1(st, st) = g in the case At → At+1 is a row splitting. When At is upper triangular in its cycle component diagonal blocks, it follows from Proposition C.1 that At+1 is as well, and that there are unipotent upper triangular matrices Ut,Vt such that

(Ut,Vt):(I − At) → (I − At+1) is a positive ElP (H)-equivalence. By induction, each cycle component <1> block of A is upper triangular, and there is a positive ElP (H)- equivalence 0 <1>0 (U1,V1):(I − A ) → (I − A ) where A0 is a stabilization of A, A<1>0 is a stabilization of A<1>, and for every i ∈ C the matrices U1{i, i}, V1{i, i} are unipotent upper triangular. Next consider the restriction of the Step 2 move M → M(s) to a diagonal block M{i, i} with i ∈ C. Clearly, if M{i, i} is upper triangular, then the trimming matrices implementing the positive equivalence (I − M) → (I − Ms) are unipotent upper triangular, and M(s){i, i} is upper triangular. Because A<1>{i, i} is upper triangular, it follows by induction that A<2>{i, i} is upper triangular and that there is a positive ElP (H)-equivalence <1>00 <2>00 (U2,V2):(I − A ) → (I − A ) where A<1>00 is a stabilization of A<1>, A<2>00 is a stabilization of A<2>, and for every i ∈ C the matrices U2{i, i}, V2{i, i} are unipotent upper triangular. By the argument for Step 1, we will likewise have that A<3> upper triangular in each cycle component block and that there is a positive ElP (H)-equivalence <2>000 <3>000 (U3,V3):(I − A ) → (I − A ) where A<2>000 is a stabilization of A<2>, A<3>000 is a stabilization of <3> A , and for every i ∈ C the matrices U3{i, i}, V3{i, i} are unipotent upper triangular. Putting this together we get there is a stabilization 0000 o <3>0000 o A ∈ MP (C, s, H) of A and a stabilization A ∈ MP (C, s, H) of <3> A such that there is a positive ElP (s, H)-equivalence 0000 <3>0000 (6.4) (UA,VA):(I − A ) → (I − A ) such that for every i ∈ C the matrices UA{i, i}, VA{i, i} are unipotent upper triangular. By the same argument, there are stabilizations B0000, B<2.5>0000 ∈ o 0 0 0 <2.5> 0 0 MP0 (C , s , H ) of B,B and a positive ElP0 (s , H )-equivalence 0000 <2.5>0000 (UB,VB):(I − B ) → (I − B ) FLOW EQUIVALENCE OF G-SFTS 31

0 such that for every i ∈ C the matrices UB{i, i}, VB{i, i} are unipotent upper triangular. Step 6: Clearing out diagonal C blocks in (U, V ). 0000 <3>0000 o From Step 5 we have a stabilizations A ,A ∈ MP (C, s, H) of <3> A, A and the positive ElP (s, H)-equivalence (6.4) such that for every i ∈ C the matrices UA{i, i}, VA{i, i} are unipotent upper triangular. In this step we will show there is a positive ElP (s, H)-equivalence (I − A0000) → (I − A0000) such that precomposing (U, V ) with this equivalence produces an equivalence

0000 <3>0000 (UeA, VeA):(I − A ) → (I − A ) such that for all i in C, the diagonal blocks UeA{i, i} and VeA{i, i} are the identity matrix. So, consider i ∈ C. Restricted to the block (I −A0000){i, i} := (I −M), 0000 <3>0000 our equivalence UA(I − A )VA = (I − A ) has the following block triangular form, with central block 1 × 1:         U11 U12 U13 I 0 0 V11 V12 V13 I 0 0  0 U22 U23 0 1 − g 0  0 V22 V23 = 0 1 − g 0 . 0 0 U33 0 0 I 0 0 V33 0 0 I The form is determined by placing the unique entry 1 − g as the central block. Suppose {s, t} ⊂ Ii, s < t, s 6= si 6= t and E is a basic elementary matrix of size matching A0000 with E(s, t) = ±h for some k h in G. (For E in ElP (H), h must be g for some k.) Then, because −1 0000 0000 A satisﬁes condition C1,(E,E ):(I − A ) → (I − A ) is a positive ElP (n, H)-equivalence. After precomposing (U, V ) with a suitable composition of these, we may assume U11 = I, U33 = I and U13 = 0. Our matrix equivalence now has the following form (6.5)         I x 0 I 0 0 V11 V12 V13 I 0 0 0 1 U23 0 1 − g 0  0 1 y  = 0 1 − g 0 0 0 I 0 0 I 0 0 V33 0 0 I which multiplies out to give     V11 V12 + x(1 − g) V13 + x(1 − g)y I 0 0  0 1 − g (1 − g)y + U23V33 = 0 1 − g 0 . 0 0 V33 0 0 I

Consequently we can rewrite the left side of (6.5) as

I x 0  I 0 0 I x(g − 1) x(g − 1)y 0 1 (g − 1)y 0 1 − g 0 0 1 y  . 0 0 I 0 0 I 0 0 I FLOW EQUIVALENCE OF G-SFTS 32

This equivalence (U, V ):(I − M) → (I − M) is a composition of two equivalences, (U1,V1) followed by (U2,V2), where I x 0 I x(g − 1) 0 U1 = 0 1 0 V1 = 0 1 0 0 0 I 0 0 I I 0 0  I 0 0 U2 = 0 1 (g − 1)y V2 = 0 1 y . 0 0 I 0 0 I We will see how these equivalences are related to positive equivalences (I − M) → (I − M). Consider a term −h (h ∈ G) which is part of an entry of y in V2, say the (si, t) entry of V . Recall Es,t(δ) denotes a basic elementary matrix with oﬀ-diagonal entry δ in position (s, t). We deﬁne now n × n matrices E1,...,E4. E1(r, t) = −M(r, si)h if r∈ / {si, t}; in other entries, E1 = I. E2 = Esi,t(−gh); E3 = Esi,t(−h); E4 = Esi,t(h). Then E4E2E1(I − M)E3 = (I − M), and applying the multiplications in the order indexed gives a positive ElP (H)-equivalence. It may be easiest to see the argument by restricting to a 4 × 4 principal submatrix. For this, suppose 2 = si, 3 = t and M(1, 2) 6= 0 6= M(2, 4). Then these principal submatrices (with names unchanged for simplicity) have the forms 1 − x −w 0 −u 1 0 −wh 0  0 1 − g 0 −z 0 1 0 0 I − M =   E1 =    0 0 1 0  0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 −gh 0 0 1 −h 0 −1 E2 =   E3 =   = (E4) . 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1

For the n×n matrices, we have (E4E2E1,E3) = (E4Esi,t((g−1)h),Esi,t(−h)). For the case the term is h, there is similarly a positive equivalence −1 −1 F4F3F1(I −M)F2 = (I −M), in which F1 = E3, F2 = (E3) , F3 = E2 −1 and F4 = E1 . In the 4 × 4 sample, this has the form 1 0 0 0 0 1 −h 0 −1 F1 =   = F 0 0 1 0 2 0 0 0 1 1 0 0 0 1 0 wh 0 0 1 gh 0 0 1 0 0 F3 =   F4 =   0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 FLOW EQUIVALENCE OF G-SFTS 33

Here (F4F3F1,F2) = (F4Esi,t((g − 1)h),Esi,t(h)). A suitable composition of the above equivalences is an equivalence which in the {i, i} block matches (U1,V1). Precomposing (UA,VA) with the inverse of this composition gives the required matrix (UeA, VeA). 0 0 Similarly, there is a positive ElP0 (s , H )-equivalence 0000 <2.5>0000 (UeB, VeB):(I − B ) → (I − B ) 0 such that for every i ∈ C the matrices UeB{i, i}, VeB{i, i} are the identity matrix. Step 7: Cohomology. <2.5> <3> In this step we will show that there are stabilizations B1 ,B1 ∈ o 0 0 0 <2.5> <3> 0 0 MP0 (C , s1, H ) of B , B and a positive ElP0 (s1, H )-equivalence 0 0 <2.5> <3> (Ue1, Ve1 ):(I − B1 ) → (I − B1 ) 0 0 0 such that for every i ∈ C the matrices Ue1{i, i}, Ve1 {i, i} are the iden- <3> tity matrix, and we will show that there are stabilizations A1 ,M1 ∈ o <3> r −1 r −1 <3> r r MP (C, s1, H) of A ,(Dγ) (Qα) B QαDγ and a positive ElP (s1, H)- equivalence <3> (Ue1, Ve1):(I − A1 ) → (I − M1) such that for every i ∈ C the matrices Ue1{i, i}, Ve1{i, i} are the identity matrix. <3> 0 −1 <2.5> 0 0 0 Recall that B = (P ) B P where P ∈ MP0 (r , Z+) is a permutation matrix. Since B satisfies condition C1, 0000 <2.5>0000 (UeB, VeB):(I − B ) → (I − B ) 0 0 0 is a positive ElP0 (s , H )-equivalence such that for every i ∈ C the <2.5> matrices UeB{i, i}, VeB{i, i} are the identity matrix, and B is non- <2.5> degenerate, it follows that B satisfies condition C1, and thus that P 0{i, i} = 1 for every i ∈ C0. It therefore follows from Proposition <2.5> <3> <2.5> <3> B.1 that there are stabilizations B1 , B1 of B , B and a 0 0 positive ElP0 (s1, H )-equivalence 0 0 <2.5> <3> (Ue1, Ve1 ):(I − B1 ) → (I − B1 ) 0 0 0 such that for every i ∈ C the matrices Ue1{i, i}, Ve1 {i, i} are the identity matrix. r −1 <3> r r −1 m0 −1 <3> m0 r Recall that (DDγ) A (DDγ) = (Dγ) (Qα ) B Qα Dγ. Since A satisfies condition C1, 0000 <3>0000 (UeA, VeA):(I − A ) → (I − A ) is a positive ElP (s, H)-equivalence such that for every i ∈ C the matri- <3> ces UeA{i, i}, VeA{i, i} are the identity matrix, and A is nondegener- <3> ate, it follows that A satisfies condition C1. It then follows from the proof of Theorem 4.7 of [12] that γ ∈ GN can be chosen such that for r each i ∈ C the diagonal matrix (DDγ)i is 1. Applying Proposition A.1, FLOW EQUIVALENCE OF G-SFTS 34

<3> <3> r −1 r −1 <3> r r we get stabilizations A1 , M1 of A ,(Dγ) (Qα) B QαDγ and a positive ElP (s1, H)-equivalence

<3> (Ue1, Ve1):(I − A1 ) → (I − M1) such that for every i ∈ C the matrices Ue1{i, i}, Ve1{i, i} are the identity matrix. Step 8: Conclusion. 0000 <2.5>0000 By composing the equivalences (UeB, VeB):(I −B ) → (I −B ) 0 0 <2.5> <3> <3> and (Ue1, Ve1 ):(I−B1 ) → (I−B1 ), we get stabilizations B2,B2 ∈ o 0 0 0 <3> 0 0 MP0 (C , s2, H ) of B, B and a positive ElP0 (s2, H )-equivalence 0 0 <3> (Ue2, Ve2 ):(I − B2) → (I − B2 )

0 0 0 such that for every i ∈ C the matrices Ue2{i, i}, Ve2 {i, i} are the identity 0 0 s2 −1 s2 −1 matrix. By multiplying this equivalence with (Dγ ) (Qα ) on the left 0 0 s2 s2 o and Qα Dγ on the right we get stabilizations C2,M2 ∈ MP (C, s2, H) s2 −1 s2 −1 <3> s2 r ∗ 0 of C,(Dγ ) (Qα ) B Qα Dγ (where α (s2) = s2) and a positive ElP (s2, H)-equivalence

(Ue2, Ve2):(I − C2) → (I − M2) such that for every i ∈ C the matrices Ue2{i, i}, Ve2{i, i} are the identity matrix. By composing the inverse of this equivalence with the equiva- 0000 <3>0000 <3> lences (UeA, VeA):(I − A ) → (I − A ) and (Ue1, Ve1):(I − A1 ) → <0> <0> o (I − M1), we get stabilizations Ae , Ce ∈ MP (C, t, H) of A, C and a positive ElP (S3, H)-equivalence

(U,e Ve):(I − Ae<0>) → (I − Ce<0>) such that for every i ∈ C the matrices Ue{i, i}, Ve{i, i} are the identity matrix. It remains to obtain the equivalence in C1 form. Let I be the index <0> <0> set of Ae (and Ce ), and let Isec be the set of elements s ∈ I such that i(s) ∈ C and Ae<0>(s, t) = Ae<0>(t, s) = 0 for all t ∈ I. Since Ue(I − Ae<0>)Ve = I − Ce<0>, Ue{i, i} and Ve{i, i} are the identity matrix <0> <0> for every i ∈ C, and Ae and Ce satisfy condition C1+ (because A and C satisfy condition C1), it follows that Isec is equal to the set of s ∈ I such that i(s) ∈ C and Ce<0>(s, t) = Ce<0>(t, s) = 0 for all t ∈ I. −1 Let Iprim be the complement in I of the Isec. Let Wf = Ve and write the equivalence in the form

(6.6) Ue(I − Ae<0>) = (I − Ce<0>)W.f Let A<0>, C<0>, U, W be the principal submatrices of Ae<0>, Ce<0>, <0> <0> Ue, Ve with index set Iprim. Then A ,C are C1 stabilizations in o <0> <0> MP (C, m, H) of A, C. If we can show U(I − A ) = (I − C )W , FLOW EQUIVALENCE OF G-SFTS 35 then we have the required C1-equivalence. For a veriﬁcation, suppose t, u ∈ Iprim. Then U(I − A<0>)(t, u) = U(t, u) − (UA<0>)(t, u) X = U(t, u) − U(t, s)A<0>(s, u)

s∈Iprim X = Ue(t, u) − Ue(t, s)Ae<0>(s, u)

s∈Iprim X = Ue(t, u) − Ue(t, s)Ae<0>(s, u) s∈I <0> = Ue(I − Ae ) (t, u). Likewise, (I − C<0>)W (t, u) = (I − Ce<0>)Wf (t, u). The required equality now follows from (6.6).

7. The Factorization Theorem: setting In this section we present the Factorization Theorem 7.2, which we shall use to prove the implication (2) =⇒ (1) in Theorem 5.3, and establish the setting for the proof of Theorem 7.2. As before, G is a finite group, P = {1,...,N} is a finite poset given by a partial order relation satisfying i j =⇒ i ≤ j, C is a subset of P, and H = {Hij}i,j∈P is a (G, P) coset structure. For the class o MP (C, n, H), recall Definition 4.21. o Definition 7.1. A matrix A in MP (C, n, H) satisfies condition C2 if the following holds: if i ∈ P and i is not a cycle component of A, then there are matrices Ui,Vi in El(ni,Hi) such that Ui(I − Ai)Vi is a block diagonal matrix with one summand a 2 × 2 identity matrix. Theorem 7.2 (Factorization Theorem). Suppose A and A0 are ma- o trices in MP (C, n, H) which satisfy conditions C1 and C2. Then the following are equivalent. 0 (1) (U, V ):(I − A) → (I − A ) is an ElP (n, H)-equivalence. 0 (2) (U, V ):(I − A) → (I − A ) is a positive ElP (n, H)-equivalence.

We do not have a sharp statement as to which general ElP (H)- equivalences are positive ElP (H) equivalences. However, the restriction above to matrices satisfying C1 and C2 is rather mild. Condition C2 is a harmless technical condition (achievable by replacing A with a larger stabilization) which is needed below to apply the Factorization Theo- rem proved in [12] for the case that the presenting matrix A over Z+G is essentially irreducible. Any G-SFT can be presented by a matrix in FLOW EQUIVALENCE OF G-SFTS 36

o some MP (C, n, H) (by Proposition 4.23) which satisfies C1 and C2 (by Proposition 7.11). Before turning to the setting of Theorem 7.2, we remark on a possible future application. Remark 7.3. Let S be a nontrivial mixing SFT defined by a matrix A over Z+. There is a “Bowen-Franks” representation, a homomorphism β from the mapping class group of S to the group of automorphisms of cok(I − A). This map β is fundamental to understanding the mapping class group; for example, quite possibly its kernel is a simple group. That β is surjective is a corollary of the Factorization Theorem of [4] (as proved in [4]), because every automorphism of cok(I −A) is induced by an elementary equivalence over Z. The mapping class group of a G-SFT, an SFT T with G acting by automorphisms, is naturally defined as the centralizer in the mapping class group of T of the subgroup of elements induced by the G-action. For a mixing G-SFT we likewise have a map β from this centralizer into the automorphism group of the ZG-module cok(I − A). The Factor- ization Theorem tells us that the automorphisms in the range of β are those induced by elementary ZG equivalences. The problem of characterizing these has not to our knowledge been addressed. Whatever analogous algebraic structures are developed for the case of reducible G-SFTs, Theorem 7.2 will be a tool for characterizing the range of invariants. It will be convenient to have a setting in which we work with equivalences (A − I) → U(A − I)V (rather than (I − A) → U(I − A)V ), within a class of matrices A − I which are “as positive as possible”. We develop the apparatus for this next.

Deﬁnition 7.4. Given H and i ≺ j, Dij = Dij(H) is the set of (Hi,Hj) double cosets contained in Hij, and Rij = Rij(H) is the set of D ∈ Dij such that

(7.5) i ≺ k ≺ j =⇒ HikHkj ∩ D = ∅ . We also deﬁne C (7.6) R = {(i, j, D): D ∈ Rij, i ∈ C and j ∈ C} . o When H is the coset structure of a matrix A in MP (n, H), and g ∈ D ∈ Rij, then g cannot be the weight of a path which goes from Ii to Ij without passing some other Ik.

Example 7.7. With G = S3, for the G-SFT given by e e e + (12) 0 e e  0 0 (123) we have C = {1, 2, 3}, H1 = H2 = {e}, H3 = A3, H12 = {e}, H13 = S3, H23 = A3. H13 contains both (H1,H3) double cosets, namely D1 = A3 C C and D2 = S3\A3. We have (1, 3,D1) ∈/ R and (1, 3,D2) ∈ R . FLOW EQUIVALENCE OF G-SFTS 37

If D is a nonempty subset of G, then πD is the projection X X πD : ngg 7→ ngg . g∈G g∈D P An element g∈G ngg is D-positive if ng ≥ 0 for any g and ng > 0 precisely when g ∈ D. The terms are used for matrices when the conditions hold entrywise.

++ Deﬁnition 7.8. MP (C, n, H) is the set of matrices M in MP (n, H) whose blocks M{i, j} satisfy the following conditions: o (1) M + I ∈ MP (C, n, H). (2) M{i, i} is Hi-positive if i∈ / C . (3) If i ≺ j and D ∈ Dij, then C (i)(i, j, D) ∈/ R =⇒ πDM{i, j} is D-positive C (ii)(i, j, D) ∈ R =⇒ πDM{i, j} > 0 .

By deﬁnition, the condition πDM{i, j} > 0 means that every entry of πDM{i, j} is nonnegative and nonzero. Note, Condition 1 above implies that M{i, i} has the form (gi − 1) for some gi ∈ G if i ∈ C. ++ Deﬁnitions 7.9. An elementary positive equivalence in MP (C, n, H) 0 is an ElP (n, H) equivalence (U, V ): B → B = UBV such that the following hold: 0 ++ (1) B,B ∈ MP (C, n, H); (2) one of U, V equals I; (3) one of U, V is a basic elementary matrix. ++ A positive equivalence in MP (C, n, H) is a composition of elementary ++ positive equivalences in MP (C, n, H). For such an equivalence, we use notations such as 0 (U,V ) 0 (U, V ): B −−−→ B or B −−−→ B0 or B −−−→ B . + + +

0 o Observation 7.10. Suppose A, A are in MP (C, n, H); B = A − I, B0 = A0 − I; and (U, V ): B −−−→ B0 + . 0 Then (U, V ): I − A → I − A is a positive ElP (n, H)-equivalence.

o Proposition 7.11. Suppose A ∈ MP (C, n, H). <0> o (1) Suppose A is the stabilization of A in MP (C, k, H), where ki = ni +2 if i∈ / C and ki = ni if i ∈ C. Deﬁne m = (k1, . . . , kn) 0 by mi = ki if i∈ / C, mi = 1 if i ∈ C. Then there is a matrix A in o MP (C, m, H), satisfying C1 and C2, such that the 1-stabilization o 0 in MP (C, k, H) of I − A is positive ElP (k, H)-equivalent to I − A<0>. FLOW EQUIVALENCE OF G-SFTS 38

0 (2) Suppose A satisﬁes C1 and C2. Then there is a matrix A , sat- 0 ++ isfying C1 and C2, such that A − I ∈ MP (C, n, H) and there 0 is a positive ElP (n, H)-equivalence from I − A to I − A .

We postpone the proof of Proposition 7.11 to later in this section. Proposition 7.11, along with Observation 7.10, tells us that (after accepting that we might need to pass to slightly larger matrices to satisfy C2, and then to smaller matrices to satisfy C1) we have reduced the problem of showing every ElP (n, H)-equivalence of matrices I − A o (with A from MP (C, n, H)) is positive to the problem of showing every ++ ElP (n, H)-equivalence of matrices M (with M from MP (C, n, H)) is ++ a positive equivalence in MP (C, n, H). Unfortunately, as we see in Example 7.12 below, Proposition 7.11 ++ would not be true if in the deﬁnition of MP (C, n, H) the condition 3 “πDM{i, j} > 0” in 3(ii) were strengthened to D-positivity. This complication accounts for a good deal of the diﬃculty of arguments to come.

Example 7.12. With G = S3, deﬁne matrices over Z+G, e e e + (12) (12) e A = and A = 0 e e . 1 0 (12) 2   0 0 (123)

For A1,(G, P) = (S3, P2); H is given by h(12)i = H1 = H2 = H12; C = {1, 2}; and D = H12 is an (H1,H2) double coset, with (1, 2,D) ∈ C R . A basic positive ElP (n, H)-equivalence (I −A) → (I −B) must be given by right or left multiplication of (I − A) by one of four matrices, 1 ±g 0 1 with g ∈ S3. The orbit of I − A1 under such equivalences (12)(12) consists of just I − A1 and I − A3, where A3 = 0 (12) . Neither A1 − I nor A3 − I is D-positive. (Alternately, we can see that a matrix (12) x B = 0 (12) defining a G-SFT which is G-flow equivalent to TA1 , with x = n1e + n2(12) ∈ Z+G, must satisfy n1 + n2 = 1, because 1 1 1 n1+n2 the augmentations A1 = ( 0 1 ) and B = ( 0 1 ) must define flow equivalent SFTs.) C For A2, continuing from Example 7.7 we have (1, 3,D2) ∈ R .A nonidentity basic positive ElP (n, H)-equivalence can only be given by multiplication by a matrix Est(±g), with g ∈ S3 and s < t. It is straightforward to check that if I − B is in the orbit of I − A1 under such positive equivalences, then B(s, t) ∈ Z+A3 if (s, t) 6= (1, 3), and for B(1, 3) = P n g we have P n = 1. Thus, B − I cannot be g∈G g g∈ /A3 g D2-positive.

3From its proof, one sees that Prop. 7.11 can be strengthened such that, given C 0 P k ∈ N, for D ∈ Dij \R , entrywise A {i, j} ≥ k g∈D g. Such ﬂexibility is not available for D in RC. FLOW EQUIVALENCE OF G-SFTS 39

Before giving the proof of Proposition 7.11, we introduce further notation. Deﬁnition 7.13. We deﬁne X δij = g if i ≺ j

g∈Hij X δi = g

g∈Hi

Pκ(gi)−1 m If i ∈ C, then δi = m=0 (gi) , where κ(g) is the order of g in G.

Definition 7.14. Let S = {(i, j) ∈ P × P : i ≺ j}. Define S0 = ∅. Inductively, given Sm, define Sm+1 to be the set of (i, j) in S \ ∪r≤mSr such that m [ i ≺ k ≺ j =⇒ {(i, k), (k, j)} ⊂ Sk . k=0

Deﬁne ρ(i, j) = m if (i, j) ∈ Sm.

To make the sets Sm more concrete, we prove the next proposition. For this, we deﬁne a path of length ` in P from i0 to i` to be a string (i0, i1, . . . , i`) such that it−1 ≺ it for 1 ≤ t ≤ `. Such a path is maximal if (it−1, it) ∈ S1 for 1 ≤ t ≤ `. For i ≺ j, deﬁne ρ(i, j) to be the maximum length of a path from i to j.

Proposition 7.15. (i, j) ∈ Sm if and only if ρ(i, j) = m. Proof. The proof is by induction on m. The basis step m = 1 is straightforward. Assume the claim is true for m. If ρ(i, j) = m+1, then clearly (i, j) ∈ ∪t≤m+1St; by the induction hypothesis, (i, j) ∈/ ∪t≤mSt, and therefore (i, j) ∈ Sm+1. Now suppose (i, j) ∈ Sm+1. Then for every maximal path (i, i1, . . . , ik, j), we have by the induction hypothesis that ρ(i, ik) ≤ m, and therefore k ≤ m and the length of (i, i1, . . . , ik, j) is at most m+1. Therefore ρ(i, j) ≤ m+1. By the induction hypothesis, ρ(i, j) > m, and therefore ρ(i, j) = m + 1. Note, i ≺ k ≺ j implies max{ρ(i, k), ρ(k, j)} < ρ(i, j). This is a key point for the proof by induction below.

Example 7.16. With P = P4, we have S partitioned into S1 = {(1, 2), (2, 3), (3, 4)}, S2 = {(1, 3), (2, 4)} and S3 = {(1, 4)}. <0> Proof of Prop. 7.11. (1) Clearly A satisfies C2, as does any matrix <0> ElP (k, H)-equivalent to A . Now apply the initial part of the proof of Proposition 4.23 to A<0>, iterating the trimming move of Example 3.8.1 to produce a matrix A1 in which a diagonal entry is zero iff the row and column through that entry are zero. By induction, these o trimming moves are positive ElP (k, H)-equivalences in MP (C, n, H), o because A1 ∈ MP (C, n, H). If i ∈ C, then the nonzero diagonal entry FLOW EQUIVALENCE OF G-SFTS 40 of A1{i, i} must be unique. It follows that A1 is the stabilization of a matrix A0 as claimed. o For the proof of (2), suppose A ∈ MP (C, n, H), satisfying C1 and C2. Without loss of generality, suppose also that a diagonal entry in A is zero iff the row and column through that entry are zero. Given i∈ / C, let Ci = A{i, i}. Steps 0,3,4,5,6 of [12, Lemma 6.6] give an explicit string of basic positive ZHi-equivalences which send I−Ci to a matrix I−Bi such that Bi − I is Hi-positive. These basic positive equivalences are given by row and column cuts (recall subsections 3.7,3.8) and therefore the elementary row and column operations defining them give a string of basic positive ElP (n, H)-equivalences from I−A to some matrix I−B in o MP (C, n, H) such that (B{i, i}−I) is Hi-positive. After applying such equivalences for every i∈ / C, and renaming, we may assume without loss of generality that A satisfies all conditions of Definition 7.8 except perhaps condition (3). To arrange for condition (3), we consider double cosets D ∈ Dij by induction, addressing the sets Dij in an order with ρ(i, j) nondecreas- ing. The positive equivalences below are compositions of ElP (n, H)- equivalences, of the form (I − B) → (I − C) with B ≤ C. So, all the matrices C continue to satisfy Conditions 1 and 2. Also, if Condition 3 is satisfied by B for D ∈ Di0j0 , then this remains true for C. In particular, when considering D ∈ Dij, if e.g. ρ(i, k) < ρ(i, j) then we may assume Condition 3 holds for D in Dik. We begin with a claim. Claim 1. Suppose i∈ / C or j∈ / C;(s, t) ∈ Ii × Ij; g ∈ Hij; and A(s, t) ≥ g ∈ Hij. Let D = HigHj. Then there is a positive equivalence (I − A) → (I − A0) where A0 ≥ A and A0{i, j} is D-positive.

Proof of Claim 1. We assume j∈ / C (the proof for the case i∈ / C is similar). Let E = Est(g). There is a positive equivalence (I − A) → (I − B) (a (g, s, t) row cut equivalence, as in Sec 3.7) implemented by I − B = E(I − B). Here row s of B{i, j} is gHj-positive, because

B(s, t) = A(s, t) − g + gA(t, t) ≥ gδj , 0 0 0 0 0 B(s, t ) = A(s, t ) + gA(t, t ) ≥ gδj if t ∈ Ij and t 6= t , and the inequalities hold because (A − I){j, j} is Hj-positive. For use in Subcase 2, after performing a second (g, s, t) row cut equivalence (if necessary), we obtain B0 with

0 0 0 0 0 0 B (s, t ) = B(s, t ) + gB(t, t ) ≥ A(s, t ) + 2gδj if t ∈ Ij . There are two cases for the rest of the proof. 0 0 Subcase 1: i∈ / C. Suppose t ∈ Ij; B(s, t ) ≥ h; and E = Est0 . Choose gh ∈ gHj. Then there is a positive equivalence to (I − B) → (I −C) (a (gh, s, t0) column cut equivalence, as in Sec 3.8) implemented 0 by (I − C) = (I − B)E. Here column t of C{i, j} is HigHj positive, FLOW EQUIVALENCE OF G-SFTS 41 because 0 0 0 C(s, t ) = B(s, t ) − gh + B(s, s)A(s, t ) ≥ δigδj , 0 0 0 0 0 0 0 0 C(s , t ) = B(s , t ) + B(s , s)A(s, t ) ≥ δigδj , if s ∈ Ii and s 6= s , where the inequalities hold because (B − I){i, i} is Hi-positive. So, 0 0 after implementing (h, s, t ) column cuts for each t in Ij, we pass to I − A where A{i, j} is D-positive. 0 Subcase 2: i ∈ C. Now Ii = {s}. Let B(s, s) = gi. Suppose t ∈ Ij. ` 0 k 0 Let (gi) = 1. Starting from B , apply ((gi) , s, t ) column cuts, for 0 0 k = 0, 1, . . . , ` − 1, to produce C0,...,C`−1. Because B (s, t ) ≥ 2gδj, we have 0 0 0 0 0 C0(s, t ) = B (s, t ) − g + giB (s, t ) > gδj + gig2δj 0 0 0 2 C1(s, t ) = C0(s, t ) − gig + giC0(s, t ) > gδj + gigδj + (gi) 2gδj ... 0 `−1 C`−1(s, t ) > gδj + gigδj + ··· + (gi−1) (δj − 1) = δigδj .

Now we consider D ∈ Dij. If D ∈ Rij, then there must be some g in D and some (s, t) in Is × It such that A(s, t) ≥ g. If {i, j} ⊂ C, then A{i, j} has only one entry, and therefore Condition 3(ii) holds for D. If {i, j} is not contained in C, then by Claim 1 there is a positive equivalence to some A0 with A0{i, j} D-positive. From here, suppose (i, j, D) ∈/ R. Because H is a coset structure for A, we deduce that there exists k such that i ≺ k ≺ j and D ⊂ HikHkj. So, given i ≺ k ≺ j, we will ﬁnish by producing a positive equivalence 0 0 0 replacing A with a matrix A , A ≥ A, such that A {i, j} is HikHkj- positive. This argument goes by cases. For an element i of C, we let gi be the unique entry of A{i, i}. Case 1: k∈ / C, or {i, j} ∩ C = ∅. Suppose (s, r) ∈ Ii × Ik. Perform an (A(s, r), s, r) row cut equivalence on A to produce a matrix B such that

B(s, r) = A(s, r)B(r, r) ≥ δik

B(s, t) = A(s, t) + A(s, r)B(r, t) ≥ δikδkj if t ∈ Ij , where the inequalities hold because, by the induction hypothesis, (A − I){i, k} is Hjk positive and (A − I){k, j} is Hkj positive. Thus row s of B{i, j} is now HikHkj positive. Repeat as needed for all s in Ii to 0 0 obtain A such that A {i, j} is HikHkj positive. Case 2: {i, k, j} ⊂ C. Let Ii × Ik × Ij = {(s, r, t)}. Looking only at the principal submatrices on indices {s, r, t}, let A = gi b c 0 gk d . By the induction hypothesis, for D ∈ Hik we have πD(b) > 0 0 gj 0, and for D ∈ Hkj we have πD(d) > 0. Therefore HikHkj ⊂ HibHkdHj. FLOW EQUIVALENCE OF G-SFTS 42

So, it suffices to give a string of positive equivalences producing A0 with 0 A (s, t) ≥ δibδkdδj. t Let ` = κ(gi). We apply in order (b(gk) , s, r) row cuts, for t = 0, 1, . . . , ` − 1, starting with A. The first equivalence (t = 0) is implemented by       1 b 0 1 − gi −b −c 1 − gi −bgk −c − bd 0 1 0  0 1 − gk −d  =  0 1 − gk −d  0 0 1 0 0 1 − gj 0 0 1 − gj After the last (t = ` − 1) equivalence we reach  ` 0  1 − gi −b(gk) −c I − B =  0 1 − gk −d  0 0 1 − gj ` 0 where B ≥ A (because b(gk) = b) and c = −A(s, t) − b(1 + gk + ··· + `−1 (gk) d = A(s, t)+bδkd. We follow this with a column cut equivalence,  0     00  1 − gi −b −c 1 0 bδk 1 − gi −b −c  0 1 − gk −d  0 1 0  =  0 1 − gk −d  0 0 1 − gj 0 0 1 0 0 1 − gj and then repeat the first move to reach a matrix I − C, C ≥ A, with C(s, t) = A(s, t) + (1 + gi)bδkd. Iterating this process leads to a matrix 0 00 I − D such that D(s, t) = A(s, t) + δibδkd := d . Setting δ = δibδkd, we then apply the equivalence  00  0   00  1 0 δ 1 − gi −b −d 1 − gi −b −d 0 1 0   0 1 − gk −d  =  0 1 − gk −d  0 0 1 0 0 1 − gj 0 0 1 − gj to reach a matrix I −E for which E(s, t) = A(s, t)+δabδkdgj. Iterating the move that produced E from A, we arrive at A0 with A0(s, t) = A(s, t) + δabδkdδj. Case 3: {i, k} ⊂ C, j∈ / C. Suppose t ∈ Ij. Use the Case 2 moves which led the matrix D with D(s, t) = A(s, t) + δabδkd. By the induction hypothesis, d is Hkj positive, and therefore E(s, t) is HikHkj 0 0 positive. Iterating this move over t ∈ Ij, we reach A such that A {i, j} is HikHkj positive. Case 4: {j, k} ⊂ C, i∈ / C. The proof here is similar to the proof for Case 3.

8. The Factorization Theorem: proof This section is devoted to the proof of the Factorization Theorem 7.2. We use and generalize proof techniques from [12] and [4]. We will prove three lemmas, and then use them in a short argument to ﬁnish the proof of Theorem 7.2. FLOW EQUIVALENCE OF G-SFTS 43

Let UP (n, H) be the set of matrices M in ElP (n, H) such that every diagonal block M{i, i} is the identity matrix. We will address equivalences (U, V ) for matrices U, V in UP (n, H). In Lemma 8.2, we will consider 2 × 2 upper triangular matrices. Recall P2 = {1, 2} with 1 ≺ 2.

Deﬁnition 8.1. Suppose m = (m1, m2); H is a (G, P2) coset structure; and B,B0 are in M++(m, H). A string of basic elementary positive P2

UP2 (H)-equivalences

(E ,F ) (E ,F ) (E ,F ) B −−−−→1 1 −−−−→2 2 ... −−−−→t t B0 + + + is extendable if the matrix products Ei ··· E1 and F1 ··· Fi are nonnegative, 1 ≤ i ≤ t . In this case, with (U, V ) = (Et ··· E2E1,F1F2 ··· Ft), we also say that (U, V ): B → C is an extendable positive equivalence. Our interest in extendable equivalences is the following. Suppose 0 ++ 0 M,M are in MP (C, n, H) with 2 × 2 principal submatrices B,B on the same coordinate indices s, t contained in a block {i, j} with i ≺ j. Then an extendable positive equivalence of B,B0 (with respect to the restriction of H) will give (by the same elementary operations) 0 ++ a positive equivalence from M to M in MP (C, n, H). 0 ++ Lemma 8.2. Given matrices B,B in MP (C, n, H), suppose (U, V ): 0 B → B is a UP (H)-equivalence which only diﬀers from the identity at blocks U{i, j} and V {i, j}, and that every entry of U{i, j} and V {i, j} 0 lies in Z+G. Then (U, V ): B → B is an extendable positive ElP (H) equivalence, and consequently (U, V ): B −−−→ B0 + Proof. Case 1 i 6∈ C, j 6∈ C: Since B,B0 have positive entries in all relevant diagonal blocks we can simply decompose U and V one entry at a time, thus obtaining an extendable positive Hij-equivalence at every step. Case 2 i ∈ C, j 6∈ C: As in Case 1, (U, I): B −→U B. So, without loss of generality, we + can assume (U, V ) = (I,V ). Considering compositions, it is enough to address the case that V has a single nonzero oﬀdiagonal entry, say, V (1, 2) = s 6= 0. Then, in the principal submatrices on indices g−1 r {1, 2}, the equivalence (I,V ): B → BV has the form B = 0 h → g−1 r+(g−1)s 0 h . Choose p in Z+Hij such that p > s; then ph > s, because h is Hj-positive. Applying the equivalences (E12(p),I), (I,V ), (E12(−p)) produces g−1 r+ph g−1 r+ph+(g−1)s g−1 r+(g−1)s B → 0 h → 0 h → 0 h . FLOW EQUIVALENCE OF G-SFTS 44

Let E12(p) = Ek ...E1, E12(s) = F1 ...Fk be factorizations into nonnegative matrices with oﬀdiagonal entries in Hij Then (I,V ) is the com- −1 −1 position (E1,I), ··· , (Ek,I), (I,F1), ··· , (I,Fk), (Ek ,I), ··· , (E1 ,I) and therefore (I,V ) is extendable. Case 3 i 6∈ C, j ∈ C: The proof here is essentially as for Case 2. Case 4 i ∈ C, j ∈ C: Note ﬁrst that in this case, the nontrivial matrices U{i, j} and V {i, j} are 1 × 1. Let p be the entry of U{i, j} and s the entry of V {i, j}; we have assumed that p, s ∈ Z+G. The proof is by induction on K = p + s, and the lemma is true for K = 0. Suppose p + s = K > 0 and the lemma holds if p + s < K. Here the submatrix of the equivalence UBV = B0 containing any change has the form 1 p g − 1 r 1 s g − 1 r0 (8.3) = . 0 1 0 h − 1 0 1 0 h − 1 where (8.4) r0 = r + p(h − 1) + (g − 1)s

1 x We use E(x) to denote a matrix ( 0 1 ); e.g., U = E(p). For any x, y, z, (8.5) 1 x g − 1 y 1 z g − 1 y + x(h − 1) + (g − 1)z = 0 1 0 h − 1 0 1 0 h − 1 so here the pair (E(x),E(z)) acts by adding x(h − 1) + (g − 1)z to the (1, 2) entry. The equivalence given by (U, V ) is a composition of basic elementary equivalences, given by (I,E(w)) or (E(w),I), with w ∈ G a summand of p or s. Such an equivalence acts by adding a term w0 −w to the 1, 2 position, where w0 is wh or gw. Case 4(i): 0 P 0 P 0 Assume r 6= r . Let r = w nww and r = w nww. The images r and r0 under the augmentation must be equal. So, there must be some 0 w ∈ G such that nw > nw. Therefore w must be a summand of p or s, and (I,E(w)) or (E(w),I) applied to B is a positive equivalence in ++ MP (C, n, H). Now the equivalence given by (U, V ) is this positive equivalence followed by one satisfying the induction hypothesis. A composition of extendable equivalences is extendable. This completes the inductive step if r 6= r0. Case 4(ii): Assume r = r0 and note that in this case (8.6) p + s = ph + gs according to (8.4). Suppose w0 is a summand of p + s. Then w0 ∈ H12, since (U, V ): 0 ++ B → B is a UP (H)-equivalence. Because B ∈ MP (C, n, H), there FLOW EQUIVALENCE OF G-SFTS 45

i j must be a summand x of r and i, j such that w0 = g xh . We then have a positive equivalence (E,F ): B → B0 deﬁned by g − 1 r (I,E(x)) (I,E(gx)) (I,E(gi−1x)) −−−−→ −−−−−→ ··· −−−−−−−→ 0 h − 1 + + + i j (E(gix),I) (E(gixh),I) (E(gixhj−1),I) g − 1 r + g xh − x −−−−−−→ −−−−−−→ ··· −−−−−−−−→ + + + 0 h − 1

Let (Et,Ft) be the tth of these basic positive equivalences, so, (E,F ) = (Ei+j ··· E1,F1 ··· Fi+j). Deﬁne ( 0 0 (E(w0),I) if w0 is a summand of p (E1,F1) = (I,E(w0)) otherwise .

0 0 0 0 Now (E1,F1): B0 −→ E1B0F1 =: B1, with B1(1, 2) = B0(1, 2)+w1 −w0, + 0 0 where w1 = w0h if E1 = E(w0) and w1 = gw0 if F1 = E(w0). In 0 0 either case, according to (8.6) and the deﬁnition of (E1,F1), w1 is a summand of p + s. Since B1(1, 2) = B0(1, 2) + w1 − w0, if w1 6= w0 then w1 must be a summand of p + s − w0, and we may construct a 0 0 positive equivalence (E2,F2): B1 → B2 as before, with w1 in place of w0 and B2(1, 2) = B1(1, 2) + w2 − w1 = B0(1, 2) + w2 − w0. Since p + s 0 is ﬁnite and r = r , this process must reach some wm = w0, and we 0 0 0 0 0 0 set (E ,F ) = (Em ··· E1,F1 ··· Fm). Because Bm = B0, we have by composition a positive equivalence

0 0 −1 −1 0 0 (E,F ) (E ,F ) (E ,F ) (E ,F ): B −−−→ B0 −−−−→ B0 −−−−−−→ B. + + +

(E,F ) (E0,F 0) The equivalence B −−−→ B0 −−−−→ B0 is extendable because the ma- 0 0 trices Et,Ft,Et,Ft are nonnegative. Extendability through the remaining basic equivalences holds because

 −1 −1 0 0 −1 −1 Et ··· E1 E E,FF F1 ··· Ft

 0 −1 −1 −1 −1 0 = E Et ··· E1 E,FF1 ··· Ft F

0 0 = E Et+1 ··· Ei+j,Ft+1 ··· Fi+jF and all the matrices in the last line are nonnegative. The equivalence (U(E0)−1, (F 0)−1V ): B → B has the form (E(p0),E(s0)) with p0 + s0 = p + s − m < p + s, and therefore is extendable by the induction hypothesis. This ﬁnishes the inductive step for Case 2(ii). 0 Lemma 8.7. Suppose U and V are matrices in UP (n, H), B and B are in M++(C, n, H), and UBV = B0. Then (U, V ): B −→ B0. P +

Proof. Recalling Deﬁnition 7.14, let (i1, j1),..., (ir, jr) be an enumera- tion of elements of S such that t ≤ s =⇒ ρ(it, jt) ≤ ρ(is, js). FLOW EQUIVALENCE OF G-SFTS 46

We will deﬁne various matrices by induction, beginning with B0 = 0 0 0 B,B0 = B ,U0 = U, V0 = V . For 1 ≤ s ≤ r, given Bs−1,Bs−1,Us−1,Vs−1 0 ++ with Bs−1,Bs−1 ∈ MP (C, n, H) and Us−1,Vs−1 ∈ UP (n, H), we choose matrices Ps,Qs in UP (n, H), equal to I outside block {is, js}, such that the following Positivity Conditions hold:

(1) For some nonnegative integer Ms, every entry of Ps{is, js} and every entry of Qs{is, js} equals Msδiδijδj. (2) The blocks (PsUs−1){is, js} and (Vs−1Qs){is, js} have all entries in Z+Hij.

We note that by taking Ms large in (1), we can achieve (2). We then deﬁne matrices Ws,Xs in UP (n, H), equal to I outside block {is, js}, by setting

Ws{is, js} = (PsUs−1){is, js}

Xs{is, js} = (Vs−1Qs){is, js} .

Finally we deﬁne

−1 Us = PsUs−1Ws Bs = WsBs−1Xs −1 0 0 Vs = Xs Vs−1Qs Bs = PsBs−1Qs

0 ++ Then Us,Vs ∈ UP (n, H) and Bs,Bs ∈ MP (C, n, H), 0 ≤ s ≤ r. For 1 ≤ s ≤ r, we will verify the following claims by induction.

0 (a) UsBsVs = Bs

0 (Ps,Qs) 0 (b) Bs−1 −−−−→ Bs +

(Ws,Xs) (c) Bs−1 −−−−−→ Bs + −1 −1 (d) Us = Ps ··· P1UW1 ··· Ws and −1 −1 Vs = Xs ··· X1 VQ1 ··· Qs

(e) Us{it, jt} = 0 = Vs{it, jt} if 1 ≤ t ≤ s .

Before proving (a)-(e), suppose all these claims hold. Deﬁne P = PrPr−1 ··· P1 and Q = Q1Q2 ··· Qr. From (b), we have

0 0 (P1,Q1) 0 (P2,Q2) 0 (Pr,Qr) 0 0 B = B0 −−−−→ B1 −−−−→ B2 ··· −−−−→ Br = PB Q + + +

0 (P,Q) 0 and therefore B −−−→ PB Q. Similarly, let W = Wr ··· W1 and X = + X1 ··· Xr. From (c) we have

(W1,X1) (W2,X2) (Wr,Xr) B = B0 −−−−−→ B1 −−−−−→ B2 ··· −−−−−→ Br = WBX + + + FLOW EQUIVALENCE OF G-SFTS 47

(W,X) and therefore B −−−→ WBX. Because {Ur,Vr} ⊂ UP (n, H), from + (e) with s = r we get Ur = I = Vr. Using (d) at s = r, we then get −1 −1 −1 I = Ur = Pr ··· P1UW1 ··· Wr = PUW −1 and similarly I = Vr = X VQ. Therefore (P U, V Q) = (W, X) and

(P U,V Q) (P −1U,V Q−1) B −−−−−→ PUBVQ = B0Q −−−−−−−−→ B0 . + + This shows (U, V ): B → B0 is a positive equivalence. To ﬁnish the proof it remains to verify (a)-(e) for 1 ≤ s ≤ r. 0 Proof of (a). We have U0B0V0 = B0. Suppose 0 < s ≤ r and (a) holds at s − 1. Then

−1 −1 UsBsVs = PsUs−1Ws WsBs−1Xs Xs Vs−1Qs = Ps Us−1Bs−1Vs−1 Qs 0 0 = PsBs−1Qs = Bs . 0 0 ++ Proof of (b). We have B0 = B ∈ MP (C, n, H). Suppose 1 ≤ 0 ++ 0 s ≤ r and Bs−1 ∈ MP (C, n, H). An entry in Bs−1 could decrease in PsBs−1Qs only as a result of addition of a term (Msδiδijδj)(gj − 1) or (gi − 1)(Msδiδijδj). But, these terms vanish, as (gi − 1)δi = 0 = δj(gj − ++ 1). Therefore PsBs−1Qs ≥ Bs−1. Because Bs−1 ∈ MP (C, n, H), this ++ implies PsBs−1Qs ∈ MP (C, n, H). Now enumerate the coordinates of the nonzero oﬀ-diagonal entries of Qs as (a1, b1),..., (aT , bT ). For 1 ≤ t ≤ T , let Et be the basic elementary matrix such that Et(at, bt) = Qs(at, bt). Because these en- QT tries lie in blocks {is, j} with is ≺ j, we have Qs = t=1 Et. This 0 0 (I,Q): Bs−1 → Bs−1Q is a composition of equivalences

0 0 (I,E1) 0 (I,E2) (I,ET ) 0 0 Bs−1 := Bs−1,0 −−−→ Bs−1,1 −−−→ · · · −−−−→ Bs−1,T = Bs−1Qs 0 ++ By induction, for 1 ≤ t ≤ T , Et is nonnegative and Bs−1,t ∈ MP (C, n, H). It then follows from Lemma 8.2 that each (I,Et) gives a positive equiva- 0 (I,Qs) 0 0 (Ps,I) 0 lence. Thus Bs−1 −−−→ Bs−1Qs, and similarly Bs−1Qs −−−→ PBs−1Qs. + + 0 (Ps,Qs) 0 By composition, Bs−1Q −−−−→ PsBs−1Qs. + ++ Proof of (c). We have B0 ∈ MP (C, n, H). Now suppose 1 ≤ s ≤ r ++ and Bs−1 ∈ MP (C, n, H). The matrices Ws and Xs are in UP (n, H), with all entries in Z+G, and Bs−1 ≤ WsBs−1Xs := Bs. Therefore ++ Bs ∈ MP (C, n, H). An argument very similar to the proof of claim 0 (Ws,Xs) (b) now shows that Bs−1 −−−−−→ Bs. + Proof of (d). The claim (d) follows by induction from the deﬁnitions −1 −1 U0 = U, V0 = V , Us = PsUs−1Ws and Vs = Xs Vs−1Qs. FLOW EQUIVALENCE OF G-SFTS 48

Proof of (e). Suppose 1 ≤ s ≤ r and (e) holds at s−1. (At s−1 = 0, −1 (e) is an empty statement.) We have Us = PsUs−1Ws , with Ps and −1 Ws equal to I outside block {is, js}. On account of the zero block structure of matrices in UP (n, H), we have Us = Us−1 except possibly in blocks {i, j} such that i is or j js. At (is, js), we have

 −1 Us{is, js} = (PsUs−1)Ws {is, js} −1 = (PsUs−1){is, js} + Ws {is, js}

= (PsUs−1){is, js} − Ws{is, js} = 0 .

Now suppose i ≺ is. Then Us{i, j} = Us−1{i, j} except possibly in the case j = js, where −1 (8.8) Us{i, js} − Us−1{i, js} = Us−1{i, is}Ws {is, js} .

The right side of (8.8) can be nonzero only if i ≺ is ≺ js = j. In this case, ρ(is, js) < ρ(i, j), so (i, j) cannot equal (it, jt) for any t less than s. Thus if 1 ≤ t < s, then Us{it, jt} = Us−1{it, jt}, which is zero by the induction hypothesis. The analogous argument for the case js ≺ j ﬁnishes the proof.

We make contact to the case with U, V ∈ UP (H) from the general case using the following lemma in combination with a key result from [12].

Lemma 8.9. Suppose i∈ / C, E is a basic elementary matrix in ElP (H); 0 ++ E{j, k} = I{j, k} when (j, k) 6= (i, i); B,B ∈ MP (H); and (E{i, i},I): B{i, i} −→ B0{i, i} . +

Then there exists V in UP (n, H) such that (E,V ): B −→ EB0V. + Similarly, if (I,E{i, i}): B{i, i} −→ B0{i, i} + then there exists U in UP (n, H) such that (U, E): B −→ UB0E. + Proof. We will consider the equivalence (E,I); the other case is similar. Let E(s, t) = v be the nonzero oﬀ-diagonal entry of E. E acts on B from the left to add v times row t of B to row s of B. If each block {i, `} of EB is Hi`-positive (e.g., if v ≥ 0), then set V = I. Otherwise, pick r an index for a column through the {i, i} block. For a positive integer L, let V be the matrix in UP (n, H) such that (i) if FLOW EQUIVALENCE OF G-SFTS 49 i ≺ `, then every entry of V {i, `} equals Lδi` and (ii) in other entries, V agrees with I. Then for (s, q) in block {i, `},

(EBV )(s, q) ≥ (EB)(s, q) + (B(s, r) − B(t, r))(Lδi`) .

Because (EB){i, i} is Hi-positive, for suﬃciently large L the displayed (I,V ) sum must for each such ` be Hi`-positive. Then B −−−→ BV and + ++ EBV ∈ MP (C, n, H), so

(I,V ) (E,I) B −−−→ BV −−−→ EBV + + as required.

Proof of Theorem 7.2. It follows from Observation 7.10 and Proposi- tion 7.11 that to prove Theorem 7.2 we may assume that B,B0 ∈ ++ MP (H). 0 Thus let (U, V ): B → B be the given ElP (n, H) equivalence, with 0 ++ 0 0 B,B ∈ MP (H). Set U = ⊕iU{i, i} and V = ⊕iV {i, i}. If i ∈ C, then ni = 1 and U{i, i} = (1) = V {i, i}. If i∈ / C, then by [12, Theorem 6.1] we have that

(U{i, i},V {i, i}): B{i, i} → B0{i, i} is a positive ZHi-equivalence through matrices which are Hi-positive. So, there is a string (E1,F1),..., (ET ,FT ) of elementary ElP (n, H)- equivalences which accomplishes the elementary positive equivalence decomposition inside the diagonal blocks, such that each Et and Ft equals the identity outside diagonal blocks {i, i} with i∈ / C. By Lemma 8.9, we may ﬁnd (U1,V1),..., (Ut,Vt) with each Us and Vs in UP (n, H) such that (U ,F ) (E ,V ) (U ,F ) (E ,V ) B −−−−→1 1 −−−−→1 1 ··· −−−−→t t −−−−→t t B∗ . + + + +

Let X = EtUt ··· E2U2E1U1. Let Y = F1V1F2V2 ··· FtVt. Then for all −1 i in P, X{i, i} = U{i, i} and Y {i, i} = V {i, i}, so UX ∈ UP (n, H) −1 and Y V ∈ UP (n, H). It then follows from Lemma 8.7 that

(UX−1,Y −1V ) B∗ −−−−−−−−→ B0 . +

Thus (U, V ): B → B0 is the composition

(X,Y ) (UX−1,Y −1V ) B −−−→ B∗ −−−−−−−−→ B0 + + and therefore (U, V ): B −→ B0. + FLOW EQUIVALENCE OF G-SFTS 50

9. Conclusion of proofs We begin with the promised proof of Theorem 5.3. (1) =⇒ (2): This implication follows directly from Theorem 6.2. (2) =⇒ (1): Suppose (2) holds. Applying first Proposition 7.11 if needed, it follows from Theorem 7.2 that there is a positive ElP (m, H)- equivalence from I − A<0> to I − C<0>. There is therefore a positive ZG-equivalence from I − A<0> to I − C<0>. Since every positive ZG- equivalence induces a G-flow equivalence (see Section 3), it follows that TA<0> and TC<0> are G-flow equivalent. Since TA<0> = TA and TC<0> = TC , we thus have that TA and TC are G-flow equivalent. It follows in a similar way from Proposition A.1 and Proposition B.1 that TB and TC are G-flow equivalent. Thus, we have that TA and TB are G-flow equivalent as wanted. Next we describe which equivalence classes of matrices arise in the equivalence classes we use as G-flow equivalence invariants. (The in- variance of these classes under stabilization was discussed in Section 3.11.)

Theorem 9.1. Given G, P, C, H, and n with ni = 1 if and only if i ∈ C, suppose B is a matrix in MP (n, H). Then the following are equivalent.

(1) There is a k ≥ n, with ki = 1 if and only if i ∈ C, and a matrix o <0> A in MP (C, k, H), such that I − A is ElP (k, H) to I − B , <0> where B is the 0-stabilization of B in MP (k, H). (2) The following hold: (a) If i ∈ C, then the 1×1 ith diagonal block of B has the form [g], with Hi generated by g. (b) If i ≺ j, D ∈ Rij and {i, j} ⊂ C and the 1 × 1 ij block of P P B is g∈G ngg, with each ng in Z, then g∈D ng > 0. ++ Moreover, given (2), the matrix A can be chosen from MP (C, m, H), where mi = 1 if i ∈ C and mi = ni + 1 otherwise.

Proof. (2) =⇒ (1): With m as defined in the “Moreover” statement, 0 o 0 let B be the stabilization of B in MP (C, m, H). Let M = B − I, with diagonal blocks Mi. It suffices to apply ElP (m, H) equivalences ++ to M which produce a matrix in MP (C, m, H) (recall Definition 7.8). By [12, Proposition 8.8], for i not in C the matrix Mi is El(ZHi)- 0 equivalent to an Hi-positive matrix, Mi . After applying a block diagonal ElP (m, H) equivalence, we may assume for i∈ / C that Mi is Hi-positive. For these i, in increasing order: for i ≺ j, as needed multiply from the right by matrices in ElP (m, H) zero outside the ij block to put all entries of the ij block of M into ZHij, with strictly positive coefficients. Then similarly for j in decreasing order: for i ≺ j, as needed multiply from the left to achieve this positivity. FLOW EQUIVALENCE OF G-SFTS 51

o At this point, all blocks of M are in form for MP (C, m, H) except perhaps the 1 × 1 ij blocks with {i, j} ⊂ C. First, for each D ∈ Rij: pick an element x from D, and multiply from the left and right by basic elementary matrices, of the form (gixgj) in the ij block, to eﬀect the P P replacement of g∈D ngg with ( g∈D ng)x, which by (2)(b) is positive. P For D not in Rij, the coeﬃcients of g∈D ngg are made positive by elementary multiplications as in the (i, j, D) ∈/ RC step in the proof of Proposition 7.11. We will refrain from reentering the details of this step. (1) =⇒ (2): Suppose (1) holds. Condition (2) holds with A in o place of B, because A ∈ MP (C, k, H) with ki = 1 for i ∈ C. Let <0> I − B = U(I − A)V be the assumed ElP (k, H)-equivalence. For i ∈ C, letting A{i, i} = (gi), we have (I − B){i, i} = (I − B<0>){i, i} = U{i, i}(I − A){i, i}V {i, i} = (1)(1 − gi)(1) . Therefore (2a) holds for B. Given {i, j} ⊂ C, let a, b, u, v denote the entries of the singleton {i, j} subblocks of A, B, U, V . For D ∈ Rij, <0> πD (1 − b) = πD (I − B ){i, j} = πD U{i, i}(I − A){i, i}V {i, i} = πD (1 − a) + u(1 − gj) + (1 − gi)v .

Clearly πD(u(1 − gj)) = 0 = πD((1 − gi)v), and therefore πD(1 − b) = πD(1 − a), and therefore (2b) holds for B. Remark 9.2. In Theorem 9.1, I − B<0> is a 1-stabilization of the matrix L = I−B. The realization can be stated in terms of 1-stabilizations of a matrix L by replacing “B has the form g” in 2(a) with “L has the P P form 1 − g”, and replacing g∈D ng > 0 with g∈D ng < 0 in 2(b). Lastly, we prove a finiteness result. Given G an abelian group, P = o {1,...,N} a poset and A ∈ MP (Z+G), let Ak be the kth diagonal block of A and let d(A) be the N-tuple (det(I − A1),..., det(I − AN )). Up to reordering, d(A) is an invariant of G-flow equivalence of the G-SFT TA defined by A. Theorem 9.3. Let G be a finite abelian group , P = {1,...,N} a poset, H a (G, P) coset structure, and d = (d1, . . . , dN ) an N-tuple of elements of ZG which are regular. Then there are only finitely many ElP (H)-equivalence classes of matrices in the set M(d) := {I − A : o A ∈ MP (H), d(A) = d}. Consequently there are only finitely many flow equivalence classes of G-SFTs TA with d(A) = d. Proof. For A in M(d), the set C of cycle components must be empty, and for each i, the matrix I − Ai is injective and the ZHi-module cok(I − Ai) has finite size, determined by det(I − Ai). We will use FLOW EQUIVALENCE OF G-SFTS 52 some facts from [12, Section 9], which contains more detail. A theorem of Fitting shows that if I −A and I −B are injective matrices over ZHi with isomorphic cokernels, then there are m, n such that (I − A) ⊕ Im and (I − B) ⊕ In are GL(ZHi)-equivalent [12, Lemma 9.1]. Because Hi is finite abelian, the group SK1(ZHi) is finite [22]; then by [12, Corollary 9.9], there are only finitely many El(ZHi)-equivalence classes of matrices with determinant the regular element di. Given such choices o for 1 ≤ i ≤ N, fix A in MP (n, H) with diagonal blocks I − Ai in the given El(ZHi) classes. o Suppose B ∈ MP (H) with I − Ai and I − Bi are El(Hi)-equivalent for each i. We first claim that I − B is ElP (H)-equivalent to a matrix o in MP (n, H) with the same diagonal blocks as I − A. To show this, for each i let k(i), `(i), m(i) be nonnegative integers such that there are Ui,Vi in El(mi,Hi) such that (I − Ai) ⊕ Ik(i) = Ui (I − Bi) ⊕ I`(i) Vi .

0 0 Let m = (m1, . . . , mN ) and let A ,B be the stabilizations of A, B in o MP (m, H). Let U = U1 ⊕ · · · ⊕ UN and V = V1 ⊕ · · · ⊕ VN . Set 0 0 I − C = U(I − B )V . Then I − C and I − B are ElP (m, H) equivalent and the ith diagonal block of (I − C) equals (I − Ai) ⊕ Ik(i). After adding multiples of rows and columns from the Ik(i), we may produce a matrix I − D, ElP (m, H)-equivalent to I − B, such that D is zero outside its principal submatrix (P , say) on the indices used to define A. Now I − P is ElP (H)-equivalent to I − B and its diagonal blocks equal those of I − A. To finish, it suffices to show I −P is ElP (n, H)-equivalent to a matrix with bounded entries. The ith diagonal block of I − P is the ni × ni ni matrix I − Ai. Let Ri be the image of the space of row vectors (ZHi) under the map v 7→ v(I − Ai). Let κi ∈ N be the index of Ri in ni ni (ZHi) . Then Ri contains κi(ZHi) . In the order j = 2, 3,...,N do the following: for i ≺ j, as needed, multiply I − P from the left by matrices of ElP (n, H) which are equal to I outside the ijth block to reduce all Z coefficients in that block to lie in the interval [0, κj). This shows I − P is ElP (n, H)-equivalent to one of a bounded set of matrices, as required.

Appendix A. Cohomology as positive equivalence The next proposition was proved in [12], with (much) worse control over m, using the positive K-theory polynomial strong shift equivalence equations from [13]. The elementary argument below gives a better bound on m; and for the proof of Theorem 6.2, we use the case where mi is controlled to be ni. The identity element of G is denoted e.

Proposition A.1. Suppose D is an n × n diagonal matrix over Z+G such that for each s, D(s, s) = gs ∈ G. Suppose A is an n × n matrix FLOW EQUIVALENCE OF G-SFTS 53

−1 over Z+ and B = D AD. Then there is an m ≤ n + 1 and m × m stabilizations A0,B0 of A, B such that there is a positive ZG equivalence (I − A0) → (I − B0). o Now suppose in addition that A ∈ MP (C, n, H) and for all i in P o that gs ∈ Hi whenever s ∈ Ii. Then B ∈ MP (C, n, H), and there are 0 0 o m and stabilizations A ,B of A, B in MP (C, m, H) such that there is 0 0 a positive ElP (m, H)-equivalence (I − A ) → (I − B ). The vector m can be chosen such that for all i ∈ P,

(1) mi ≤ ni + 1, and (2) if gs = e for all s ∈ Ii then mi = ni.

o Proof. In the second case, B will be in MP (C, n, H) because H is a coset structure. We will describe given s a positive ZG-equivalence which has the −1 effect of multiplying row s from the left by gs and multiplying column s from the right by gs. The equivalence will satisfy the stabilization bounds and in the second case be a positive ElP (H)-equivalence. Applying such an equivalence for each s proves the proposition. For concreteness, suppose s = 1 and g1 = g. We will describe the equivalence as a finite sequence of the row and column cuts from Section 3. We first consider the special case that A(1, 1) = 0. The target matrix will be named A0. To lighten the notation (avoiding no technical difficulty), we will suppose a nonzero entry of A is a single element of G; e.g., A(s, 1) = a means an edge from vertex s to vertex 1 is labeled by a, as in the graph I below.

(A.2) s t s / t s / t @ ab ? ab

a b b 1 1 1

I II III

Multiplying I − A from the left by Es1(a) eﬀects the row cut of the edge from s to 1 labeled a in I and produces the change I→II. Do this for every edge into 1, producing a graph in which 1 has no incoming edge. Then column cut every edge out of 1; the eﬀect is to remove those edges, as in III, leaving 1 an isolated vertex. Let A00 be the matrix produced from A by these moves. Note that applying this procedure to the matrix A0 produces the same matrix A00:

s t s / t s / t @ ag−1gb @ ab

ag−1 gb gb 1 1 1 FLOW EQUIVALENCE OF G-SFTS 54

The positive equivalence for A → A00 postcomposed with the inverse of the positive equivalence for A0 → A00 gives the required positive equivalence A → A0 for this case. For the case that A(1, 1) = c 6= 0, we introduce an additional isolated vertex, named v. (If for some vertex s there is no edge from s to itself, then by applying the move corresponding to I→III in (A.2) we could isolate s and avoid increasing the number of vertices.) The argument again is described by a ﬁnite sequence of evolving graphs.

(A.3) v v v O g−1c g g−1c s / 1 / t s / 1 / t s / 1 / t a Y b a Y b a b c c I II III

g−1cg g−1cg g−1cg

v Ô v Ô v Ô −1 −1 −1 O g cb ag > O g cb ag = g cb g g b s / 1 / t s 1 / t s 1 t a b 8 7 ab ab IV V VI

g−1cg

v Ô = −1 ag g b

s 1 ! t

VII Here is a list of the corresponding positive equivalences. • II→I. Column cut the edge v → 1. • III→II. Row cut the edge 1 → v. • III→IV. Row cut the edge v → 1. • IV→V. Row cut all incoming edges to 1. • V→VI. Column cut all outgoing edges from 1. • VII→VI. Row cut each outgoing edge from v to a diﬀerent vertex. At this point, the move from I to VII in (A.3) has replaced the given matrix A with a matrix A00 which satisﬁes our conditions, except that the vertex v is playing in A00 the role we require for vertex 1. To remedy FLOW EQUIVALENCE OF G-SFTS 55 this, apply the procedure I→VII above to A00, but with (s, t, v, 1, e) in place of (s, t, 1, v, g). We end up with the required matrix A0 , with the additional isolated vertex v (i.e., row v and column v of A0 are zero). Appendix B. Permutation similarity as positive equivalence

Suppose A is an n × n matrix over Z+G and P is an n × n permutation matrix and B = P −1AP . Then A, B are elementary strong shift −1 −1 equivalent over Z+G, as B = (P )(AP ) and A = (AP )(P ), and therefore A and B are ZG positive equivalent [13]. In the next proposition we show that we can obtain this positive equivalence through o (n + 1) × (n + 1) matrices. We also show that if A ∈ MP (n, H) and P ∈ MP (n, Z+G), then we get a positive ElP (H)-equivalence (I − A) → (I − B). Proposition B.1. Let A, B, P, G be as above. Suppose there is an index s with A(s, s) = 0. Then there is a positive ZG-equivalence from A to B through n × n matrices. In any case there are stabilizations A0,B0 of A, B which are positive ZG-equivalent through (n+1)×(n+1) matrices. o Now suppose in addition that A ∈ MP (C, n, H) and P ∈ MP (n, Z+). o 0 0 Then B ∈ MP (C, n, H), and there are m and stabilizations A ,B of o A, B in MP (C, m, H) such that there is a positive ElP (m, H)-equivalence (U, V ):(I − A0) → (I − B0). The vector m can be chosen such that mi ≤ ni + 1 for all i ∈ P. If P {i, i} = I where i ∈ P, then U and V can be chosen such that U{i, i} and V {i, i} are the identity matrix. Proof. Assume first that there is an index s with A(s, s) = 0. Let t be an index different from s. We will describe a positive ZG-equivalence which has the effect of permuting s and t. If t1, t2 are arbitrary indexes, then we get a positive ZG-equivalence which has the effect of permuting t1 and t2 by first permuting s and t1, then permuting t1 and t2, and then finally permuting t2 and s. Since every permutation of {1, . . . , n} is the product of transpositions, it will follow that there is a positive ZG-equivalence (I − A) → (I − B). The procedure described in (A.2) shows that there is a positive ZG- equivalence (I−A) → (I−A1) such that s is an isolated index in GA1 . If also A(t, t) = 0, then there is a positive ZG-equivalence (I−A1) → (I−

A2) such that t is an isolated index in GA2 , and if we then postcompose the equivalence (I −A) → (I −A2) with its inverse but with the role of s and t interchanged, then we get a positive ZG-equivalence (I − A) → (I −A3) where A3 is the matrix obtained from A by permuting s and t. If A(t, t) 6= 0, then the procedure described in (A.3) with g = e shows 0 0 that there is a positive ZG-equivalence (I − A1) → (I − A2) where A2 is obtained from A2 by permuting s and t. By postcomposing with the FLOW EQUIVALENCE OF G-SFTS 56 inverse of the equivalence (I −A) → (I −A1) but with the role of s and 0 t interchanged, we get a positive ZG-equivalence (I − A) → (I − A3) 0 where A3 is the matrix obtained from A by permuting s and t. If there is no index s ∈ GA with A(s, s) = 0, then we add a zero row and a zero column to A and B to obtain matrices A0 and B0, and then it follows from the argument above that there is a positive ZG-equivalence (I − A0) → (I − B0). o Now suppose in addition that A ∈ MP (C, n, H) and P ∈ MP (n, Z+G). o Then P {i, j} = 0 if i 6= j. It follows that B ∈ MP (C, n, H). We let ∗ ∗ Pi denote the matrix in MP (n, Z+G) such that Pi {i, i} = P {i, i}, ∗ ∗ 0 0 0 0 Pi {j, j} = I for j 6= i, and Pi {i , j } = 0 for i 6= j . Then P = P ∗ P ∗ ··· P ∗ . Let A = (P ∗ )−1AP ∗ , A = (P ∗ )−1A P ∗ ,. . . ,A = i1 i2 iN 1 i1 i1 2 i2 1 i2 N (P ∗ )−1A P ∗ = B. By the ﬁrst half of the proposition, there are iN N−1 iN 0 0 0 0 0 stabilizations A ,A1,A2,...,AN = B of A, A1,A2,...,AN = B and positive ZG-equivalences 0 0 0 0 0 0 (I −A ) → (I −A1) → (I −A1) → (I −A2) → · · · → (I −AN ) = (I −B ) 0 0 o and that B can be chosen such that B ∈ MP (C, m, H) with mi ≤ ni + 1 for all i ∈ P. It is not diﬃcult to check that the described 0 0 equivalence (U, V ):(I − A ) → (I − B ) is a positive ElP (m, H)- equivalence, and that if P {i, i} = I where i ∈ P, then U and V can be chosen such that U{i, i} and V {i, i} are the identity matrix.

Appendix C. Resolving extensions o 0 Proposition C.1. Suppose A is a matrix in MP (C, n, H) and A is a matrix obtained from A by splitting a row s into two rows. Let the rows of A0 be in the same order as corresponding rows of A, with the interpolation of a new row s0 directly following s. Let A0 have the natural P blocking: s0 is in the block of s, and every other index is in the block of the row from which it was copied. Let Ae be the matrix of size and blocking from n0 obtained by interpolating a zero s0 row and column into A. 0 o 0 0 Then A,e A ∈ MP (C, n , H) and there is a positive ElP (n , H)-equivalence (U, V ):(I − Ae) → (I − A0). If A is upper triangular and A(s, s) = A0(s, s), then A0 is upper triangular and the matrices U, V can be chosen to be unipotent upper triangular. Moreover, the same conclusion holds if in the above statements “row” is replaced by “column” and “following” is replaced by “preceding”.

0 o 0 Proof. Let us ﬁrst check that A ∈ MP (C, n , H) (it is obvious that o 0 0 o Ae ∈ MP (C, n , H)). It is easy to check that A ∈ MP (C, n, Z+G) ∩ MP (n, H), so we just need to show that H is a (G, P) coset structure for A0. Since H is a (G, P) coset structure for A, there is a family of vertices {v(i)}i∈P such that v(i) belongs to the irreducible core of A{i, i} for each i ∈ P, and Hij is the set of weights of paths from v(i) FLOW EQUIVALENCE OF G-SFTS 57 to v(j) in GA. Let i, j ∈ P. We aim to show that the set of weights of paths from v(i) to v(j) in GA0 is equal to Hij. Notice that if p is a path in GA not starting at s, then there is a path in GA0 starting and ending at the same vertices as p and with the same weight as p. Notice also that if p is a path in GA starting at s, then 0 there is a path in GA0 starting at either s or s and ending at the same vertex as p and with the same weight as p. Similarly, if p is a path in GA0 , then there is a path in GA which has the same weight as p and which starts and ends at the same vertices as p (except of course if p 0 starts/ends at s in which case the path in GA starts/ends at s instead). It follows that if v(i) 6= s, then the set of weights of paths from v(i) to v(j) in GA0 is equal to Hij. Suppose that v(i) = s and that p is a path in GA starting at s and 0 that there is a path in GA0 starting at s and ending at the same vertex as p and with the same weight as p. Suppose that there is a path from 0 s to s in GA0 (if there is no path in GA0 , then there must be a path from s0 to s because s = v(i) in the irreducible core of A{i, i}, and the we just interchange the role of s and s0). Let γ be the weight of this path. Since the set of weights of paths in GA0 from s to s is equal to the set 0 of weights of paths in GA0 from s to s and is a group (because it is a ﬁnite semigroup), it follows that there is a path in GA0 from s to s with −1 weight γ , and thus that there is there is a path in GA0 starting at s and ending at the same vertex as p and with the same weight as p. It follows that the set of weights of paths from s = v(i) to v(j) in GA0 is 0 equal to Hij. This shows that H is a (G, P) coset structure for A and 0 o 0 thus that A ∈ MP (C, n , H). 0 We then show that there is a positive ElP (n , H)-equivalence (U, V ): I − Ae → I − A0. We write A in a 3 × 3 block form, giving

q r 0 t  q r t  u v 0 w A = u v w A =   .   e 0 0 0 0 x y z   x y 0 z

(We omit the easier proof for the case that s is a ﬁrst or last index of A, and the block form is smaller.) The central index set of A is {s}, so v is the 1 × 1 matrix A(s, s). The matrix A0 then has the block form

 q r r t  0 u1 v1 v1 w1 A =   u2 v2 v2 w2 x y y z FLOW EQUIVALENCE OF G-SFTS 58

0 with A nonnegative and (u1, v1, w1) + (u2, v2, w2) = (u, v, w). We then have a string of positive equivalences:

1 0 0 0 1 − q −r 0 −t  0 1 −1 0  −u 1 − v 0 −w  I − Ae → E1(I − Ae) =     0 0 1 0  0 0 1 0  0 0 0 1 −x −y 0 1 − z 1 − q −r 0 −t   −u 1 − v −1 −w  =   := I − A1 .  0 0 1 0  −x −y 0 1 − z

1 − q −r 0 −t   1 0 0 0  −u 1 − v −1 −w   0 1 0 0 I − A1 → (I − A1)E2 =      0 0 1 0  −u2 0 1 0 −x −y 0 1 − z 0 0 0 1 1 − q −r 0 −t   −u1 1 − v −1 −w  =   := I − A2  −u2 0 1 0  −x −y 0 1 − z

1 − q −r 0 −t  1 0 0 0  −u1 1 − v −1 −w  0 1 0 0 I − A2 → (I − A2)E3 =      −u2 0 1 0  0 −v2 1 0 −x −y 0 1 − z 0 0 0 1 1 − q −r 0 −t   −u1 1 − v1 −1 −w  =   := I − A3  −u2 −v2 1 0  −x −y 0 1 − z

1 − q −r 0 −t  1 0 0 0   −u1 1 − v1 −1 −w  0 1 0 0  I − A3 → (I − A3)E4 =      −u2 −v2 1 0  0 0 1 −w2 −x −y 0 1 − z 0 0 0 1 1 − q −r 0 −t   −u1 1 − v1 −1 −w1  =   := I − A4  −u2 −v2 1 −w2  −x −y 0 1 − z FLOW EQUIVALENCE OF G-SFTS 59

1 − q −r 0 −t  1 0 0 0  −u1 1 − v1 −1 −w1  0 1 1 0 (I − A4) → (I − A4)E5 =      −u2 −v2 1 −w2  0 0 1 0 −x −y 0 1 − z 0 0 0 1 1 − q −r −r −t   −u1 1 − v1 −v1 −w1  0 =   = I − A .  −u2 −v2 1 − v2 −w2  −x −y −y 1 − z

0 This exhibits the equivalence (U, V ): I −Ae → I −A , with U = E1 and 0 V = E2E3E4E5. It is clear that E1,E2,E3,E4,E5 ∈ ElP (n , H), and o 0 it is not diﬃcult to check that A1,A2,A3,A4 ∈ MP (n , H). It follows 0 0 that (U, V ): I − Ae → I − A is a positive ElP (n , H)-equivalence. In general the matrices U, V will not be upper triangular, because in general E2 and E3 are not upper triangular. However, if A is upper 0 triangular, then u = 0, so u1 = u2 = 0; and if A (s, s) = A(s, s), then 0 v1 = v, so v2 = v − v1 = 0. Thus under the additional assumptions, A is upper triangular and the matrices U = E1 and V = E2E3E4E5 are unipotent upper triangular as required. The argument for the “Moreover” claim is essentially the same, and we omit it.

Appendix D. G-SFTs following Adler-Kitchens-Marcus The purpose of this appendix is to integrate our matrix approach with the classification of G-SFTs with the group actions framework of Adler-Kitchens-Marcus [2, 1]. This appendix is not necessary for the statements or proofs of the flow equivalence results of earlier sections. Throughout, G is a finite group. We make no conceptual advance on [2] (which in turn acknowledges a huge debt to the ergodic-theoretic work [27] of Rudolph). Still, we give a detailed presentation of this framework (with some additional details), as the ideas in [2] are intermingled with that paper’s focus on almost topological conjugacy, and the paper does not isolate all the explicit statements we want. The paper [23] of Parry also covers much, but not all, of this framework. A matrix A is G-primitive if its entries lie in Z+G and in addition there is a positive integer m such that every entry of Am is G-positive. An SFT is nonwandering if it has no wandering orbit; equivalently, it is the disjoint union of finitely many irreducible SFTs (its irreducible components). A nonwandering/irreducible/mixing G-SFT is a G-SFT (Y,T ) which as an SFT is nonwandering/irreducible/mixing. A nonwandering G-SFT was defined to be G-transitive [1, Section 4] if the FLOW EQUIVALENCE OF G-SFTS 60

G action on irreducible components is transitive. A nonwandering G- SFT is G-transitive if and only if the canonical factor map collapsing G-orbits maps each irreducible component onto the same irreducible SFT. Clearly the classification of nonwandering G-SFTs reduces to the classification of G-transitive nonwandering G-SFTs. Let G be a finite group and let (Y,T ) be a nonwandering G-transitive G-SFT. We take this left G-SFT (Y,T ) to be Y = X × G with T : (x, g) → (σx, gτ(x)) with τ : X → G continuous, and left G action by g :(x, h) → (x, gh), as in Section 3 (recall our Standing Convention 3.3.1). Let C be an irreducible component of Y , with cyclically moving subsets C0,...,Cp−1 . For g ∈ G, let gC := {(x, gh):(x, h) ∈ C}. Then gC is an irreducible component of Y . The map (x, h) 7→ (x, gh) sending C to gC is a topological conjugacy of SFTs (but not of G- SFTs, when G is not abelian). The stabilizer of C is the subgroup −1 HC = H = {g ∈ G : gC = C}. For g ∈ G, we have HgC = gHC g . The next result explains how to reduce the classification of nonwandering G-SFTs to the classification of irreducible K-SFTs, for subgroups K of G. Theorem D.1. Suppose G is a finite group and (Y,T ) and (Y 0,T 0) are nonwandering G-transitive G-SFTs, containing irreducible components 0 C,C (resp.). Let H denote the stabilizer HC of C. Then the following are equivalent. (1) (Y,T ) and (Y 0,T 0) are G-conjugate. (2) There exists g in G such that the stabilizer of gC0 is H (i.e., the stabilizers of C and C0 are conjugate subgroups in G) and the irreducible H-SFTs C and gC0 are H-conjugate. Proof of Theorem D.1. (1) =⇒ (2): A G-conjugacy (Y,T ) → (Y 0,T 0) will restrict to an H-conjugacy from C to some irreducible component 0 D of Y with stabilizer HD = H. By the G-transitivity, there is some g ∈ G such that D = gC0. (2) =⇒ (1): 0 Let D = gC and let ϕC : C → D be a conjugacy of H-SFTs. Pick elements gi of G, 0 ≤ i < |G/H|, such that g0 = e and the giH are the distinct left cosets of H in G. Define ϕ : G → G by setting ϕ(gix) = giϕC (x) for gix in giC. Then for x ∈ C and g = gih ∈ giH, we have

ϕ(gx) = ϕ(gihx) = giϕC (hx) = gihϕC (x) = gϕ(x) .

For gix ∈ giC and g ∈ G, it follows that ϕ(g(gix)) = ggiϕ(x) = 0 gϕ(gix). Thus, ϕg = gϕ for all g. Then, ϕT = T ϕ, because for gix ∈ giC, 0 ϕ(T (gix)) = ϕ(gi(T x)) = giϕC (T x)) = giT (ϕC x) 0 0 = T (gi(ϕC x)) = T (ϕ(gix)) . FLOW EQUIVALENCE OF G-SFTS 61

For a reduction of the classiﬁcation of irreducible G-SFTs to a structure on mixing SFTs, we continue the Adler-Kitchens-Marcus analysis. Suppose H is a ﬁnite group and T : Y → Y is an irreducible H-SFT (we use H to match letters in [2]) with period p > 1. Let C0,...,Cp−1 denote the cyclically moving subsets of Y : the Ci are disjoint; T maps Ci onto Ci+1 (superscripts interpreted mod p); and for each i, the restriction of T p to Ci is a mixing SFT. For 0 ≤ i < p, set

i i 0 i 0 i HC = H = {g ∈ G : gC = C } = {g ∈ H : gC ∩ C 6= ∅} = {g ∈ G : gCj = Ci} , for any j ∈ {0, 1, . . . , p − 1} .

H0 is a normal subgroup of H. The sets Hi are disjoint, with union H; each Hi, if nonempty, is a left coset of H0. Identifying H0 and Hp, i deﬁne κT = min{i ∈ N : 0 < i ≤ p, H 6= ∅}. Then κT divides p, and i for 0 < i < p, H is nonempty if and only if κT divides i. 1 4 It can happen that H is empty (i.e., κT > 1). For example, if H = {e} and T is a single orbit of length p, then κT = p. If H = 3 Z2 = {e, g}, T is a single orbit of length 6, and g acts by T , then 0 3 H = {e},H = {g} and κT = 3. However, there is a straightforward interpretation of κT , as follows. (Recall, for a property P , T is totally P if T k is P for all k > 0.) Proposition D.2. Suppose (Y,T ) is an irreducible H-SFT. Let T be the irreducible SFT which is the factor of T obtained by collapsing H- orbits. Then κT is the period of T . The following are equivalent.

(1) κT = 1. (2) T is totally H-transitive. (3) T is mixing. Proof. For (1) ⇐⇒ (2), note H-transitivity of all powers of T is equivalent to transitivity of the H-action on cyclically moving subsets, which is equivalent to κT = 1. For (2) ⇐⇒ (3), note every power of T is H-transitive if and only if ever power of T is irreducible. This happens if and only if the irreducible SFT T is mixing. Now let k be the period of T . κT is the smallest positive integer j such that T j is the disjoint union of j irreducible SFTs, each of which j has κ = 1. For j < k, irreducible components of T are not mixing, so irreducible components of T j cannot have κ = 1. On the other hand, (T )k is the disjoint union of k mixing SFTs, so T k is the disjoint union of k H-SFTs, each of which has κ = 1. Thus κT = k.

4The statements [2, Part (iii) of the p.22 Lemma and p.23 Corollary] neglect the possibility H1 = ∅, but the p.23 proof addresses it. FLOW EQUIVALENCE OF G-SFTS 62

Reduction to the case κT = 1 clariﬁes issues and simpliﬁes nota- k tion. Note, for k = κT , the restriction of T to any of its irreducible components is H-invariant and therefore an H-SFT. Proposition D.3. Suppose (Y,T ) and (Y 0,T 0) are irreducible H-SFTs with period p > 1, with κT = κT 0 := k > 1. Then the restriction of T k or (T 0)k to any of its irreducible components has κ = 1. If C is an irreducible component for T k, then a conjugacy of H-SFTs T,T 0 restricts to a conjugacy of H-SFTs T k|C, (T 0)k|C0, for some irreducible component C0 of T k. Conversely, any conjugacy of H-SFTs T k|C, (T 0)k|C0 extends uniquely to a conjugacy of H-SFTS T,T 0. 0 Proof. We will prove the extension claim. Suppose ϕ0 : C → C is a k 0 k 0 conjugacy of H-SFTs T |C,(T ) |C . The unique extension of ϕ0 to a 0 i 0 i topological conjugacy of T,T as SFTs is given by ϕ : T x 7→ (T ) (ϕ0x), for x ∈ C and 0 ≤ i < k. For h ∈ H, 0 ≤ i < k, x ∈ X and y = T ix, we i i 0 i 0 i then have ϕ(hy) = ϕ(hT x) = ϕ(T hx) = (T ) ϕ0(hx) = (T ) hϕ0(x) = 0 i i h(T ) ϕ0(x) = hϕ(T x) = hϕ(y). Now suppose T is an irreducible H-SFT of period p > 1 with H1 = H1(T ) nonempty. Pick c in H1. We have H0 a normal subgroup of H, Hi = ciH0 and the disjoint sets Hi are the cosets of H0. We call the subgroup H0 of H the primitive stabilizer of C. We call H1 the stabilizer coset (“primitive stabilizer coset” would be more ac- curate, but lengthier). Adler-Kitchens-Marcus are not responsible for the terms “primitive stabilizer” and “stabilizer coset”. 0 0 Given irreducible H-SFTs T,T of period p, let T0, T0 be mixing SFTs given by restriction of T p, (T 0)p to some cyclically moving subset 0 0 of T,T . If p > 1 and κT = 1, then T0 and T0 are not H-SFTs, but they are H0-SFTs. However, a natural candidate reduction fails badly, as in the following example.

Example D.4. Let H = Z2 = {e, g}. We define two irreducible, period 2 G-SFTs, T and T 0, with H0 = {e} and H1 = {g}, such that 0 2 0 2 the restrictions T0,T0 of T and (T ) to irreducible components are 0 conjugate H0-SFTs, but T and T are not conjugate as H-SFTs. Let 1 2 0 0 2 1 T = TA for A = g ( 2 1 ), and similarly define T from A = g ( 1 2 ). The SFTs g−1T and g−1T 0 are not conjugate, having different numbers of 0 2 0 2 fixed points. But, T0 and T0 are conjugate, because A = (A ) . To find some reduction of the classification of irreducible H-SFTs to a classification defined on mixing SFTs, one is forced to consider the α-skew H-SFTs. For this, we continue below to follow [2]. Suppose H is a finite group and α : H → H is a group automorphism. In [2], Adler, Kitchens and Marcus defined a Z ⊗α H action on an SFT T (we will call this pair an α-skew H-SFT) to be an embedding of H as a group of homeomorphisms such that T h = α(h)T (i.e. T h(x) = α(h)T (x) for all x and h). (The H-SFT case is the case that α is the FLOW EQUIVALENCE OF G-SFTS 63 identity.) They showed (see [2, Observation 1, p.4]) that an α-skew H-SFT can be presented very concretely, as a one-step SFT with an embedding h 7→ πh of H into the group of permutations of the alphabet, such that for x = (xn) and h ∈ H, hx is defined by (hx)n = παn(h)(xn). They also showed (see [2, Theorem 1]) these skew SFTs are abundant: for every H and α, every positive entropy irreducible SFT admits a (not necessarily free) α-skew H-SFT, which (by [2, Theorem 3]) is then a 1-1 a.e. factor of an α-skew H-SFT which is free (the H-orbit of x has cardinality |H|, for every x). Let T be an irreducible H-SFT of period p > 1, with H1(T ) nonempty. Fix c in H1(T ), and define S = c−1T (i.e., S(x) = c−1T (x)). If c is in the center of H, then S is an H-SFT; otherwise, it is not. Two α-skew H-SFTs are by definition topologically conjugate if they are conjugate as SFTs by a conjugacy intertwining the H actions.

Theorem D.5. Suppose T is an irreducible H-SFT of period p > 1, with nonempty coset H1(T ). Fix c in H1(T ), and define S = c−1T . i Let the cyclically moving subsets of T be C , 0 ≤ i < p. Let Si be the restriction of S to Ci. Define α : h → c−1hc (with domain given by context). Then the following hold. (1) With the given H action, S is a free α-skew H-SFT (and by restriction of the action, S is an α-skew H0-SFT). i i (2) Each Si : C → C is a free mixing α-skew H0-SFT. (3) Suppose T 0 is another irreducible H-SFT. Assume the period of T 0 is also p, and H1(T 0) = H1(T ) = H1 6= ∅. (These are necessary conditions for conjugacy of T,T 0 as H-SFTs.) Let C0i, 0 ≤ i < p, be the cyclically moving subsets of T 0. Define 0 −1 0 0 0 0i S = c T and Si = S |C . Then the following hold. (a) A conjugacy ϕ of H-SFTs T,T 0 restricts to a conjugacy of 0 −i mixing α-skew H0 SFTs S0,Si, for some i. Then ϕ ◦ T 0 is a conjugacy of the mixing α-skew H0-SFTs S0,S0. 0 00 (b) Given a conjugacy ϕ0 : C → C of the α-skew H0-SFTs 0 0 S0,S0, there is a unique conjugacy ϕ of H-SFTs T,T such 0 that ϕ = ϕ0 on C .

Proof. (1) Suppose cr = e. Because cS = Sc, it follows that Sr = (c−1T )r = T r, an SFT. As a root of an SFT, S must be an SFT. Next, given h ∈ H, we have S(hx) = c−1T (hx) = c−1hT (x) = c−1hc(c−1T )(x) = α(h)(Sx). S is free because an H-SFT is by deﬁnition (in this paper) free. i i pr (2) Clearly C is mapped to C by Si and H0. Also, (Si) = −1 pr p r (c T ) |Ci = (T ) |Ci . Roots and powers of mixing SFTs are mixing SFTs, so Si is a mixing α-skew H0-SFT. (3) The claim (a) follows from parts (1) and (2). Now suppose ϕ0 0 i i is given as in (b). For x ∈ C and 0 ≤ i < p, deﬁne ϕ(c x) = c ϕ0(x). FLOW EQUIVALENCE OF G-SFTS 64

This ϕ is the only possible extension of ϕ0 to a conjugacy of the H-SFTs T,T 0. We claim S0ϕ = ϕS. For this, suppose y = cix, with x ∈ C0 and 0 ≤ i < p. Note cS = Sc and cS0 = S0c. So, we have ϕ(Sy) = ϕ(S(cix)) = i i i 0 0 i 0 ϕ(c S(x)) = c ϕ0(Sx) = c S0ϕ0(x) = S (c ϕ0(x)) = S ϕ(y). Next, we claim cϕ = ϕc. For x ∈ Ci with 0 ≤ i < p − 1, this is p−1 p−1 p p clear. For y = c x ∈ C , we have ϕ(cy) = ϕ0(c x) = c ϕ0(x) = p−1 p−1 c(c ϕ0(x)) = cϕ(c x) = cϕ(y). We now have ϕT = T 0ϕ, because ϕT = ϕcS = cϕS = cS0ϕ = 0 T ϕ. For h in H0 and y ∈ Y , it remains to check ϕ(hy) = hϕy. Let y = cix ∈ Ci, 0 ≤ i < p. Then ϕ(hy) = ϕ(hcix) = ϕ(ci(c−ihcix)) = i −i i i −i i i c ϕ0(c hc x) = c (c hc )ϕ0x = hc ϕ0x = hϕ(y). Theorem D.5 reduces the problem of classifying irreducible H-SFTs, for all H, to the problem of classifying mixing α-skew H-SFTs (with H acting freely), for all α and H. For mixing H-SFTs, there is a reasonably satisfactory framework of invariants arising from a theory of strong shift equivalence of matrices over ZH (or elementary equivalence of matrices over Z[t]. One naturally has an analogous (somewhat ill defined) problem: Problem D.6. Find a satisfactory classification scheme for mixing α-skew H-SFTs. Finally, we turn to relating matrix properties to the Adler-Kitchens- Marcus setting. Let A be a square matrix over Z+G with augmentation A over Z+ as in Section 3. Let aij = A(i, j) and define nonnegative k P integers aijkg by A (i, j) = g∈G aijkgg; let aijg denote aij1g. We recall some terminology. A is irreducible/primitive if A is irreducible/primitive. A is nondegenerate if it has no zero row and no zero column. The nondegenerate core of A is its maximum nondegenerate principal submatrix. For a property P, A is essentially P if its nondegenerate core is P. A is G-primitive if there exists k > 0 such that aijkg > 0 for all i, j, g.

Deﬁnition D.7. Let A be a square matrix over Z+G. For an index i of A, the weights group (at i) is

Wi(A) = {g ∈ G : ∃k > 0 such that aiikg > 0} and the ratio group (at i) is −1 ∆i(A) = {gh : ∃k such that aijkg > 0 and aijkh > 0} −1 = {gh : ∃k such that aiikg > 0 and aiikh > 0}.

Wi(A) is clearly a ﬁnite semigroup, and hence a group. ∆i(A) is a j group, because given (g2) = e, we have −1 −1 j−1 j −1 g1h1 g2h2 = (g1h1 g2)(h2h1) . FLOW EQUIVALENCE OF G-SFTS 65

∆i(A) is named after a “ratio group” which plays an analogous role in the theory of Markov shifts [21, 25]. Our next task is to compute the stabilizer data for TA from the matrix A. First we recall a standard reduction; see the citation for a proof. Proposition D.8. [12, Proposition 4.4] Let A be an irreducible matrix over Z+G. Let i be an index of A and let H = Wi(A). Then there is a diagonal matrix D over Z+G with each diagonal entry in G (i.e., −1 D = I) such that every entry of DAD lies in Z+H. (It suﬃces for each j to set D(j, j) = g for some g such that for some k, aijkg > 0.) An important technical point for proofs below is the following observation. Remark D.9. Suppose A is an essentially irreducible square matrix over Z+G,(w, g), (x, h) ∈ XA × G, g = h, and the initial vertices of x0 and w0 are equal. Then (w, g) and (x, h) are in the same cyclically moving subset of the same irreducible component of XA × G.

Proposition D.10. Let A be an irreducible matrix over Z+G. Let x be a point in XA with x0 beginning at index i. Let C be an irreducible 0 p−1 component of TA containing (x, e) and choose C ,...,C such that (x, e) ∈ C0. Then the following hold.

(1) The stabilizer HC is the weights group Wi(A). −1 (2) The matrix B = DAD over Z+H from Theorem D.8 deﬁnes an irreducible H-SFT TB which is G-cohomologous to the H- SFT T |C . 0 (3) The primitive stabilizer HC is the ratio group ∆i(A).

Proof. (1): Suppose g ∈ HC ; then (x, g) ∈ C. By irreducibility of C, there must then be s ome path z0 . . . zk−1 in XA from i to i with weight g. Therefore g ∈ Wi(A). Conversely, suppose g ∈ Wi(A). Then there is k > 0 and a periodic point w in XA with `(w0)`(w1) ··· `(wk−1) = g (here `(wn) denotes the label of the edge wn in GA, see Section 3) such that i equals the initial vertex of w0 and the terminal vertex of xk−1. The point (w, e) must be in C. Therefore (w, g) ∈ C. Thus gC ∩ C 6= ∅, so gC = C and g ∈ HC . (2): The diagonal matrix D gives the G-cohomology. TB is irreducible because C is irreducible. (3): Given g, h, k as in the deﬁnition of ∆i(A), using Remark D.9 k −1 0 one can see g and h are in HC , and therefore gh ∈ HC . Conversely, 0 suppose g ∈ HC . Then g there is k > 0 and a path w0 ··· wkp−1 from i to i with weight g. Then for arbitrarily large j > 0 there is a path (by concatenations) of length jp from i to i with weight g. For suﬃciently large j, there is also a path of length jp from i to i with weight e, 0 p because the HC -SFT (TA) |C0 is mixing. FLOW EQUIVALENCE OF G-SFTS 66

Proposition D.11. Let G be a ﬁnite group and A a square matrix over Z+G, deﬁning a G-SFT TA as in Section 3. The following hold. (1) A is essentially G-primitive ⇐⇒ TA is mixing. (2) A is essentially irreducible with a weights group Wi(A) = G ⇐⇒ the G-SFT TA is irreducible. (3) A is essentially irreducible ⇐⇒ TA is G-transitive and nonwandering. (4) A is essentially irreducible, with a weights group Wi(A) = G and with A essentially primitive ⇐⇒ TA is irreducible with κ = 1. Proof. (1) is [10, Cor. B.7]. (2) is clear. (3) follows from (2) and part (1) of Proposition D.10. Then (4) follows from Proposition D.2. Finally, we give algorithmically a presentation (essentially following [25]) for the classifying α-skew H0 SFT, from a given presenting matrix c 1 A over Z+H. For an example, let Z2 = {1, c} and A = ( 1 c ). Then 0 1 c 0 W1(A) = Z2, ∆1(A) = {1} = H and H = {c}; with D = ( 0 1 ) and 1 1 −1 A = ( 1 1 ), we have and DAD = cA.

Proposition D.12. Suppose A is an irreducible matrix over Z+H, with A primitive, and H equal to a weights group Wi(A) such that 0 Wi(A) 6= ∆(A) (i.e., H 6= H , so TA is not mixing). Let β be the weight of a point of period n + 1 from i to i; let γ be the weight of a point of period n from i to i; let c = γ−1β. Pick N such that for each j, there is a path of length N from i to j; choose dj the weight of some such path. Let D be the diagonal matrix with D(j, j) = dj. −1 Then DAD = cA, where A is a H0-primitive matrix. −1 0 −1 c T acts on C in X × G by the rule (x, g) 7→ (σx, (c gc)τA(x)), where τA(x) is the A label of the edge x0. −1 For c in the center of H, the H0-SFT c TA is conjugate to TA. If H1 contains an element of the center of H, then this element can be chosen as c.

Proof. If z is a point of period j from i to i in XA, with weight h, then h must be in Hj (j interpreted mod p). This is because T j(z, e) = (z, h), and h takes (z, e) to (z, h). It follows that the chosen c lies in H1. Also, let g be the weight of a path from k to i; then −1 −1 −1 DAD (j, k) = (djajk)(dk) = (djajkg)(dkg) .

Interpreting (djajkg) and (dkg) as weights of paths from i to i, we see every entry of DAD−1 lies in H1. Likewise, every entry of c−1DAD−1 lies in H0. Finally, given another element c0 of H1, let c0 = ch with h ∈ H0; then we may pass from cA to (ch)(h−1A). Finally, A being −1 H0-primitive follows from c T being mixing, as the latter means that for a large n, T n maps some point (x, e) with initial vertex i to a point with (z, h) where z has initial vertex j. Here, h = c−neg such that aijng > 0. This forces A to be H-primitive. FLOW EQUIVALENCE OF G-SFTS 67

−1 The image of (x, g) ∈ X × G under c TA computes to be −1 −1 −1 −1 (σx, c gτA(x)) = (σx, (c gc)c τA(x)) = (σx, (c gc)τA(x)) . The final claim follows from −1 −1 −1 c TA :(x, g) 7→ (σx, c gτA(x)) = (σx, gc τA(x)) = (σx, gτA(x)) . Appendix E. A special case Recall, G denotes a finite group. The general theorems of this paper reduce the classification of G-SFTs to an algebraic problem which is far beyond the scope of this paper. Still, in this section we study the algebraic invariants of a natural initial class, including a complete solution in a meaningful (though very special) case. The argument points to algebraic challenges for the general case. Throughout, P is the poset P2, and H is the coset structure for which H11 = H12 = H22 = G. To facilitate a quicker overview, proofs of some results do not immediately follow statements. p x ++ Given p, q, x in ZG, let M(p, q, x) = 0 q . Let M (p, q, x) be o the set of matrices A in MP (n, Z+G) with coset structure H such that I − A and M(p, q, x) are ElP (ZG)-equivalent (i.e., they have 1- stabilizations which for some n are ElP (n, ZG)-equivalent). When G is abelian, this forces det(I − A{1, 1}) = p and det(I − A{2, 2}) = q. ++ ++ Let M (p, q) = ∪xM (p, q, x). In this appendix, we study algebraic invariants of G-flow equivalence for the G-SFTs defined by ++ ++ M (p, q) = ∪xM (p, q, x): for general G, then abelian G satisfying a K-theory constraint, and finally for G = Z2, where we give a solution which is complete and algorithmically practical. For example, ++ when G = Z2 and M (p, q) consists of finitely many G-FE classes, we can count them (Theorem E.16). Before turning to the algebra, we note the following consequence of Proposition 7.11 (or a simple exercise), which shows the algebraic study corresponds to actual G-SFTs.

Proposition E.1. Suppose G is a ﬁnite group. For all p, q, x in ZG, M++(p, q, x) is nonempty.

Now we consider an arbitrary finite G. Given p in ZG we define Ue(p) to be the set of y in ZG such that µy : v 7→ yv induces an automorphism of ZG/(pZG). The induced map is a right G-module homomorphism; it is a right G-module automorphism if and only if it an abelian group automorphism. The map µy induces an automorphism if there exist v, w, x in ZG such that vy = 1 + pw and yv = 1 + px. When G is abelian, this means simply that [y] is a unit in the quotient ring ZG/p(ZG). Similarly, given q ∈ ZG we define Wf(q) to be the set of z in ZG such that v 7→ vz defines a group automorphism (equivalently, FLOW EQUIVALENCE OF G-SFTS 68 a left G-module automorphism) of ZG/(ZG)q. This means there exist v, w, x in ZG such that yv = 1 + wq and vy = 1 + xq. We need a little more. Let Mk(p, q, x) be the 1-stabilization of M(p, q, x) in MP (n, ZG) with n = (k, k). So, for M = Mk(p, q, x), M = I except that M1,1 = p, Mk+1,k+1 = q and M1,k+1 = x.A matrix A has a 1-stabilization El(ZG)-equivalent to a 1-stabilization of M(p, q, x) if and only if for all/any sufficiently large k, A has a 1- eq stabilization El(ZG)-equivalent to Mk(p, q, x). Now we define Ue (p) to be the set of y in Ue(p) such that for some k, there is an El(k + 1, ZG) equivalence U(p⊕Ik)V = (p⊕Ik) such that U(1, 1) = y. It will be convenient to write this equivalence in the form U(p⊕Ik) = (p⊕Ik)W (W =V −1). Similarly, we define Wfeq(q) to be the set of z in Wf(q) such that for some k, there is an El(k +1, ZG) equivalence U(q ⊕Ik) = (q ⊕Ik)W such that W (1, 1) = z. Finally, given p, q in ZG we define the abelian group L(p, q) = {pc + dq ∈ ZG : c ∈ ZG, d ∈ ZG}. When G is abelian, L(p, q) is also an ideal in ZG. For G not abelian, L(p, q) need not be even a onesided ideal.

Theorem E.2. Suppose G is a ﬁnite group, p, q, x ∈ ZG, and A, A0 are matrices in M++(p, q, x), M++(p, q, x0) respectively. Then the following are equivalent.

(1) The G-SFTs TA, TA0 are G-flow equivalent. 0 (2) M(p, q, x) and M(p, q, x ) have 1-stabilizations which are ElP (ZG)- equivalent. (3) There exist y ∈ Ueeq(p) and z ∈ Wfeq(p) such that yx − x0z ∈ L(p, q). Suppose G is abelian, and consider quotient groups as rings. Then Ue(p) is the set of x in ZG such that [x] is a unit in ZG/ hpi (and similarly for Ue(q)). Condition (3) holds if and only if there exist elements y ∈ Ueeq(p), z ∈ Ueeq(q) such that [y][x] = [z][x0] in ZG/L(p, q). Given p, q, with L = L(p, q) Theorem E.2 tells us5 that the G- flow equivalence classes of G-SFTs defined from matrices in M++(p, q) are in bijective correspondence with the set of full orbits of points in the abelian group ZG/L under the action of the semigroup of homomorphisms v + L 7→ yv + L and v + L 7→ vz + L coming from y ∈ Ueeq(p), z ∈ Weq(q). Often the group ZG/L(p, q) is finite, and in this case, with G abelian, that orbit relation on ZG/L(p, q) can be computed mechanically. We now turn to some proofs.

5The relation (3) in the statement of Theorem E.2 is a special case version of an adaptation to ZG of the invariant introduced by Huang in [18] for ﬂow equivalence of reducible SFTs with two irreducible components. FLOW EQUIVALENCE OF G-SFTS 69

Proof of Theorem E.2. (1) ⇐⇒ (2): This follows from the classifying Theorem 5.1, by our choice of H, because the only permutation of {1, 2} respecting the P2 relation is the identity. (2) =⇒ (3): For some k, we have an equivalence UMk(p, q, x) = 0 Mk(p, q, x )W of 2k × 2k matrices, which we can write in block form as   p 0 ( x 0 ) U{1, 1} U{1, 2} 0 Ik−1 0 0 (E.3)   0 U{2, 2} ( 0 0 ) q 0 0 0 0 Ik−1   p 0 ( x0 0 ) 0 Ik−1 0 0 W {1, 1} W {1, 2} =   . ( 0 0 ) q 0 0 W {2, 2} 0 0 0 Ik−1 Computation of the 1, k + 1 entry of (E.3) produces 0 U1,1x + U1,k+1q = pW1,k+1 + x Wk+1,k+1 . 0 Set y = U1,1 and z = Wk+1,k+1. If follows that yx − x z ∈ L(p, q). k T Next, view (ZG) as a set of column vectors, and let v = (v1, . . . , vk) denote an element of ( G)k. Because U{1, 1} p 0 = p 0 W {1, 1}, Z 0 Ik 0 Ik the map v 7→ U{1, 1}v induces a group automorphism (also a right G-module automorphism) of cok p 0 = ( G)k+1/ p 0 ( G)k+1. Z 0 Ik Z 0 Ik Z The map π : v 7→ v induces an isomorphism cok p 0 → cok(p) = 1 0 Ik G/p(ZG). Here π pushes the automorphism of cok p 0 down to the Z 0 Ik automorphism of ZG/p(ZG) induced by v1 7→ yv1. A similar argument, using the equivalence U{2, 2} q 0 = q 0 W {2, 2} and considering 0 Ik 0 Ik the cokernel of q 0 = q 0 with respect to the action on row vec- 0 Ik 0 Ik tors, shows that the rule vk+1 7→ vk+1z induces an automorphism of ZG/(ZG)q. This finishes the proof that (2) =⇒ (3). (3) =⇒ (2): By assumption, for large enough k we have an elementary equivalence G p 0 H = p 0 such that G = y. 0 Ik−1 0 Ik−1 1,1 Then     p 0 ( x 0 ) p 0 yx 0 G 0 0 Ik−1 0 0 H 0 0 Ik−1 w 0   =   0 Ik ( 0 0 ) q 0 0 Ik ( 0 0 ) q 0 0 0 0 Ik−1 0 0 0 Ik−1 where w is a size k − 1 column vector. Adding suitable multiples of columns 2, . . . , k to column k + 1 to zero out w implements a unipotent (hence elementary) equivalence to M(p, q, yx), which therefore is elementary equivalent to M(p, q, x). Similarly M(p, q, x0) is elementary equivalent to M(p, q, x0z). Because yx − x0z ∈ L(p, q), say yx + pr = x0z + sq, another application of unipotent equivalence shows 0 M(p, q, yx) and M(p, q, x z) are ElP (ZG)-equivalent. The final claims, for abelian G, are easily checked. We recall some facts from the book [22, pages 1-4] of Oliver. Sup- pose the finite group G is abelian. Then SK1(ZG) is the subgroup of FLOW EQUIVALENCE OF G-SFTS 70

K1(ZG) represented by matrices with determinant 1; it is the torsion subgroup of K1(ZG), and it is ﬁnite. The meaning of SK1(ZG) being trivial is precisely that every matrix over ZG with determinant 1 has a 1-stabilization which is an elementary matrix over ZG. Being abelian, G is the direct sum of its Sylow p-subgroups, and SK1(ZG) is trivial if 6 and only if one of the following hold (in which Ck denotes the cyclic group of order k).

(1) G is a direct sum of copies of C2. (2) Each Sylow p-subgroup of G has the form Cpn or Cp ⊕ Cpn .

Proposition E.4. Suppose G is abelian and SK1(ZG) is trivial. Then Ueeq(p) = Ue(p) and Wfeq(q) = Wf(q). Proof. Because G is abelian, obviously it is enough to prove Ueeq(p) = Ue(p). So, suppose y ∈ Ue(p); we will show y ∈ Ueeq(p). Because x 7→ yx induces an automorphism of ZG/p(ZG) there exist a in ZG (implementing the inverse automorphism) and b in ZG such that ay = 1+bp. Thus there is an SL(2, ZG)-equivalence y p p 0 a −1 yp p a −1 ypa−bp2 0 p 0 ( b a ) 0 1 −bp y = bp a −bp y = 0 −bp+ay = 0 1 and so, for arbitrary k > 0 there is also an SL(k + 2, ZG) equivalence y p 0 p 0 0 a −1 0 p 0 0 b a 0 0 1 0 −bp y 0 = 0 1 0 . 0 0 Ik 0 0 Ik 0 0 Ik 0 0 Ik Because SK1(ZG) is trivial, for some k this SL(k + 2, ZG)-equivalence eq is an El(k + 2, ZG)-equivalence. This shows y ∈ Ue . To finish, we work out the classification in complete detail for the case G = Z2. From here, G denotes Z2. We use Zn to denote Z/nZ and G = {e, g}. ∼ The ring R = ZG and its ideals. Let R denote the subring of Z2 consisting of all (α, β) with α ≡ β mod 2. The map ZG → R given by ae + bg 7→ (a + b, a − b) is a well known ring isomorphism α+β α−β (with inverse (α, β) 7→ 2 e + 2 g). We will work with R rather than ZG. We define E to be the even elements of R, i.e. the (α, β) with α and β even integers. The set R \ E is the set of odd elements. Let E+ = {(α, 0) ∈ R} = h(2, 0)i. Let E− = {(0, β) ∈ R} = h(0, 2)i. J will always refer to an ideal in R. Given J, let J+,J− be the ideals and j+, j− the nonnegative integers such that J+ = E+ ∩ J = j+E+ = h(2j+, 0)i and J− = E− ∩ J = j−E− = h(0, 2j−)i. If (α, β) ∈ J, then (2α, 0) ∈ J+ and (0, 2j−) ∈ J− From this we deduce that either J = J+ ⊕ J− or |J/(J+ ⊕ J−)| = 2. J has rank 2 (as an abelian group) if and only if j+ 6= 0 6= j−. If |J/(J+ ⊕ J−)| = 2, then J is a rank two principal ideal, J = hji with j = (j+, j−). For a nonnegative vector v = (a, b), γv : R → Za ⊕ Zb denotes the obvious map (α, β) 7→ ([α], [β]). 6 See [22, pp. 1-19] for an overview of K1(ZG) and its history. FLOW EQUIVALENCE OF G-SFTS 71

Proposition E.5. For J an ideal of R, by cases the following maps ρ give a presentation of the abelian group epimorphism R → R/J (i.e., they deﬁne group epimorphisms with kernel J).

(1) J = hji, j odd, j = (j+, j−). Then ρ : R → Zj+ ⊕ Zj− . (a) ρ :(α, β) 7→ γj(α/2, β/2) if (α, β) ∈ E, (b) ρ : x 7→ γj(x − j) if (α, β) ∈ R \ E. (2) J = j+E+ ⊕ j−E−. Then ρ = γ2j : R → γ2jR, where γ2jR =

{(a, b) ∈ Z2j+ ⊕ Z2j− : a ≡ b mod 2}. (3) J = hji rank 2 (i.e. j+ 6= 0 6= j−), j = (j+, j−) even. Let j = γ2j(j). Then ρ : R → γ2jR/{0, j}. Here ρ is γ2j followed by the quotient map from γ2jR with kernel the two-point subgroup 0 0 {0, j}. Moreover, ρ(x) = ρ(x ) if and only if γ2j(x) = γ2j(x ) or 0 γ2j(x − j) = γ2j(x ). 0 Proof. Let J = h(2j+, 0), (0, 2j−)i. Case 1. If x is even, then x ∈ J iﬀ x ∈ J 0. If x is odd, then x ∈ J 0 0 0 iﬀ x − r ∈ J . Let ρ (x) = γ2j(x) if x ∈ E, and ρ (x) = γ2j(x − j) 0 if x ∈ R \ E. Then ρ : R → Z2j+ ⊕ Z2j− is a homomorphism and ker(ρ0) = J. The image of ρ0 is the subgroup H of ([α], [β]) with α and β even. The map ([α], [β]) 7→ ([α/2], [β/2]) is a group isomorphism

H → Zj+ ⊕ Zj− . Case 2. This is clear, because J = J 0.

Case 3. {0, j} is a group, because 2j = 0 in Z2j+ ⊕ Z2j− . Then −1 0 0 ker(ρ) = γ2j {0, j} = J ∪ (j + J ) = J. The ﬁnal “Moreover” statement follows.

In Case 1 above, on account of the ﬁnal map H → Zj+ ⊕ Zj− , the group epimorphism ρ is not a ring homomorphism (unless J = R). Given v ∈ R, we let µv denote the map R → R given by w 7→ vw, and also (for a lighter notation) the homomorphism induced on a quotient group of R. For an ideal J, let Ue(J) be the set of v in R such that µv is an automorphism of R/J. For elements v1, . . . , vk of R, Ue(v1, . . . , vk) denotes Ue(J) for J = hv1, . . . , vki. Theorem E.6. Ue(J) is characterized by cases.

(1) J = hji, j = (j+, j−) odd. Then (α, β) ∈ Ue(J) if and only if gcd(α, j+) = 1 = gcd(β, j−) . (2) J ⊂ E. Then (α, β) ∈ Ue(J) if and only if (α, β) is odd with gcd(α, j+) = 1 = gcd(β, j−) . In Case 1, (2k, 2`) ∈ Ue(J) if k and ` are positive. In Case 2, if J = hji with j even, then the requirement that (α, β) must be odd is redundant; if J = j+E+ ⊕ j−E−, then J ⊂ E, and when (j+, j−) is odd the requirement is not redundant. For k nonzero in Z, gcd(k, 0) = |k|. So, if j+ 6= 0 = j−, Theorem E.6 gives Ue(J) = {(α, ±1) ∈ R: gcd(α, j+) = FLOW EQUIVALENCE OF G-SFTS 72

1}. Likewise, if j+ = 0 6= j−, then Ue(J) = {(±1, β) ∈ R: gcd(α, j+) = 1}. If J = h0i, then Ue(J) = {±1, ±1}, the units of R. If J = R, then Ue(J) = R. Proof of Theorem E.6. Let v = (a, b) ∈ R. We use the maps ρ from Proposition E.5 which present R → R/J.

Case 1. Here ρ : R → Zj+ ⊕ Zj− . For x = (α, β) ∈ E the induced map ρ(x) 7→ ρ(vx) is [α/2], [β/2] 7→ [aα/2], [bβ/2] = a[α/2], b[β/2] , and for x ∈ R \ E the map is [(α − j+)/2], [(β − j−)/2] 7→ [(aα − j+)/2], [(bβ − j−)/2] = a[(α − j+)/2], b[(β − j−)/2] where the last equality holds because (j+, j−) is odd. It follows that the induced map µv is an automorphism if and only the gcd conditions hold. Case 2. First suppose J = j+E+ ⊕ j−E−. Here ρ = γ2j : R →

Z2j+ ⊕ Z2j− . The map ρ(x) 7→ ρ(vx) is ([α], [β]) 7→ ([aα], [bβ]), so it is an automorphism iﬀ gcd(a, 2j+) = 1 = gcd(b, 2j−). Lastly, suppose J = hji, with j = (j+, j−) even and j+ 6= 0 6= j−.

Here ρ : R → (Z2j+ ⊕Z2j− )/{0, j} presents R → R/J. If v is even or the gcd condition fails, then µv as an endomorphism of (Z2j+ ⊕Z2j− )/{0, j} has a nontrivial kernel. If v is odd, then µv as an endomorphism of (Z2j+ ⊕ Z2j− ) ﬁxes j, so it deﬁnes an automorphism of (Z2j+ ⊕

Z2j− )/{0, j} if and only if deﬁnes an automorphism of (Z2j+ ⊕ Z2j− ), which is equivalent to the gcd conditions. Corollary E.7. For elements p, q of R, let J = hp, qi. Then

Ue(p) ∪ Ue(q) ⊂ Ue(p, q) = {vw : v ∈ Ue(p), w ∈ Ue(q)} .

Proof. For an ideal J and element x of R, x ∈ Ue(J) iff its image in R/J, considered as a ring, is a unit. As a unit in R/ hpi pushes down to a unit in R/ hp, qi, we have Ue(p) ⊂ Ue(p, q), and likewise Ue(q) ⊂ Ue(p, q). It remains to show Ue(p, q) ⊂ {vw : v ∈ Ue(p), w ∈ Ue(q)}. We appeal to Theorem E.6, by cases. Define d+ = gcd(αp, αq) if at least one of αp, αq is nonzero, and d− = gcd(βp, βq) if at least one of βp, βq is nonzero. Case I. J = hri with r = (αr, βr) odd. By Lemma E.13, r = (d+, d−). If e.g. gcd(α, αr) = 1 and α is an odd prime, then (α, 1) ∈ Ue(p) ∪ Ue(q). We see that an odd element u of Ue(p, q) is the product of elements of the form (α, 1), (1, β) from Ue(p)∪Ue(q). If k and ` are positive, then (2k, 2`) ∈ Ue(p)∪Ue(q), because at least one of p and q must be odd (because r is odd). Thus Ue(p)∪Ue(q) FLOW EQUIVALENCE OF G-SFTS 73 generates Ue(p, q). A product of several elements from Ue(p) ∪ Ue(q) is a product of a single element of Ue(p) and a single element of Ue(q). Case II. J ⊂ E. Here p and q must be even, and there can be no odd element in any of Ue(p), Ue(q), Ue(J). The rest of the argument proceeds as in Case I by considering gcd and generating units (α, 1), (1, β). Definition E.8. Given elements x, x0 of a ring S, we say x ∼ x0 in S if there is a multiplicative unit u in S such that ux = x0.

In the product ring Zm ⊕Zn, the units are the pairs ([a], [b]) such that gcd(a, m) = 1 = gcd(b, n), and these are the units deﬁning ∼ in part (3) of the statement of Theorem E.9 below. (The map ρ in part (3a) is not a ring homomorphism, but the logic of the proof does not need it to be.) Also, M++(p, q, x) was deﬁned with {p, q, x} ⊂ ZG. In Theorem E.9, we use the corresponding elements in R, without introducing more notation. E.g., x = ae + bg becomes x = (a + b, a − b).

Theorem E.9. Let J be the ideal hp, qi in R, with J ∩ E = j+E+ ⊕ 0 j−E−. Suppose x, x are in R. Then the following are equivalent. (1) Matrices in M++(p, q, x) and M++(p, q, x0) deﬁne G-ﬂow equivalent G-SFTs. (2) [x] ∼ [x0] in R/J. (3) The conditions below hold, according to the type of J. 0 (a) J = hji with j = (j+, j−) odd. Then ρ(x) ∼ ρ(x ) in

Zj+ ⊕ Zj− , where (i) ρ(x) = γj(x/2) if x ∈ E, and (ii) ρ(x) = γj((x − j)/2) if x∈ / E. 0 (b) J = j+E+ ⊕ j−E− . Then γ2j(x) ∼ γ2j(x ) in Z2j+ ⊕ Z2j− . (c) J = hji, j = (j+, j−) ∈ E, j+ 6= 0 6= j−. 0 0 Then γ2j(x) ∼ γ2j(x ) or γ2j(x − j) ∼ γ2jx in Z2j+ ⊕ Z2j− .

Proof. Because SK1(Z2) is trivial, it follows from Proposition E.4 and Theorem E.2 that (1) holds if and only there are elements y ∈ Ue(p) and z ∈ Ue(q) such that [y][x] = [z][x0] in R/J. By Corollary E.7, these elements y, z exist if and only if [x] ∼ [x0] in R/J. We have shown (1) ⇐⇒ (2). We next show (1) ⇐⇒ (3). Let x = (α, β) and x0 = (α0, β0) be elements of R. We will use the map ρ of Proposition E.5 which presents R → R/J. Then

[x] ∼ [x0] in R/J ⇐⇒ ∃(a, b) ∈ Ue(J) , [(aα, bβ)] = [(α0, bβ)] in R/J ⇐⇒ (E.10) ∃(a, b) ∈ Ue(J) , ρ((aα, bβ)) = ρ((α0, β0)) . FLOW EQUIVALENCE OF G-SFTS 74

Case (a): J = hji with j = (j+, j−) odd. Here ρ : R → Zj+ ⊕

Zj− . By Theorem E.6 and Corollary E.7, (a, b) ∈ Ue(J) if and only if gcd(a, j+) = 1 = gcd(b, j−). Suppose (a, b) ∈ Ue(J) and (α, β) ∈ R. Then ( γ (α/2, β/2) if (α, β) ∈ E, ρ(α, β) = j γj((α − j+)/2, (β − j−)/2)) if (α, β) ∈ R \ E ;

 γj(aα/2, bβ/2) if (α, β) ∈ E,    γ (aα/2, bβ/2) if (α, β) ∈/ E and ρ(aα, bβ) = j  (a, b) ∈ E,  γj((aα − j+)/2, (bβ − j−)/2)) if (α, β) ∈/ E and   (a, b) ∈/ E. In each case, we can check that ρ(aα, bβ) equals the image of ρ(α, β) under the automorphism of Zj+ ⊕ Zj− deﬁned by ([m], [n]) 7→ ([am], [bn]). 0 Therefore (E.10) is equivalent to ρ(x) ∼ ρ(x ) in Zj+ ⊕ Zj− . Case (b): J = j+E+ ⊕ j−E− = h(2j+, 0), (0, 2j−)i. Here ρ = γ2j. By Theorem E.6 and Corollary E.7, (a, b) ∈ Ue(J) if and only if gcd(a, 2j+) = 1 = gcd(b, 2j−). Thus (E.10) is equivalent to (α, β) ∼ 0 0 (α , β ) in Z2j+ ⊕ Z2j− . Case (c): J = hji, j = (j+, j−) ∈ E, j− 6= 0 6= j+ . By Proposition 0 0 E.5, here ρ(x) = ρ(x ) if and only if γ2j(x) = γ2j(x ) or γ2j(x + j) = 0 γ2j(x ). So, (E.10) holds iﬀ there exists (a, b) ∈ Ue(J) such that 0 0 0 0 (E.11) γ2j((aα, bβ)) = γ2j(α , β ) or γ2j((aα, bβ)+j) = γ2j(α , β ) .

As in Case (b), (a, b) ∈ Ue(J) if and only if gcd(a, 2j+) = 1 = gcd(b, 2j−). For (a, b) ∈ Ue(J), γ2j((aα, bβ) + j) = γ2j((a(α + j+), b(β + j−)) be- 0 cause (a, b) is odd. Thus (E.11) is equivalent to γ2j(x) ∼ γ2j(x ) or 0 γ2j(x + j) ∼ γ2j(x ) in Z2j+ ⊕ Z2j− . The next two propositions explain how to compute the ideal J in the form of Theorem E.9 from its generating polynomials p, q. The (trivial) Proposition E.12 is stated to give a complete list of possibilities.

Proposition E.12. Let p = (αp, βp) and q = (αq, βq). Suppose J = hp, qi with {p, q} ⊂ E+ ∪ E−} (i.e., αpβp = 0 = αqβq). (1) J = 0 if and only if p = q = 0. (2) Suppose βp = 0 = βq and at least one of αp, αq is nonzero. Let d+ = gcd(αp, αq). Then J has rank 1, and J = h(d+, 0)i = j+E+ with j+ = d+/2. FLOW EQUIVALENCE OF G-SFTS 75

(3) Suppose αp = 0 = αq and at least one of βp, βq is nonzero. Let d− = gcd(βp, βq). Then J has rank 1, and J = h(0, d−)i = j−E− with j− = d−/2. (4) Suppose one of p, q is (2m, 0) 6= 0 and the other is (0, 2n) 6= 0 with m 6= 0 6= n. Then J has rank 2, and J = j+E+ ⊕ j−E− with (j+, j−) = (m, n).

Proposition E.13. Suppose elements p = (αp, βp) and q = (αq, βq) generate a rank 2 ideal J = hp, qi, and at least one of p, q lies outside E+ ∪ E−. Set d+ = gcd(αp, αq) and d− = gcd(βp, βq). Then one of the following holds.

d+ d− (1) J = 2 E+ ⊕ 2 E− = h(d+, 0), (0, d−)i . (2) J is the principal ideal h(d+, d−)i.

Write αp = rpd+, βp = spd−, αq = rqd+ and βq = sqd−. Then J is principal if and only if

(E.14) rp ≡ sp mod 2 and rq ≡ sq mod 2 .

Proof. The claim when {p, q} ⊂ E+ ∪ E− is obvious. Now suppose p∈ / (E+ ∪ E−) (the argument when q∈ / (E+ ∪ E−) is essentially the same). Then αp 6= 0 6= βp. Let H be the subgroup of R generated by the groups Hp = αpE+ ⊕ βpE− and Hq = αqE+ ⊕ βqE−. Then

H = d+E+ ⊕ d−E−

= h(2d+, 0), (0, 2d−)i ⊂ J ⊂ h(d+, 0), (0, d−)i .

There are integers `1, `2 such that d+ = `1αp + `2αq. Let v = `1p + `2q. Then

v = `1(αp, βp) + `2(αq, βq)

= `1(rpd+, spd−) + `2(rqd+, sqd−) = (`1rp + `2rq)d+, (`1sp + `2sq)d− = d+, c1d− for some integer c1. Likewise for some w in J we have w = (c2d+, d−) ∈ J. So, (d+, d−) ∈ J, and (d+, 0) ∈ J iﬀ (0, d−) ∈ J. We can now see that J is principal if and only the following holds for all integers `1, `2:

(E.15) (`1rp + `2rq) is odd ⇐⇒ (`1sp + `2sq) is odd .

Clearly (E.14) implies (E.15). For the converse, assume (E.15). If rp, rq are both odd, then (E.14) holds, because sp, sq are not both even, as gcd(sp, sq) = 1. Suppose rp is even; then rq is odd. Let integers `1, `2 give `1rp + `2rq = 1. Here, `2 must be odd, but we may choose `1 even or odd, because (`1 + rq)rp + (`2 − rp)rq = 1. Because `1sp + `2sq must be odd in either case, it follows that sp must be even, which forces sq to be odd, and therefore (E.14) holds. The argument for the case that rq is even is essentially the same. FLOW EQUIVALENCE OF G-SFTS 76

To use Theorem E.9, we recall some elementary number theory. Sup- 0 0 pose n is a nonnegative integer. For integers α and α ,[α] ∼ [α ] in Zn means there exists an integer a such that gcd(a, n) = 1 and [aα] = [α0]. If n = 0, then [a] ∼ [a0] means a = ±a0. If n = 1 then [a] ∼ [a0] holds for all a, a0. Suppose n = pm with p prime and m > 0. Write Pm−1 j y in Zpm as y = j=0 yjp with each yj in {0, 1, . . . , p − 1}; y is 0 a unit if and only if y0 6= 0. Now, [y] ∼ [y ] in Zpm if and only if 0 0 y = 0 = y or min{j : yj 6= 0} = min{j : yj 6= 0}. Equivalently, max{k : 1 ≤ k ≤ m, pk|y} = max{k : 1 ≤ k ≤ m, pk|y0}. For the equiv- 0 alence relation [a] ∼ [a ] in Zpm , there are exactly m + 1 equivalence classes. Qk mi Finally, suppose n > 1 with prime power factorization n = i=1 pi . Qk As a ring, Zn is isomorphic to i=1 Zpmi . In this presentation, let 0 0 0 0 Qk mi x = (x1, . . . , xk) and x = (x1, . . . , xk). Then x ∼ x in i=1 pi if and 0 only if xi ∼ x in mi , for 1 ≤ i ≤ k. To express this another way, i Zpi k for a in Z and p = pi, let δp,n(a) = max{k : 1 ≤ k ≤ mi : p |a}. Then 0 0 [a] ∼ [a ] in Zn if and only if δp,n(a) = δp,n(a ), for each prime p dividing n. Given a, a0, n this is straightforward to compute. Qk mi Qk For n = i=1 pi as above, deﬁne κ(n) = i=1(mi + 1), and also deﬁne κ(1) = 1. Then for positive integers m, n it follows that for the equivalence relation ∼ in the product ring Zm ⊕ Zn, the number of equivalence classes is κ(m)κ(n). We will need to count ∼ classes in Zm ⊕ Zn represented by elements k+ of γ2jR = {([a], [b]) : a + b ≡ 0 mod 2}. Let j+ = 2 g+, with g+ an odd integer. The ∼ classes in Z2j+ arising from odd and even integers are disjoint. The number of classes arising from odd integers is κ(g); the number arising from even integers is (k+ + 1)κ(g). The analogous k− statements hold for j− = 2 g−, with g− an odd integer. So, in γ2jR, the number of ∼ classes arising from odd elements of R is κg+ κg− and the number arising from even elements of R is (k+ + 1)(k− + 1)κg+ κg− .

Theorem E.16. Let J = hp, qi be rank 2 in R, i.e. J ∩ E = j+E+ ⊕ k+ k− j−E− with j+ 6= 0 6= j−. Write j+ = 2 g+, j− = 2 g+ with g+, g− odd integers. Let fp,q be the number of distinct G-ﬂow equivalence classes deﬁned by matrices in M(p, q). There are three cases: (1) J = hji, with j odd. (2) J = j+E+ ⊕ j−E− = h(2j+, 0), (0, 2j−)i . (3) J = hji, with j even.

The computation of fp,q in these cases is

(1) fp,q = κ(g+)κ(g−) = κ(j+)κ(j−). (2) fp,q = κ(g+)κ(g−) (k+ + 1)(k− + 1) + 1 . (3) fp,q = κ(g+)κ(g−)(k+k− + 2) . Proof. The cases (1) and (2) follow immediately from Theorem E.9 and the discussion preceding the theorem. Now suppose we are in case (3). FLOW EQUIVALENCE OF G-SFTS 77

For x ∈ R, let C(x) be the ∼ class of γ2j(x) in γ2jR. By Theorem E.9, we must compute the number of distinct sets of the form C(x)∪C(x+j).

For any integer α and odd prime p, δp,2j+ (α + j+) = δp,2j+ (α). If

δ2,2j+ (α) = i, then

δ2,2j+ (α + j+) = i if i < k+

= k+ + 1 if i = k+

= k+ if i = k+ + 1 .

The analogous statements hold for j−, k−. Thus C(x)∪C(x+j) = C(x) if x = (α, β) with δ2,2j (α) < k+ and δ2,2j (β) < k−. The number of + − sets C(x) of this form is κ(g+)κ(g−) (k+ −1)(k− −1)+1). (Note k+ ≥ 1 and k− ≥ 1 because j is even.)

If x = (α, β) with δ2,2j+ (α) ∈ {k+, k+ + 1}, then C(x) and C(x + j) are disjoint; also, α is an even number, so (α, β) ∈ R iﬀ β is even. + Let Ci be the collection of sets C(x) for which x = (α, β) ∈ R with δ (α) = i. Then |C+ | = |C+ | = κ(g )κ(g )(k +1) . The collec- 2,2j+ k+ k++1 + − − tions C+ ,C+ are disjoint, and the rule C(x) 7→ C(x + j) induces a k+ k++1 bijection C+ ↔ C+ . So, C+ ∪C+ gives rise to κ(g )κ(g )(k +1) k+ k++1 k+ k++1 + − − distinct sets of the form C(x) ∪ C(x + j). We can reverse the roles of + and − and say likewise that C− ∪C− gives rise to κ(g )κ(g )(k +1) k− k−+1 − + + distinct sets of the form C(x) ∪ C(x + j). Finally, to compute fp,q we add the three counts above, and correct for the doublecount: fp,q = κ(g+)κ(g−) (k+ − 1)(k− − 1) + 1 + k+ + 1 + k− + 1 − 2

= κ(g+)κ(g−)(k+k− + 2) . Example E.17. We will work out the classification of G-FE classes in M++(p, q) for the example p = (12, 96), q = (8, 24). (These elements of R correspond to the elements 54e−42g, 16e−8g in ZG.) Here d+ = gcd(12, 8) = 4 and d− = gcd(96, 24) = 24. We compute (12, 96) = (rpd+, spd−) = (3(4), 4(24)), and see rp 6≡ sp mod 2. By Proposi- tion E.13, it follows that hp, qi = h(2j+, 0), (0, 2j−)i with (2j+, 2j−) = ++ ++ 0 (d+, d−) = (4, 24). By Theorem E.9, matrices in M (p, q, x), M (p, q, x ) define G-FE G-SFTs iff γ(4,24)(x) ∼ γ(4,24)(x) in Z4 ⊕ Z24. We will write this as x ∼ x0. Write x = (α, β), x0 = (α0, β0). As 4 = 22 and 3 1 0 0 24 = 2 3 , we have x ∼ x iff the following hold: δ2,4(α) = δ2,4(α ), 0 0 δ2,24(β) = δ2,24(β ), and δ3,24(β) = δ3,24(β ). The allowed combinations of possible values of these invariants is displayed in Table 1, with a few examples. We can check that e.g. (0, 6) ∼ (0, 18) 6∼ (0, 12) and (1, 3) ∼ (3, 15) 6∼ (3, 11). From Table 1 we see that 14 ∼ classes arise here for elements of R, consistent with Theorem E.16(2), which in our case FLOW EQUIVALENCE OF G-SFTS 78 Table 1.

(α, β) δ2,4(α) δ2,24(β) δ3,24(β) odd 0 0 0,1 even 1,2 1,2,3 0,1 (0,18) 2 1 1 (2,6) 1 1 1 (3,11) 0 0 0 gives

k+ k− 1 3 (j+, j−) = (2, 12) = (2 g+, 2 g−) = (2 (1), 2 (3))

fp,q = κ(g+)κ(g−)((k+ + 1)(k− + 1) + 1 ) = (1)(2) ( (1 + 1)(2 + 1) + 1) ) = 14 . Next, we work to Proposition E.19, which shows how to determine when a matrix A lies in some M++(p, q, x), and then compute p, q, x. Two matrices are El(R)-equivalent when there exists k such that they have 1-stabilizations of size k × k which are El(k, R)-equivalent. If B is El(R)-equivalent to a 1 × 1 matrix (p), then p must be det(B). Lemma E.18. Given an n × n matrix B over R, there is an algorithm which determines whether B is El(n, R)-equivalent to some 1×1 matrix (p), and computes an explicit El(n, R)-equivalence from B to (p⊕In−1), when this is the case. Proof. A matrix over R (even a 2×2 matrix) need not be GL(R) equivalent to a diagonal matrix, or even a triangular matrix (see e.g.[12, Example 8.7]). But, for the special case we consider here, a modi- ﬁcation of the Smith form argument applies to produce the explicit El(n, R)-equivalence required for (1), or a failure which shows there is no such equivalence. For detail, see [12, Lemma 8.2, Remark 8.3], and the algorithmic proof of [12, Lemma 8.2].

Proposition E.19. Suppose B ∈ MP (n,R), with n = (n1, n2) and P = P2. There is an algorithm to determine whether there exist p, q, x p x such that B is ElP (n,R)-equivalent to the matrix M(p, q, x) = 0 q , and in this case to compute p, q, x. Proof. The required elementary equivalence will exist only if there are El(R)-equivalences of the diagonal blocks of B to (p) and (q). By Lemma E.18, we can decide whether this occurs, and in the case it does we can construct elementary equivalences, UB{1, 1}V = (p)⊕In1−1 and 0 0 U B{2, 2}V = (q) ⊕ In2−1. Given this data, we deﬁne an elementary equivalence of B,   p 0 Y U 0 B{1, 1} B{1, 2} V 0 0 In1−1 0 0 =   . 0 U 0 B{2, 2} 0 V 0 q 0 0 In2−1 FLOW EQUIVALENCE OF G-SFTS 79

Then for suitable blocks Z,Z0,

 p 0   p 0  Y 0 ( x 0 ) IZ 0 In −1 IZ 0 In −1 0 0  1  =  1  0 I 0 q 0 0 I 0 q 0 0 In2−1 0 In2−1 where x is the upper left entry of Y . Note, for B in Proposition E.19: if B is not El(n, R)-equivalent to a 1-stabilization of the matrix (p) ⊕ In−1, then no 1-stabilization of B is El(n, R)-equivalent to a 1-stabilization of the matrix (p) ⊕ In−1.

References [1] R. Adler, B. Kitchens, and B. Marcus. Almost topological classification of finite-to-one factor maps between shifts of finite type. Ergodic Theory Dynam. Systems, 5(4):485–500, 1985. [2] R. L. Adler, B. Kitchens, and B. H. Marcus. Finite group actions on shifts of finite type. Ergodic Theory Dynam. Systems, 5(1):1–25, 1985. [3] S. E. Arklint, G. Restorff, and E. Ruiz. Classification of real rank zero, purely infinite C∗-algebras with at most four primitive ideals. J. Funct. Anal., 271:1921–1947, 2016. [4] M. Boyle. Flow equivalence of shifts of finite type via positive factorizations. Pacific J. Math., 204(2):273–317, 2002. [5] M. Boyle. Positive K-theory and symbolic dynamics. In Dynamics and ran- domness (Santiago, 2000), volume 7 of Nonlinear Phenom. Complex Systems, pages 31–52. Kluwer Acad. Publ., Dordrecht, 2002. [6] M. Boyle, T. M. Carlsen, and S. Eilers. Flow equivalence and isotopy for sub- shifts. Dyn. Syst., 32(3):305–325, 2017. [7] M. Boyle, T. M. Carlsen, and S. Eilers. Flow equivalence of sofic shifts. Israel J. Math., 225(1):111-146, 2018. [8] M. Boyle and S. Chuysurichay. The mapping class group of a shift of finite type. J. Mod. Dyn., 13:115-145, 2018. [9] M. Boyle and D. Huang. Poset block equivalence of integral matrices. Trans. Amer. Math. Soc., 355(10):3861–3886 (electronic), 2003. [10] M. Boyle and S. Schmieding. Finite group extensions of shifts of finite type: K- theory, Parry and Livˇsic. Ergodic Theory Dynam. Systems, 37(4):1026–1059, 2017. [11] M. Boyle and B. Steinberg. Decidability of flow equivalence and isomorphism problems for graph C*-algebras and quiver representations. arXiv:1812.04555, 2018. [12] M. Boyle and M.C. Sullivan. Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings. Proc. London Math. Soc. (3), 91(1):184–214, 2005. [13] M. Boyle and J. B. Wagoner. Positive algebraic K-theory and shifts of finite type. In Modern dynamical systems and applications, pages 45–66. Cambridge Univ. Press, Cambridge, 2004. [14] U. Fiebig. Periodic points and finite group actions on shifts of finite type. Ergodic Theory Dynam. Systems 13(3): 485–514, 1993. [15] M. Field. Equivariant diffeomorphisms hyperbolic transverse to a G-action. J. London Math. Soc. (2), 27(3):563–576, 1983. FLOW EQUIVALENCE OF G-SFTS 80

[16] M. Field and M. Golubitsky. Symmetry in chaos. Society for Industrial and Ap- plied Mathematics (SIAM), Philadelphia, PA, second edition, 2009. A search for pattern in mathematics, art, and nature. [17] M. Field and M. Nicol. Ergodic theory of equivariant diffeomorphisms: Markov partitions and stable ergodicity. Mem. Amer. Math. Soc., 169(803):viii+100, 2004. [18] D. Huang. Flow equivalence of reducible shifts of finite type. Ergodic Theory Dynam. Systems, 14(4):695–720, 1994. [19] B.P. Kitchens. Symbolic dynamics; One-sided, two-sided and countable state Markov shifts. Springer-Verlag, Berlin, 1998. [20] D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. [21] B. Marcus and S. Tuncel. The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains. Ergodic Theory Dynam. Systems, 11(1):129–180, 1991. [22] R. Oliver. Whitehead groups of finite groups, volume 132 of London Mathe- matical Society Lecture Note Series. Cambridge University Press, Cambridge, 1988. [23] W. Parry. Notes on coding problems for finite state processes. Bull. London Math. Soc., 23(1):1–33, 1991. [24] W. Parry. The Livˇsicperiodic point theorem for non-abelian cocycles. Ergodic Theory Dynam. Systems, 19(3):687–701, 1999. [25] W. Parry and K. Schmidt. Natural coefficients and invariants for Markov-shifts. Invent. Math., 76(1):15–32, 1984. [26] W. Parry and D. Sullivan. A topological invariant of flows on 1-dimensional spaces. Topology, 14(4):297–299, 1975. [27] D.J. Rudolph. Counting the relatively finite factors of a Bernoulli shift. Israel J. Math., 30(3):255–263, 1978. [28] K. Schmidt. Remarks on Livˇsic’ theory for nonabelian cocycles. Ergodic Theory Dynam. Systems, 19(3):703–721, 1999. [29] D.S. Silver and S.G. Williams. An invariant of finite group actions on shifts of finite type. Ergodic Theory Dynam. Systems, 25(6):1985–1996, 2005. [30] M.C. Sullivan. Errata to: “An invariant of basic sets of Smale flows”. Ergodic Theory Dynam. Systems, 18(4):1047, 1998. [31] M.C. Sullivan. An invariant of basic sets of Smale flows. Ergodic Theory Dy- nam. Systems, 17(6):1437-1448, 1997. [32] M.C. Sullivan. Invariants of twist-wise flow equivalence. Discrete Contin. Dy- nam. Systems 4(3):475-484, 1998. [33] M.C. Sullivan. Twistwise flow equivalence and beyond ... . Contemp. Math. 385:171-186, 2005.

Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA E-mail address: [email protected]

Department of Science and Technology, University of the Faroe Islands, Vestara Bryggja 15, FO-100 Torshavn,´ The Faroe Is- lands E-mail address: [email protected] FLOW EQUIVALENCE OF G-SFTS 81

Department of Mathematical Sciences, University of Copen- hagen, DK-2100 Copenhagen Ø, Denmark E-mail address: [email protected]