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Weakly surjunctive groups and symbolic group varieties

In this paper, we introduce the classes of weakly surjunctive and linearly surjunctive groups which include all sofic groups and more generally all surjunctive groups. We investigate various properties of such groups and establish in particular a reversibility and invertibility theorem for injective endomorphisms of symbolic group varieties over weakly surjunctive group universes with algebraic group alphabets in arbitrary characteristic. We also obtain novel evidence related to Kaplansky's stable finiteness conjecture.


Introduction
To state the main results, we fix some basic notions of symbolic dynamics. Fix a set A called the alphabet, and a group G, the universe. A configuration c ∈ A G is a map c : G → A. The Bernoulli shift G × A G → A G is defined by (g, c) → gc, where (gc)(h) := c(g −1 h) for all g, h ∈ G and c ∈ A G .
Introduced by von Neumann [18], a cellular automaton over the group G and the alphabet A is a map τ : A G → A G admitting a finite memory set M ⊂ G and a local defining map µ : A M → A such that (τ (c))(g) = µ((g −1 c)| M ) for all c ∈ A G and g ∈ G.
Now let X be an algebraic group over a field k. Denote by X(k) the set of rational points of X. The ring CA algr (G, X, k) of algebraic group cellular automata consists of cellular automata τ : X(k) G → X(k) G which admit a memory M ⊂ G and a local defining map µ : X(k) M → X(k) induced by some k-homomorphism of algebraic groups f : X M → X, i.e., µ = f | X(k) M , where X M is the fibered product of copies of X indexed by M .
A cellular automaton τ : A G → A G is reversible if it is bijective and the inverse map τ −1 : A G → A G is also a cellular automaton. It is wellknown that if the alphabet A is finite then every bijective cellular automaton τ : A G → A G is reversible (cf., e.g. [6,Theorem 1.10.2]).
In [19,Theorem 1.1], the author proved the following surjunctivity and invertibility result for endomorphisms of symbolic group varieties: Theorem 1.1. Let G be a sofic group. Let X be a connected algebraic group over an algebraically closed field k of characteristic zero. Suppose that τ ∈ CA algr (G, X, k) is injective. Then τ reversible and τ −1 ∈ CA algr (G, X, k).
Here, the wide class of sofic groups was first introduced by Gromov [11] as a common generalization of residually finite groups and amenable groups.
Let G and A be groups. Then A G admits a natural group structure with pointwise operations: for x, y ∈ A G , the product z = xy is given by z(g) = x(g)y(g) for all g ∈ G. We call a map τ : A G → A G a group cellular automaton if τ is a cellular automaton and that τ (xy) = τ (x)τ (y) for all configurations x, y ∈ A G . Equivalently, it is clear that a cellular automaton τ : A G → A G is a group cellular automaton if τ admits for some finite memory set M ⊂ G a local defining map µ : A M → A which is a homomorphism of groups.
In this paper, we obtain a generalization of Theorem 1.1 where we can eliminate the connectedness hypothesis on the alphabet and the universe is now only required to be a weakly surjunctive group defined as follows.
Definition 1.2. We call a group G a weakly surjunctive group if for every finite group alphabet A, all injective group cellular automata τ : Surjunctive groups were introduced by Gottschalk in [15]. It is clear from the definition that every surjunctive group is weakly surjunctive. Note that every sofic group is surjunctive by the Gromov-Weiss theorem ( [11], [24]). We show in Lemma 3.1 that a group G is weakly surjunctive if and only if so is every finitely generated subgroup of G.
The main result of the paper is the following: Theorem A. Let G be a weakly surjunctive group and let X be an algebraic group over an uncountable algebraically closed field k. Suppose that τ ∈ CA algr (G, X, k) is injective. Then τ is reversible and if k has characteristic zero, one has τ −1 ∈ CA algr (G, X, k).
As an immediate application of the notion of weakly surjunctive groups, we mention here the following direct finiteness property for endomorphisms of symbolic group varieties: Corollary A. Let G be a weakly surjunctive group and let X be an algebraic group over an algebraically closed field k. Suppose that σ, τ ∈ CA algr (G, X, k) satisfy σ • τ = Id. Then one also has τ • σ = Id.
When G is required to be surjunctive while we only need to suppose that τ, σ ∈ CA alg (G, X, k) (see, e.g. [7], [8], [9], [23]), the statement of Corollary A was proven in [23]. In fact, a proof of Corollary A can be easily obtained by a straightforward modification of the proof of [23,Theorem C]. In Section 5, we explain the details and give another proof using Theorem A.
We present briefly an application of Corollary A on Kaplansky's stable finiteness conjecture. A ring R stably finite every one-sided invertible square matrix with coefficient in R is also two-sided invertible. Then Kaplansky's stable finiteness conjectures [16] asserts that for every group G and every field k, the group ring k[G] is stably finite.
We obtain the following slight generalization of [23,Theorem D] where the group G is required to be surjunctive.
Corollary B. Let G be a weakly surjunctive group and let R be the endomorphism ring of a commutative algebraic group over an algebraically closed field. Then the group ring R[G] is stably finite.
Proof. (See [23, Section 5.1]) Let k be an algebraically closed field and let Y be a commutative k-algebraic group. Denote by R = End k (Y ) the endomorphism ring of k-endomorphisms of Y . Fix n ∈ N and let X = Y n . Then Mat n (R) = End k (X) by [19,Lemma 9.5] and we have an isomorphism of rings End k (X)[G] ≃ CA algr (G, X, k) by [19,Proposition 9.3]. Corollary A implies that CA algr (G, X, k) is directly finite. It follows that is also a directly finite ring. Hence, we conclude from the ring isomorphism Mat n (R[G]) ≃ Mat n (R)[G] given by [19,Lemma 9.4] that Mat n (R[G]) is a directly finite ring.
To this point, we find that it is natural to ask the following question: Question 1. Does there exist a non weakly surjunctive group?
The paper is organized as follows. We collect in Section 2 basic lemmata in algebraic group theory. In Section 3, we investigate basic properties of the class of weakly surjunctive groups. Then Section 4 presents a short proof of Theorem A with ingredients from algebraic group theory and some ideas from the papers [23] and [11]. Note that in the case of sofic groups, the proof of Theorem 1.1 given in [19] is more direct and quite different in nature from our proof of Theorem A. Section 5 contains two different proofs of Corollary A. We define the class of linearly surjunctive groups in Section 6 which includes all weakly surjunctive groups and shows in particular that a group satisfies Kaplansky's stable finiteness conjecture if and only it is a linearly surjunctive group (Theorem 6.4). Finally, weakly dual surjunctive groups, which are generalizations of dual surjunctive groups [3], will be introduced and investigated in Section 7. In particular, we extend Corollary A and Corollary B to cover the case of weakly dual surjunctive group universes (see Theorem 7.4 and Corollary 7.5 respectively).

Preliminaries
2.1. Algebraic groups. Let R be a ring. Then an R-group scheme is a data of a quadruple (X, m, inv, e) where X is an R-scheme of finite type, m : X × R X → X and inv : X → X are morphisms of R-schemes, and the neutral point e ∈ X(R) such that they verify formally the axioms of a group (cf. [17, Definition 1.1]).
When the ring R is a field, we say that X is an algebraic group. Prominent examples of algebraic groups include linear algebraic groups and abelian varieties.
An R-scheme morphism ϕ : Here, m X and m Y denote respectively the group laws of X and Y . See the references [12] and [17] for more details.
2.2. Models of homomorphisms of algebraic groups. We shall need the following standard lemma in algebraic geometry: i ∈ I, be finitely many homomorphisms of k-algebraic groups. Then there exist a finitely generated Z-algebra R ⊂ k and Rgroup schemes of finite type X R , Y R and R-homomorphisms f i,R : (

2.3.
Closed points of Z-schemes of finite type. To perform the reduction to the finite alphabet case during the proof of Theorem A, we shall need the following auxiliary lemma to describe the set of closed points of families of schemes of finite type over Z.
2.4. Inverses of injective and bijective endomorphisms. In Section 4.1, we will need the following useful preliminary results in [19].

Lemma 2.3. Given a group G and an algebraic group X over an algebraically closed field
Theorem 2.4. Given a group G and an algebraic group X over an algebraically closed field k of characteristic zero. Suppose that τ ∈ CA algr (G, X, k) is bijective. Then one has τ −1 ∈ CA algr (G, X, k).

Locally weakly surjunctive groups
Given a property (P) of groups, we say that a group G is locally (P) if every finitely subgroup of G has property (P). In [24], the author proved that a group is surjunctive if and only if it is locally surjunctive.
The below lemma shows that the similar property also holds for weakly surjunctive groups.

Lemma 3.1. A group is locally weakly surjunctive if and only if it is weakly surjunctive.
Proof. Fix a group G and a finite group A. Let τ : A G → A G be an injective group cellular automaton. Let µ : A M → A be a local defining map of τ associated with some finite memory set M ⊂ G. Since τ is a group cellular automaton, it is clear that µ is a homomorphism of groups.
Let H ⊂ G be the subgroup generated by M . Let τ H : A H → A H be the cellular automaton which admits µ : A M → A as a local defining map. Since µ is a homomorphism of groups, we find that τ H is also a group cellular automaton. By the main result of [5], we know that τ is injective, resp. surjective, if and only if so is τ H .
Suppose first that G is locally weakly surjunctive. Then H is a weakly surjunctive group. Since τ is injective, so is τ H . Therefore, τ H is surjective as H is weakly surjunctive. Hence, τ is also surjective. This proves that G is weakly surjunctive.
The other implication is similar. Suppose that G is weakly surjunctive. Let S ⊂ G be a finitely generated subgroup. Let τ S : A S → A S be an injective group cellular automaton. Let µ : A N → A be a local defining map of τ H with a memory set N ⊂ H. Note that µ is a group homomorphism.
Consider the group cellular automaton τ : A G → A G which admits µ as a local defining map. Since τ S is injective, τ is also injective. But as G is weakly surjunctive, we deduce that τ is surjective and thus so is τ S . This shows that S is weakly surjunctive. The proof is complete. Theorem 4.1. Let G be an amenable group and let X be an algebraic group over an algebraically closed field k. Suppose that τ ∈ CA algr (G, X, k) is preinjective (e.g. when τ is injective). Then τ is also surjective.
It is known (cf. [19,Theorem 1.1]) that the conclusion of the above theorem also holds if the group G is sofic but X is connected. By further requiring the base field k to be uncountable, we can prove the following general surjunctivity result for endomorphisms of symbolic group varieties: Theorem 4.2. Let G be a weakly surjunctive group and let X be an algebraic group over an uncountable algebraically closed field k. Suppose that τ ∈ CA algr (G, X, k) is injective. Then τ is also surjective.
The proof of Theorem 4.2 will occupy the rest of Section 4.1. Fixing the notations and hypotheses as in Theorem 4.2, we denote by A = X(k) the set of k-points of X and Γ = τ , we can clearly suppose without loss of generality that X is a reduced scheme.
Let us fix an arbitrary symmetric memory set M ⊂ G of τ such that 1 G ∈ M and let µ : X M → X be the homomorphism of algebraic groups which serves as the corresponding local defining map of τ .
For every finite subset E ⊂ G, we define a homomorphism of algebraic groups τ + E : . Up to replacing G by the subgroup generated by M , we can suppose without loss of generality that G is finitely generated (by using the main result of [5] as in the proof of Lemma 3.1). In particular, we can find an increasing sequence of finite subsets For each n ∈ N, we consider the following subset V n of X EnM : Observe that V n is a constructible subset of X EnM . It suffices to write: For integers m ≥ n ≥ 0, we denote by p m,n : X EmM → X EnM the canonical projection map given by x → x| EnM . Then by the construction of the maps τ + En , it is clear that we have p m,n (V m ) ⊂ V n for all m ≥ n ≥ 0. Claim: There exists n ∈ N such that V n = ∅. Indeed, suppose on the contrary that V n = ∅ for every n ∈ N. Then (V n ) n∈N forms a projective system of nonempty constructible subsets of algebraic varieties (X EnM ) n∈N with algebraic transition maps induced by the morphisms p m,n with m ≥ n ≥ 0.
Since the base field k is uncountable and algebraically closed, we infer from [7, Lemma B.2] or [9, Lemma 3.2] that lim ← −n V n = ∅. Therefore, we can choose By definition of the sets V n , we deduce that c(1 G ) = e and τ (x) = e G . In particular, c = e G but τ (c) = τ (e G ) = e G , which contradicts the injectivity of τ . The claim is thus proved. Now let us fix n ≥ 0 such that V n = ∅. Let W = Ker τ + En and U = Ker π n then it follows from (4.2) that W ⊂ U is a closed algebraic subgroup.
By Lemma 2.1, we can find a finitely generated Z-algebra R ⊂ k and Rgroup schemes of finite type X R , W R , and U R , as well as R-homomorphisms of group schemes: such that the following hold: In the condition (ii), by saying T R is induced by µ R , we mean that T R is defined by the universal proprety of fibered product by the collection of R-homomorphism of group schemes (T En,g ) g∈En where the component R-homomorphism T En,g : (X R ) EnM → (X R ) {g} , for g ∈ E n , is given by the composition of the projection (X R ) EnM → (X R ) gM followed by the We are now in position to prove Theorem 4.2.
Proof of Theorem 4.2. Let us denote S = Spec R then S is a Z-scheme of finite type. Note that by Lemma 2.2, the set of closed points of X En R is given by ∆ = ∪ p∈P,s∈Sp,d∈N H En p,s,d where P is the set of prime numbers, s ∈ S p = S ⊗ Z F p is a closed point and is a finite subset of closed points of the fibre X s = X R ⊗ R κ(s). Now fix p ∈ P, a closed point s ∈ S p , and d ∈ N. Let µ s = µ R ⊗ R k then µ s : (X s ) M → X s is a κ(s)-homomorphism of algebraic groups.
Since . It follows that τ p,s,d is a group cellular automaton.
Claim: τ p,s,d is injective. Indeed, suppose that y ∈ T G p,s,d \ {e G s } satisfies τ p,s,d (y) = e G s where e s ∈ X s is the neutral element. By G-equivariance, we can assume that y(1 G ) = e s . In particular, if we let x = y| EnM and T s = T R ⊗ R κ(s) then T s (x) = e En s . Hence, x ∈ W s = Ker T s .
Since W s ⊂ U s = Ker π s where π s = π R ⊗ R κ(s), we can deduce that y(1 G ) = x(1 G ) = e s which is a contradiction. Therefore, τ s is injective and the claim is proved.
Claim: T R is surjective. Indeed, since G is weakly surjunctive by hypothesis and as τ p,s,d is a group cellular automaton with finite group alphabet, τ p,s,d must be surjective. In particular, we have for every p ∈ P, d ∈ N, and s ∈ S p closed that It follows that T R ((X R ) EnM ) contains the set ∆ = ∪ p∈P,s∈Sp,d∈N H M p,s,d of closed points of (X R ) En . Since (X R ) En is a Jacobson scheme (cf., e.g. [23, Section 3]), ∆ is dense in every locally closed subset (i.e., every intersection of a closed subset and an open subset) of (X R ) En .
On the other hand, Chevalley's theorem [13, Théorème 1.8.4] implies that T R ((X R ) EnM ) is a constructible subset of (X R ) En , namely, a finite union of locally closed subsets. We can thus conclude that T R ((X R ) EnM ) = (X R ) En . Therefore, T R is surjective and the claim is proved. Now as surjectivity is a stable property under base change, the morphism τ + En = T R ⊗ R k is also surjective. Hence, A En = τ + En (A EnM ) = Γ En . Since M ⊂ E n by our choice, we deduce that Γ M = A M . But M can be chosen to be arbitrarily large, we find that Γ E = A E for every finite subset E ⊂ G. Thus, Γ is dense in A G with respect to the prodiscrete topology. Since Γ is closed in A G by [19, Theorem 5.1], we conclude that Γ = A G and τ is therefore surjective. The proof of Theorem 4.2 is complete.

4.2.
Reversibility. We keep the notations and hypotheses as in Theorem A and in Section 4.1. Then Theorem 4.2 implies that Γ = τ (A G ) = A G . We begin with the following lemma: There exists a finite set N ⊂ G such that for every finite subset E ⊂ G containing N , we can find a map η E : A E → A such that for all x ∈ A EM , we have η E (τ + E (x)) = x(1 G ). Proof. Let N ⊂ G be the finite subset given by Lemma 2.3. Then for every d ∈ A G , the element τ −1 (d)(1 G ) depends only on d| N ⊂ A N . Therefore, for every finite subset E ⊂ G containing N , we have a well-defined map: wherez ∈ A G is any configuration extending z ∈ A E , i.e.,z| E = z. Now, let x ∈ A EM and z = τ + E (x). Choose an arbitraryx ∈ A G such that x| EM = x. It follows thatz = τ (x) extends the configuration z. Hence, we find without difficulties that: . The proof is thus complete.
We can now prove the first part of the conclusion of Theorem A. Proof. By Lemma 4.3 and by G-equivariance of the map τ , we find immediately that the inverse map τ −1 : A G → A G is a cellular automaton which admits η N : A N → A as a local defining map. Therefore, τ is reversible.

4.3.
Invertibility. We continue with the notations and hypotheses as in Theorem A. We have seen that τ is bijective by Theorem 4.2. Hence, it suffices to apply Theorem 2.4 to see that τ −1 ∈ CA algr (G, X, k) when the base field k has characteristic zero.
The proof of Theorem A is thus complete.

5.
Direct finiteness property of CA algr (G, X, k) The goal of this section is to explain two different proofs of Corollary A presented in the Introduction.
For the proof, we fix a weakly surjunctive group G and an algebraic group X over an algebraically closed field k. Suppose that we are given σ, τ ∈ CA algr (G, X, k) with σ • τ = Id. We are going to show that τ • σ = Id as well.
First proof of Corollary A. Since every homomorphism of algebraic groups is trivially a morphism of algebraic varieties, we find that CA algr (G, X, k) ⊂ CA alg (G, X, k) where CA alg (G, X, k) denotes the monoid of cellular automata π : X(k) G → X(k) G which admit for some finite subset M ⊂ G a local defining map ρ : X(k) M → X(k) induced by a morphism of algebraic varieties χ : X M → X, that is, ρ = χ| X(k) M .
Consequently, τ, σ ∈ CA alg (G, X, k) and satisfy σ • τ = Id. Then it suffices to apply the proof and notations of [23, Theorem C] to our τ and σ and observe that all the morphisms of algebraic varieties and morphisms of R-schemes involved in the proof can be taken to be homomorphisms of algebraic groups and homomorphisms of R-group schemes by using Lemma 2.1.
Hence, with the notations in the proof of [23, Theorem C], the induced injective cellular automata τ p,s,d : T G p,s,d → T G p,s,d become injective group cellular automata. But since G is weakly surjunctive, the maps τ p,s,d are surjective and the rest of the proof of [23,Theorem C] shows us that τ • σ = Id. The proof of Corollary A is thus complete.
Second proof of Corollary A. Since we only work with A = X(k), we can suppose without loss of generality that X is reduced so that we can identify X with X(k). Let M ⊂ G be a large enough common symmetric memory set of both σ and τ such that 1 G ∈ M . Let µ, η : X M → X be the kmorphisms of algebraic varieties which induce the local defining maps of τ and σ respectively.
Let us define a k-homomorphism of algebraic groups τ + M : X M 2 → X M defined by τ + M (c)(g) = µ((g −1 c)| M ) for c ∈ A M 2 and g ∈ M . Since σ•τ = Id, it follows immediately that We embed k into some uncountable algebraically closed field K. By making the base change to K, we obtain the morphisms µ K = µ ⊗ k K and η K = η ⊗ k K and a K-algebraic group X K = X ⊗ k K. It is trivial that is the canonical projection to the factor X {1 G } . Consider the K-homomorphism of algebraic groups τ + K,M = τ + M ⊗ k K. We infer from (5.1) that η K •τ + K,M = π K . Denote by τ K , σ K ∈ CA algr (G, X K , K) the cellular automata admitting respectively µ K and η K as local defining maps. Then the relation η K • τ + K,M = π K implies that σ K • τ K = Id. Therefore, τ K is injective and Theorem A implies that τ K is bijective. Consequently, τ K • σ K = Id.
Consider the k-homomorphism of algebraic groups σ + M : Since the field k is algebraically closed, it follows from e.g. [7, Lemma 7.2] that µ • σ + M = π and we can finally conclude that τ • σ = Id. The proof of Corollary A is thus complete.

Linearly surjunctive groups and further applications
Given a group G and a vector space A over a field k. Then A G is naturally a vector space over k with pointwise operations. A cellular automaton τ : A G → A G is said to be linear if it is also a k-linear map of vector spaces.
Equivalently, we see that a cellular automaton is linear if it admits a linear local defining map.
From our main results obtained in the above sections, it is natural to define the following general class of linearly surjunctive groups. Definition 6.1. We say that a group G is a linearly surjunctive group if for every finite dimensional vector space A over a finite field k, all injective linear cellular automata τ : A G → A G are surjective.
Hence, it follows from the definitions and the Gromov-Weiss theorem ( [11], [24]) which says that all sofic groups are surjunctive, we have the following relations between various classes of groups: It is not hard to see using a similar proof of Lemma 3.1 that a group is linearly surjunctive if and only if it is locally linearly surjunctive.
By working in the smaller category of finitely generated modules over a finitely generated Z-algebra instead of the category of group schemes, we find that a straightforward modification of the proof of Theorem A shows the following result: Theorem 6.2. Let G be a linearly surjunctive group and let A be a finite dimensional vector space over an uncountable algebraically closed field k. Suppose that a linear cellular automaton τ : A G → A G is injective. Then τ is reversible and τ −1 : A G → A G is also a linear cellular automaton.
Proof. It suffices to follow, mutatis mutandis, the same proof of Theorem A in Section 4 to see that τ is reversible. We remark that the various R-group schemes and homomorphisms between them are now simply R-modules of finite type and homomorphisms of R-modules. For the second statement of the conclusion, observe that the inverse cellular automaton τ −1 : A G → A G is automatically linear as the inverse of a linear map.
As an immediate consequence, we obtain: Theorem 6.3. Kaplansky's stable finiteness theorem holds for every field k and every linearly surjunctive group G.
In [4,Definition 1.1], the authors introduced the notion of L-surjunctive groups: a group G is L-surjunctive if for every finite dimensional vector space V over a field k (not necessarily finite), all injective linear cellular automata τ : V G → V G are surjective. They showed in [4] that a group G is L-surjunctive if and only if for every field k, the group ring k[G] is stably finite. In other words, a group satisfies Kaplansky's stable finiteness conjecture for every field if and only if it is L-surjunctive. By definition, observe that every L-surjunctive group is linearly surjunctive.
As a direct application of Theorem 6.3, we deduce from the above characterization of L-surjuntive groups that:

Weakly dual surjunctive groups
Fix a group G and a set A. Two configurations x, y ∈ A G are said to be asymptotic if there exists a finite subset E ⊂ G such that x| G\E = y G\E . Following [3], a cellular automaton τ : A G → A G is said to be post-surjective if for every x, y ∈ A G such that y is asymptotic to τ (x), we can find z ∈ A G asymptotic to x such that τ (z) = y. It was shown in [3, Theorem 2] that for every sofic group G and every finite set A, every post-surjective cellular automaton τ : Following [3], we will say that a group G is dual surjunctive if for every finite alphabet A, all post-surjective cellular automaton τ : A G → A G are pre-injective. As for weakly surjunctive groups, we introduce the following notions of weakly and linearly dual surjunctive groups which include all dual surjunctive groups and hence all sofic groups.
Definition 7.1. Let G be a group. We say that G is weakly dual surjunctive if for every finite group A, every post-surjective group cellular automaton A G → A G is pre-injective. Similarly, G is linearly dual surjunctive if for every finite dimensional vector space A over a finite field, all linear postsurjective cellular automata A G → A G must be pre-injective.
Note that by [3, Theorem 1], every pre-injective, post-surjective cellular automaton over a finite alphabet must be bijective and moreover reversible.
We begin with the following basic observation: (iv) G satisfies Kaplansky's stable finiteness conjecture, that is, for every field k, the group ring k[G] is stably finite.
We also obtain the following direct finiteness property of endomorphisms of symbolic group varieties over weakly dual surjunctive group universes: Theorem 7.4. Let G be a weakly dual surjunctive group. Let X be an algebraic group over an algebraically closed field K. Suppose that τ, σ ∈ CA algr (G, X, K) satisfy σ • τ = Id. Then one has τ • σ = Id.
As a direct application, we obtain: Corollary 7.5. Let G be a weakly dual surjunctive group and let R be the endomorphism ring of a commutative algebraic group over an algebraically closed field. Then the group ring R[G] is stably finite.
Proof. It suffices to follow the exact same proof of Corollary B. The only modification needed in the proof is that instead of using Corollary A, we apply Theorem 7.4.
For the proof of Theorem 7.4, we shall establish first the following elementary result.
Lemma 7.6. Let G and A be groups and let τ, σ : A G → A G be group cellular automata such that σ • τ = Id. Then σ is post-surjective.
The proof is thus complete.
We are now in the position to give the proof of Theorem 7.4.
Proof of Theorem 7.4. As for the first proof of Corollary A, we proceed by applying the constructions and notations introduced in the proof of [23, Theorem C] to our τ and σ. Again, observe that the various morphisms of algebraic varieties and morphisms of R-schemes involved are now respectively homomorphisms of algebraic groups and homomorphisms of R-group schemes by using Lemma 2.1.
Let p be a prime, let s ∈ A s be a closed point, and let d ∈ N. Then as for τ p,s,d : T G p,s,d → T G p,s,d , we also have a well-defined group cellular automaton σ p,s,d = σ s | T G Since σ s • τ s = Id, we deduce that σ p,s,d • τ p,s,d = Id. We infer from Lemma 7.6 that σ p,s,d is a post-surjective group cellular automata. But as the group G is weakly dual surjunctive by hypothesis, σ p,s,d and τ p,s,d are mutual inverses by [3,Theorem 1]. It follows that τ p,s,d • σ p,s,d = Id. Therefore, we deduce immediately that τ is surjective and again, the rest of the proof of [23, Theorem C] allows us to conclude that τ • σ = Id.
The proof of Theorem 7.4 is thus complete.