Expansive actions on uniform spaces and surjunctive maps

We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group actions and discuss several applications. We prove in particular that for any group $\Gamma$ and any field $\K$, the space of $\Gamma$-marked groups $G$ such that the group algebra $\K[G]$ is stably finite is compact.


Introduction
A map f from a set X into itself is said to be surjunctive if it is surjective or not injective [Gro]. Thus, a non-surjunctive map is a map which is injective but not surjective. With this terminology at hand, Dedekind's characterization of infinite sets may be rephrased by saying that a set X is infinite if and only if it admits a non-surjunctive map f : X → X. Similarly, elementary linear algebra tells us that a vector space is infinite-dimensional if and only if it admits a non-surjunctive endomorphism. In fact, it turns out that the absence of non-surjunctive endomorphisms for a given mathematical object X often reflects some "finiteness" property of X.
The word surjunctive was created by W. Gottschalk [Got] who introduced the notion of a surjunctive group. Let G be a group. Given a set A, consider the set A G equipped with the prodiscrete topology, that is, with the product topology obtained by taking the discrete topology on each factor A of A G . There is a natural action of G on A G obtained by shifting coordinates via left multiplication in G (see Section 5). This action is called the G-shift and the study of its dynamical properties is the central theme of symbolic dynamics. The group G is said to be surjunctive if, for any finite set A, every G-equivariant continuous map τ : A G → A G is surjunctive. Gottschalk asked whether every group G is surjunctive. Although there is a large class of groups, namely sofic groups (see [Weis], [ES2]), which are known to be surjunctive, this question remains open. Here, it is worth mentioning that the existence of a non-sofic group is also an open problem.
In algebraic geometry, there is a famous theorem of J. Ax [Ax] which says that every endomorphism of a complex algebraic variety is surjunctive. Ax' surjunctivity theorem was extended by M. Gromov [Gro] to certain classes of "symbolic algebraic spaces" obtained by taking projective limits of algebraic varieties. In order to establish these generalizations, Gromov developed various techniques for proving surjunctivity of maps f : X → X obtained as limits of surjunctive maps f i : X i → X i . He also used one of these techniques to prove the closedness of the set of marked surjunctive groups (see [Gro] and [GG] for an alternative proof based on model theory). The purpose of our paper is to give a detailed exposition of this technique in a somewhat more general setting that the one considered in [Gro]. To be a little bit more precise, we extend results on the surjunctivity of maps commuting with expansive actions on metric spaces by replacing metric spaces by uniform spaces. It turns out that this extension is quite natural and we think it may help to a better understanding of the roles of the various mathematical concepts involved in Gromov's approach.
Uniform spaces were introduced by A. Weil [Wei] as a generalization of metric spaces and topological groups. In a uniform space X, the closeness of a pair of points is not measured by a real number, like in a metric space, but by the fact that this pair of points belong or does not belong to certain subsets of the Cartesian square X × X which are called the entourages of the uniform structure (see Section 2 for precise definitions). Every metric on a set X defines a uniform structure on X and every uniform structure defines a topology on the underlying set. In the case when the uniform structure comes from a metric, the associated topology coincides with the topology defined by the metric.
Uniformly continuous maps between uniform spaces are defined as being the maps which pullback entourages to entourages.
There is a natural uniform structure on the set P(X) of subsets of a uniform space X, which is called the Hausdorff-Bourbaki uniform structure. Its name comes from the fact that it generalizes the Hausdorff distance on the set of bounded closed subsets of a metric space and was first mentioned in exercises in the treatise of N. Bourbaki [Bou].
The classical definition of expansivity for groups actions on metric spaces may be extended to uniform spaces. More precisely, an action of a group Γ on a uniform space X is said to be expansive if there is an entourage V 0 such that there is no pair of distinct points whose orbit under the diagonal action of Γ on X × X remains inside V 0 .
One of our main results is the following theorem whose proof will be given in Section 4: Theorem 1.1. Let X be a uniform space equipped with a uniformly continuous and expansive action of a group Γ and let f : X → X be a uniformly continuous and Γ-equivariant map. Let Y be a compact subset of X. Suppose that there exists a net (Z i ) i∈I of Γ-invariant subsets of X, which converges to Y in the Hausdorff-Bourbaki topology such As an immediate consequence of the preceding theorem, we get Corollary 1.2. Let X be a uniform space equipped with a uniformly continuous and expansive action of a group Γ and let f : X → X be a uniformly continuous and Γ-equivariant Then Σ(f ) is closed in the set of all subsets of X for the Hausdorff-Bourbaki topology.
Consider now a group G and an arbitrary set A. There is a natural uniform structure on A G which is called the prodiscrete uniform structure (see Subsection 2.3). The associated topology is the prodiscrete topology on A G and the shift action is expansive and uniformly continuous. Note that if G is uncountable then the prodiscrete topology on A G is not metrizable as soon as A has at least two elements. The uniformly continuous G-equivariant maps τ : A G → A G are precisely the cellular automata over the group G and the alphabet A (see Section 5).
Let Γ be a group. The set of quotients of Γ may be identified with the set N (Γ) of normal subgroups of Γ. The set N (Γ) is called the set of Γ-marked groups. It is a closed (and hence compact) subset of the compact space P(Γ) = {0, 1} Γ for the prodiscrete topology.
By applying Theorem 1.1 to the shift on A Γ with A finite, we deduce in Section 6 the above mentioned closedness theorem for surjunctive groups: Corollary 1.3 (Gromov). Let Γ be a group. Then the set of normal subgroups N ⊂ Γ such that the quotient group Γ/N is surjunctive is closed (and hence compact) in N (Γ).
Let now K be a field. A group G is said to be L K -surjunctive if, for any finite-dimensional vector space V over K, every linear cellular automaton τ : V G → V G is surjunctive. It is known that every sofic group is L K -surjunctive (see [CC2]). In Section 7, we shall establish the closedness of the set of L K -surjunctive marked groups: Theorem 1.4. Let Γ be a group and let K be a field. Then the set of normal subgroups N ⊂ Γ such that the quotient group Γ/N is L Ksurjunctive is closed (and hence compact) in N (Γ).
The proof of the preceding theorem uses a stronger version of Theorem 1.1 (see Corollary 4.2), where the compactness hypothesis for Y is replaced by the weaker hypotheses that both Y and f (Y ) are closed in X together with the fact that the restriction of f to Y is a uniform embedding.
One of the famous conjectures of I. Kaplansky [Kap] about the algebraic properties of group algebras is that K[G] is stably finite for any group G and any field K (we recall that a ring R is said to be stably finite if every one-sided invertible square matrix over R is also twosided invertible). This conjecture remains open up to now although it was settled by Kaplansky himself for all fields of characteristic 0 and later by Elek and Szabó [ES1] (see also [CC2]) for sofic groups in any characteristic. In [CC2], it is shown that a group G is L K -surjunctive if and only if the group algebra K[G] is stably finite. We thus have: Corollary 1.5. Let Γ be a group and let K be a field. Then the set of normal subgroups N ⊂ Γ such that the group algebra K[Γ/N ] is stably finite is closed (and hence compact) in N (Γ).
The paper is organized as follows. Section 2 contains background material on uniform spaces. The definition of the Hausdorff-Bourbaki uniform structure on the set of subsets of a uniform space is recalled in Section 3. For the convenience of the reader, proofs of some well known facts about the Hausdorff-Bourbaki uniform structure that are needed in the sequel are also included in this section. In Section 4 we prove a uniform version of Lemma 4.H" in [Gro] which is used to prove Theorem 1.1. Basic properties of cellular automata are presented in Section 5. Section 6 is devoted to marked groups. We show that if G = Γ/N is a Γ-marked group, then the set A G is canonically isomorphic, as a uniform G-space, to the subset Fix(N ) ⊂ A Γ consisting of the points of A Γ which are fixed by the normal subgroup N ⊂ Γ. We also prove that the map N → Fix(N ) yields a uniform embedding of N (Γ), the space of Γ-marked groups, into P(A Γ ). This gives a symbolic representation of the space of Γ-marked groups which is used, in combination with Theorem 1.1 (resp. Corollary 1.2), in the proof of Corollary 1.3 (resp. Theorem 1.4) which is completed in Section 6 (resp. Section 7).

Uniform structures
In this section we collect some basic facts about uniform spaces, uniformly continuous maps, and uniformly continuous group actions. More details and proofs may be found for example in [Bou], [Jam], or [Kel].
Throughout the paper, we shall use the following general notation. Let X be a set. We denote by P(X) the set of all subsets of X and by ∆ X the diagonal in X × X, that is, ∆ X = {(x, x) : x ∈ X}. Let R be a binary relation on X, that is, a subset R ⊂ X × X. Given an element y ∈ X, we define the subset R[y] ⊂ X by R[y] = {x ∈ X : (x, y) ∈ R}.
The composite of two binary relations R and S is the binary relation R•S = {(x, y) : there exists z ∈ X such that (x, z) ∈ R and (z, y) ∈ S} ⊂ X×X.

Uniform spaces.
Definition. Let X be a set. A uniform structure on X is a non-empty set U of subsets of X × X satisfying the following conditions: A set X equipped with a uniform structure U is called a uniform space and the elements of U are called the entourages of X. Let It is easy to see that a base B satisfies the following properties: Conversely, given a set X and a subset B ⊂ P(X × X) satisfying conditions (B-1) -(B-4), there exists a unique uniform structure U on X admitting B as a base.
There is a topology on X which is associated with the uniform structure. This is the topology for which the neighborhoods of an arbitrary point x ∈ X consist of the sets V [x], where V runs over all entourages of X. This topology is Hausdorff if and only if the intersection of the entourages of X coincides with the diagonal ∆ X ⊂ X × X.
Examples. 1) The discrete uniform structure on a set X is the uniform structure whose entourages consist of all subsets of X × X containing ∆ X . The topology associated with this uniform structure is the discrete topology on X.
2) If (X, d) is a metric space, then there is a uniform structure on X associated with the metric. A base for this uniform structure is given by the sets V ε = {(x, y) ∈ X × X : d(x, y) < ε}, ε > 0. The topology associated with this uniform structure coincides with the topology associated with the metric.
2.2. Uniformly continuous maps. Let X and Y be uniform spaces.
A map f : X → Y is said to be uniformly continuous if it satisfies the following condition: for each entourage W of Y , there exists an Every uniformly continuous map f : X → Y is continuous but the converse fails to hold in general.
A uniform isomorphism between uniform spaces X and Y is a bijective map f : X → Y such that both f and f −1 are uniformly continuous. One says that a map f : X → Y is a uniform embedding if f is injective and induces a uniform isomorphism between X and f (X).
2.3. Product of uniform spaces. Suppose that (X λ ) λ∈Λ is a family of uniform spaces. Then the smallest uniform structure on the Cartesian product X = λ∈Λ X λ for which all projection maps π λ : X → X λ , λ ∈ Λ, are uniformly continuous is called the product uniform structure on X. A base of entourages for the product uniform structure on X is obtained by taking all subsets of X × X which are of the form If Y is a uniform space, then a map f : Y → X is uniformly continuous if and only if the maps π λ • f : Y → X λ are uniformly continuous for all λ ∈ Λ.
In the particular case when each X λ is endowed with the discrete uniform structure, the product uniform structure on X = λ∈Λ X λ is called the prodiscrete uniform structure on X.

Uniformly continuous and expansive actions.
Suppose that X is a uniform space equipped with an action of a group Γ.
One says that the action of Γ on X is uniformly continuous if the map f γ : X → X defined by f γ (x) = γx is uniformly continuous for each γ ∈ Γ.
Consider the diagonal action of Γ on X × X defined by γ(x, y) = (γx, γy) for all x, y ∈ X and γ ∈ Γ. One says that the action of Γ on X is expansive if there exists an entourage W 0 of X satisfying the following property: for any two distinct points x, y ∈ X, there exists an element γ ∈ Γ such that γ(x, y) / ∈ W 0 . In other words, the action of Γ on X is expansive if and only if there exists an entourage W 0 such that Note that if X admits a uniformly continuous and expansive action then the topology on X is necessarily Hausdorff.

The Hausdorff-Bourbaki uniform structure
In this section we briefly review the definition and basic properties of the Hausdorff-Bourbaki uniform structure on the set of subsets of a uniform space (see [Bou,ch. II exerc. 5 p. 34 and exerc. 6 p. 36]). Some proofs of well known facts that will be used in the next section are included for the convenience of the reader.
Let X be a uniform space with uniform structure U. For V ∈ U, we define the subset V ⊂ P(X) × P(X) by One easily checks that the set B = { V : V ∈ U} ⊂ P(P(X) × P(X)) satisfies conditions (B-1) -(B-4). Therefore, there exists a unique uniform structure on P(X) admitting B as a base. This uniform structure is called the Hausdorff-Bourbaki uniform structure associated with U. The topology associated with the Hausdorff-Bourbaki uniform structure is called the Hausdorff-Bourbaki topology on P(X). If (Y i ) i∈I is a net in P(X) (that is, a family of subsets of X indexed by some directed set I) and Z ∈ P(X), we shall write Y i H−B → Z to mean that the net (Y i ) i∈I converges to Z in the Hausdorff-Bourbaki topology.
Proposition 3.1. Let X be a uniform space and let Y and Z be closed Proof. Let y ∈ Y and let Ω be a neighborhood of y in X. Then there is a ⊂ Ω. This shows that y is in the closure of Z. Since Z is closed in X, we deduce that Y ⊂ Z. By symmetry, we also have Z ⊂ Y . Consequently, Y = Z.
Remarks. 1) An immediate consequence of Proposition 3.1 is that, if X is a uniform space, then the topology induced by the Hausdorff-Bourbaki topology on the set of closed subsets of X is Hausdorff.
2) Suppose that (X, d) is a metric space and let C b (X) denote the subset of P(X) consisting of all closed bounded subsets of X. For x ∈ X and r > 0, denote by B(x, r) the open ball of radius r centered at x. The Hausdorff metric is the metric δ on C b (X) defined by The uniform structure on C b (X) associated with the metric δ is the uniform structure induced by the Hausdorff-Bourbaki structure on P(X).
Proposition 3.2. Let X be a uniform space and let us equip P(X) with the Hausdorff-Bourbaki uniform structure and P(X) × P(X) with the corresponding product uniform structure. Then the map η : P(X) × P(X) → P(X) defined by η(Y, Z) = Y ∪ Z is uniformly continuous.
Proof. Let V be an entourage of X and consider the entourage V of P(X) given by ( This shows that η is uniformly continuous. Corollary 3.3. Let X be a uniform space. Let (Y i ) i∈I and (Z i ) i∈I be nets in P(X) such that Y i ⊂ Z i for all i ∈ I. Suppose that there exist closed subsets A and B of X such that Y i Proposition 3.4. Let X and Y be uniform spaces and let f : X → Y be a uniformly continuous map. Then the map f * : P(X) → P(Y ) which sends each subset A ⊂ X to its image f (A) ⊂ Y is uniformly continuous with respect to the Hausdorff-Bourbaki uniform structures on P(X) and P(Y ).
Proof. Let W be an entourage of Y and let be the associated entourage of P(Y ). Since f is uniformly continuous, there is an entourage V of X such that If a 1 ∈ A 1 , then there exists a 2 ∈ A 2 such that (a 1 , a 2 ) ∈ V and hence (f (a 1 ), f (a 2 )) ∈ W . Therefore, we have f Corollary 3.5. Let X be a uniform space and let f : X → X be a uniformly continuous map. Let Y be a subset of X such that the sets Y and f (Y ) are both closed in X. Suppose that there is a net In particular, we have: Corollary 3.6. Let X be a uniform space equipped with a uniformly continuous action of a group Γ. Let Y be a closed subset of X. Suppose that there is a net (Z i ) i∈I of Γ-invariant subsets of X such that Z i

Gromov's injectivity lemma
The following result is a uniform version of Lemma 4.H" in [Gro].
Theorem 4.1 (Gromov's injectivity lemma). Let X be a uniform space equipped with a uniformly continuous and expansive action of a group Γ and let f : X → X be a uniformly continuous and Γ-equivariant map. Suppose that Y is a subset of X such that the restriction of f to Y is a uniform embedding. Then there exists an entourage V of X satisfying the following property: if Z is a Γ-invariant subset of X such that Z ⊂ V [Y ], then the restriction of f to Z is injective.
Proof. By expansivity of the action of Γ on X, there exists an entourage W 0 of X such that It follows from the axioms of a uniform structure that we can find a symmetric entourage S of X such that Since the restriction of f to Y is a uniform embedding, there exists an entourage T of X such that (4.3) (f (y 1 ), f (y 2 )) ∈ T ⇒ (y 1 , y 2 ) ∈ S for all y 1 , y 2 ∈ Y . Let U be a symmetric entourage of X such that As f is uniformly continuous, we can find an entourage E of X such that for all x 1 , x 2 ∈ X. Let us show that the entourage V = S ∩E has the required property. Suppose that Z is a Γ-invariant subset of X such that Z ⊂ V [Y ]. Let us show that the restriction of f to Z is injective.
Corollary 4.2. Let X be a uniform space equipped with a uniformly continuous and expansive action of a group Γ and let f : X → X be a uniformly continuous and Γ-equivariant map. Let Y be a subset of X such that Y and f (Y ) are both closed in X and the restriction of f to Y is a uniform embedding. Suppose that there exists a net (Z i ) i∈I of Γ-invariant subsets of X such that f (Z i ) ⊂ Z i and the restriction maps f | Z i : Z i → Z i are surjunctive for all i ∈ I and Z i Proof. The Γ-invariance of Y directly follows from Corollary 3.6. Let V be an entourage of X as in Theorem 4.1.
As Y and f (Y ) are closed in X, we conclude that f (Y ) = Y by applying Proposition 3.1.
Proof of Theorem 1.1. First observe that the space X is Hausdorff since it admits an expansive uniformly continuous action. It follows that both Y and f (Y ) are closed in X by compactness of Y .
The fact that Y is Γ-invariant and satisfies f (Y ) ⊂ Y is a consequence of Corollary 3.6 and Corollary 3.5.
Suppose now that f | Y is injective. As Y is compact, we deduce that the restriction of f is a uniform embedding. We conclude that f | Y is surjective by applying Corollary 4.2.

Cellular automata
Let G be a group and let A be an arbitrary set that we shall call the alphabet. We consider the set A G consisting of all maps x : G → A. The elements of A G are called configurations over the group G and the alphabet A.
Given a subset S ⊂ G, we denote by π S : A G → A S the projection map.
We equip the set A G = g∈G A with its prodiscrete uniform structure (cf. Subsection 2.3) and the left action of G defined by gx(h) = x(g −1 h) for all g, h ∈ G and x ∈ A G . This action is called the G-shift on A G .
Proposition 5.1. The G-shift on A G is uniformly continuous and expansive.
Proof. Uniform continuity follows from the fact that G acts on A G by permuting coordinates. Expansiveness is due to the fact that the action of G on itself by left multiplication is transitive. Indeed, consider the entourage W 0 of A G defined by Given g ∈ G, we have (x, y) ∈ g −1 W 0 if and only if x(g −1 ) = y(g −1 ).
The space A G with the G-shift action is called the full shift on G and A. A G-invariant subset X ⊂ A G is called a subshift (note that here we do not require X to be closed in A G ).
Definition. Let X, Y ⊂ A G be two subshifts. A cellular automaton from X to Y is a map τ : X → Y satisfying the following property: there exist a finite subset S ⊂ G and a map µ : π S (X) → A such that (5.1) τ (x)(g) = µ(π S (g −1 x)) for all x ∈ X and g ∈ G.
Such a set S is then called a memory set and µ is called a local defining map for τ .
The following statement is proved in [CC3] in the particular case when X = Y = A G : Proposition 5.2. Let G be a group and let A be a set. Let X, Y ⊂ A G be two subshifts and let τ : X → Y be a map. Then the following conditions are equivalent: (a) τ is a cellular automaton; (b) τ is uniformly continuous and G-equivariant; (c) There exists a cellular automaton τ : Proof. Suppose that τ is a cellular automaton with memory set S and local defining map µ : π S (X) → A. Let µ : A S → A be a map extending µ. Then the map τ : A G → A G defined by τ (x)(g) = µ(π S (g −1 x)) for all x ∈ A G and g ∈ G is a cellular automaton over A G such that τ (X) ⊂ Y and τ (x) = τ (x) for all x ∈ X. This shows that (a) implies (c). It is shown in [CC3,Th. 1.1] that every cellular automaton τ : A G → A G is G-equivariant and uniformly continuous. Therefore (c) implies (b).
Suppose that τ : X → Y is G-equivariant and uniformly continuous. Since the map f : X → A defined by f (x) = τ (x)(1 G ) is uniformly continuous, this implies that there is a finite subset S ⊂ G such that if two elements x, y ∈ X coincide on S then f (x) = f (y). In other words, there is a map µ : π S (X) → A such that τ (x)(1 G ) = µ(π S (x)) for all x ∈ X. Now the G-equivariance of τ gives us for all x ∈ X and g ∈ G. This shows that (b) implies (a).
If X, Y ⊂ A G are two subshifts, we shall denote by CA(X, Y ; G, A) the set consisting of all cellular automata τ : X → Y . If Y = X we shall simply write CA(X; G, A) instead of CA(X, X; G, A). Note that it immediately follows from the equivalence of conditions (a) and (b) in Proposition 5.2 that CA(X; G, A) is a monoid for the composition of maps.

Marked groups
Let Γ be a group. A Γ-quotient is a pair (G, ρ), where G is a group and ρ : Γ → G is a group epimorphism. Two Γ-quotients (G 1 , ρ 1 ) and (G 2 , ρ 2 ) are said to be equivalent if there exists a group isomorphism φ : G 2 → G 1 such that the following diagram is commutative: that is, such that ρ 1 = φ • ρ 2 . An equivalence class of Γ-quotients is called a Γ-marked group. Observe that two Γ-quotients (G 1 , ρ 1 ) and (G 2 , ρ 2 ) are equivalent if and only if Ker(ρ 1 ) = Ker(ρ 2 ). Thus the set of Γ-marked groups may be identified with the set N (Γ) consisting of all normal subgroups of Γ.
Let us equip the set P(Γ) = {0, 1} Γ with its prodiscrete uniform structure and N (Γ) with the induced uniform structure. A base of entourages of N (Γ) is provided by the sets where F runs over all finite subsets of Γ. Intuitively, two normal subgroups of Γ are "close" in N (Γ) when their intersection with a large finite subset of Γ coincide.
The space P(Γ) is Hausdorff and totally disconnected since it is a product of discrete spaces. Moreover, P(Γ) is compact by the Tychonoff product theorem. The set N (Γ) is closed in P(Γ) and therefore N (Γ) is a totally disconnected compact Hausdorff space (see [CHA], [CGP] and the references therein).
Let A be a set. Consider the set A Γ equipped with its prodiscrete uniform structure and the Γ-shift action. For each N ∈ N (Γ), let denote the set of points in A Γ which are fixed by N . Observe that Fix(N ) is a closed subshift of A Γ . Since N acts trivially on Fix(N ), the Γ-shift on A Γ induces an action of the quotient group G = Γ/N on Fix(N ).
Proposition 6.1. Let N be a normal subgroup of a group Γ and let ρ : Γ → G = Γ/N denote the canonical epimorphism. Then y • ρ ∈ Fix(N ) for every y ∈ A G . Moreover, the map ρ * : A G → Fix(N ) defined by ρ * (y) = y • ρ is a G-equivariant uniform isomorphism.
Proof. An element x ∈ A Γ is in Fix(N ) if and only if x(ν −1 γ) = x(γ) for all γ ∈ Γ and ν ∈ N , that is, if and only if the configuration x is constant on each coset of Γ modulo N . This proves the first assertion and the fact that the map ρ * is surjective. The injectivity of ρ * follows from the surjectivity of ρ. Let g ∈ G and choose γ ∈ Γ such that ρ(γ) = g. For all y ∈ A G and α ∈ Γ, we have This shows that g(y • ρ) = (gy) • ρ, that is, gρ * (y) = ρ * (gy). Thus ρ * is G-equivariant.
By using the characterization of cellular automata provided by the equivalence of conditions (a) and (b) in Proposition 5.2, we immediately deduce the following: Corollary 6.2. If τ : A G → A G is a cellular automaton over A G , then the map τ * : Fix(N ) → Fix(N ) given by τ * = ρ * • τ • (ρ * ) −1 is a cellular automaton over the subshift Fix(N ) ⊂ A Γ . Moreover, the map Θ : CA(A G , G, A) → CA(Fix(N ), Γ, A) defined by Θ(τ ) = τ * is a monoid isomorphism.
Let us equip P(A Γ ) with the Hausdorff-Bourbaki uniform structure associated with the prodiscrete uniform structure on A Γ . Theorem 6.3. Let Γ be a group and let A be a set. Then the map Ψ : N (Γ) → P(A Γ ) defined by Ψ(N ) = Fix(N ) is uniformly continuous. Moreover, if A contains at least two elements then Ψ is a uniform embedding.
Proof. Let N 0 ∈ N (Γ) and let W be an entourage of P(A Γ ). Let us show that there exists an entourage V of N (Γ) such that This will prove that Ψ is continuous. By definition of the Hausdorff-Bourbaki uniform structure on P(A Γ ), there is an entourage T of A Γ such that (6.2) Since A Γ is endowed with its prodiscrete uniform structure, there is a finite subset F ⊂ Γ such that where π F : A Γ → A F is the projection map. Consider now the finite subset E ⊂ Γ defined by and the entourage V of N (Γ) given by We claim that V satisfies (6.1). To prove our claim, suppose that N ∈ V [N 0 ]. Let x ∈ Fix(N ). The fact that N ∩ E = N 0 ∩ E implies that if γ and η are elements of F with γ = νη for some ν ∈ N , then ν ∈ N 0 and therefore x(γ) = x(η). Denoting by ρ 0 : Γ → Γ/N 0 the canonical epimorphism, we deduce that we may find an element . Therefore Ψ is continuous and hence uniformly continuous by compactness of N (Γ). Suppose now that A has at least two elements. Let us show that Ψ is injective. Let N 1 , N 2 ∈ N (Γ). Fix two elements a, b ∈ A with a = b and consider the map x : Γ → A defined by x(γ) = a if γ ∈ N 1 and x(γ) = b otherwise. We clearly have x ∈ Fix(N 1 ). Suppose that Ψ(N 1 ) = Ψ(N 2 ), that is, Fix(N 1 ) = Fix(N 2 ). Then for all ν ∈ N 2 , we have ν −1 x = x since x ∈ Fix(N 1 ) = Fix(N 2 ), and hence x(ν) = ν −1 x(1 Γ ) = x(1 Γ ) = a. This implies N 2 ⊂ N 1 . By symmetry, we also have N 1 ⊂ N 2 . Therefore N 1 = N 2 . This shows that Ψ is injective. As N (Γ) is compact and P(A Γ ) is Hausdorff, we conclude that Ψ is a uniform embedding.
Proof of Corollary 1.3. Let N ∈ N (Γ) and let (N i ) i∈I be a net in N (Γ) converging to N . Suppose that the groups G i = Γ/N i are surjunctive for all i ∈ I and let us show that the group G = Γ/N is surjunctive as well.
Let A be a finite set and let τ : A G → A G be a cellular automaton over A G . Let ρ : Γ → G denote the canonical epimorphism and let ρ * : A G → Fix(N ) be the map defined by ρ * = y • ρ for all y ∈ A G (cf Proposition 6.1). By Corollary 6.2, the map τ * : Fix(N ) → Fix(N ) given by τ * = ρ * • τ • (ρ * ) −1 is a cellular automaton over the subshift Fix(N ) ⊂ A Γ . By applying Proposition 5.2, we deduce the existence of a cellular automaton τ : A Γ → A Γ such that τ (Fix(N )) ⊂ Fix(N ) and τ | Fix(N ) = τ * .
We claim that the hypotheses of Theorem 1.1 are satisfied by taking X = A Γ , f = τ , Y = Fix(N ), and Z i = Fix(N i ). First observe that the space X is compact by the Tychonoff product theorem since A is finite.
As Y is closed in X, it follows that Y is compact. Moreover, the action of Γ on X is uniformly continuous and expansive by Proposition 5.1. On the other hand, the cellular automaton f : X → X is uniformly continuous and Γ-equivariant by Proposition 5.2. We have f (Z i ) ⊂ Z i since the cellular automaton f is Γ-equivariant. Moreover, Z i = Fix(N i ) is Γinvariant by normality of the subgroup N i ⊂ Γ. Also, by Theorem 6.3, be the uniform isomorphism given by Proposition 6.1. It follows from Corollary 6.2 that the map is a cellular automaton and hence is surjunctive since G i is a surjunctive group. This implies that f | Z i = τ | Fix(N i ) is surjunctive as well. By applying Theorem 1.1, we deduce that f | Y = τ * and hence τ = (ρ * ) −1 • τ * • ρ * is surjunctive. Consequently, the group G is surjunctive.

Linear cellular automata
Let G be a group and let V be a vector space over a field K. The set V G has a natural vector space structure and the G-shift action on V G is linear (here and in the sequel "linear" stands for "K-linear").
A linear subshift of V G is a G-invariant vector subspace of V G . Let X, Y ⊂ V G be two linear subshifts. A linear cellular automaton from X to Y is a cellular automaton τ : X → Y which is linear. Note that if τ : X → Y is a cellular automaton with memory set S ⊂ G and local defining map µ : π S (X) → V , then τ is linear if and only if µ is linear.
Proposition 7.1. Let G be a group and let V be a vector space over a field K. Let X, Y ⊂ V G be two linear subshifts and let τ : X → Y be a map. Then the following conditions are equivalent: (a) τ is a linear cellular automaton; (b) τ is linear, uniformly continuous, and G-equivariant; (c) There exists a linear cellular automaton τ : Proof. This immediately follows from Proposition 5.2 after observing that if S ⊂ G then any linear map µ : π S (X) → V may be extended to a linear map µ : V S → V .
We shall denote by LCA(X, Y ; G, V ) the set consisting of all linear cellular automata τ : X → Y . If X = Y we shall simply write LCA(X; G, V ) instead of LCA(X, X; G, V ). Note that LCA(X; G, V ) has a natural structure of K-algebra, where the multiplicative law is given by composition.
If N is a normal subgroup of a group Γ and V is a vector space over a filed K, then the set Fix(N ) of points of V Γ which are fixed by N is a closed linear subshift of V Γ . From Proposition 6.1, we immediately deduce the following: Proposition 7.2. Let N be a normal subgroup of a group Γ and let ρ : Γ → G = Γ/N denote the canonical epimorphism. Let V be a vector space over a field K. Then y • ρ ∈ Fix(N ) for every y ∈ V G . Moreover, the map ρ * : V G → Fix(N ) defined by ρ * (y) = y • ρ is a linear G-equivariant uniform isomorphism.
In the proof of Theorem 1.4 we shall use the following results.
Lemma 7.4 (Closure lemma). Let G be a group and let V be a finite dimensional vector space over a field K. Let τ : V G → V G be a linear cellular automaton. Then τ (V G ) is closed in V G for the prodiscrete topology.
Proof. When G is countable this is shown in [CC1,Lemma 3.1]. The general case is treated in [CC4,Corollary 1.6].
Lemma 7.5. Let G be a group and let V be a finite dimensional vector space over a field K. Let τ : V G → V G be an injective linear cellular automaton and let Y = τ (V G ) denote the image of τ . Then the inverse map τ −1 : Y → V G is a linear cellular automaton and τ is a uniform embedding.
Proof. When G is countable this is shown in [CC2,Theorem 3.1].
We now treat the general case. Let S ⊂ G be a memory set for τ and denote by µ : V S → V the corresponding local defining map. Let H ⊂ G be the subgroup generated by S. Note that H is countable. Let G/H denote the set of all left cosets of H in G. For c ∈ G/H denote by π c : V G → V c = g∈c V the projection map. We have V G = c∈G/H V c and, for every x ∈ V G we write x = (x c ) c∈G/H , where x c = π c (x) ∈ V c .
For c ∈ G/H, g ∈ c and z ∈ V c we denote by g −1 z ∈ V H the configuration defined by (g −1 z)(h) = z(gh) for all h ∈ H. Then, if x = (x c ) c∈G/H ∈ V G and c ∈ G/H, we have for all h ∈ H, that is, For c ∈ G/H, define the map τ c : V c → V c by setting τ c (z)(g) = µ(π H S (g −1 z)) for all z ∈ V c and g ∈ c, where π H S : V H → V S is the projection map. Observe that τ H : V H → V H is a linear cellular automaton with memory set S over the group H. We then have (7.2) τ = c∈G/H τ c in the sense that π c • τ = τ c • π c for all c ∈ G/H. From (7.2) we immediately deduce that Y = c∈G/H Y c , where Y c = π c (Y ) = τ c (V c ), and that τ c is injective for all c ∈ G/H.
This shows that σ c • τ c equals the identity map Id V c : V c → V c .
We have proved that τ −1 = σ : Y → V G is a linear cellular automaton. In particular, τ −1 is uniformly continuous by Proposition 5.2 and therefore τ is a uniform embedding.
Proof of Theorem 1.4. Let N ∈ N (Γ) and let (N i ) i∈I be a net in N (Γ) converging to N . Suppose that the groups G i = Γ/N i are L K -surjunctive H−B → Y since N i converges to N . For i ∈ I, let ρ i : Γ → G i = Γ/N i denote the canonical epimorphism and let ρ * i : V G i → Fix(N i ) be the uniform isomorphism given by Proposition 7.2. It follows from Corollary 7.3 that the map τ i = (ρ * i ) −1 • τ | Z i • ρ * i : V G i → V G i is a linear cellular automaton and hence is surjunctive since G i is an L K -surjunctive group. This implies that f | Z i = τ Fix(N ) is surjunctive as well. By applying Corollary 4.2, we deduce that f (Y ) = Y . Therefore τ is surjective. This shows that the group G is L K -surjunctive.