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Simplicity of the automorphism group of fields with operators

We adapt a proof of Lascar in order to show the simplicity of the group of automorphisms fixing pointwise all non-generic elements for a class of uncountable models of suitable theories of fields with operators.


Introduction
In [14] Lascar showed, under the continuum hypothesis, the simplicity of the group of field automorphisms of the complex numbers C which fix pointwise the algebraic closure Q of the prime field. Lascar himself wrote [14, p. 249] J'ai peine à imaginer que ce fait n'est pas déjà connu (I cannot imagine that this result is not already well-known). Lascar's proof had two main ingredients: firstly, every non-trivial automorphism of C is unbounded (see Definition 5.4), or moves maximally the unique non-algebraic type, in the terminology of Tent and Ziegler ( [18]) on the simplicity of the group of isometries of the Urysohn space. Secondly, algebraic independence coincides with non-forking independence for the stable theory of the field C, which eliminates imaginaries, so types over (real) algebraically closed subsets are stationary, and thus there is a canonical way to fuse partial elementary maps defined on independent subsets.
Lascar's proof [14] used strongly that the group of automorphisms of a countable structure is naturally a Polish group, and hence many of his arguments had a topological flavor. In [15] Lascar provided an algebraic and more direct proof of the simplicity of Aut(C/Q) which circumvented the need of topological arguments and the continuum hypothesis. His second proof relies on a clever use of the aforementioned ingredients via transfinite induction.
In [9, Example 3.14], Lascar's first proof was adapted by Evans, Ghadernezad and Tent to show the simplicity of the group of automorphisms of a countable saturated differentially closed field fixing pointwise the differentially algebraic elements. Their proof used two results due to Konnerth: the extension of suitable partial elementary maps to global automorphisms of the saturated countable differentially closed field [12,Lemma 2.4] and the triviality of bounded automorphisms [12,Proposition 2.9].
Motivated by the existing work, we will present a unifying approach of Lascar's proof in [15] to the simplicity of the automorphism group of several fields with operators fixing the closure of the prime field with respect to a natural pregeometry. The description of bounded automorphisms was already generalized to several theories of fields with operators in [2,20]. A natural algebraic example of a theory of fields with operators is the theory ACFA of difference closed fields, that is, the common theory of existentially closed difference fields. The theory ACFA is simple yet unstable, so non-forking independence still captures many of the desired properties of an independence relation. However, types over algebraically closed sets need not be stationary, even if the theory admits elimination of (hyper)-imaginaries. In [5,Proposition 4.9] a useful criterion for relative stationarity was exhibited for models of ACFA 0 , that is, for difference closed fields of characteristic 0. This criterion has been crucial in order to adapt Konnerth's proof to show the existence of global extensions of suitable partial elementary maps [12,Lemma 2.4] to ACFA 0 (see Proposition 2.10) The field of complex numbers C is the unique saturated model in its cardinality of the stable theory of algebraically closed fields of characteristic 0. Uncountably saturated models exist for stable theories, but this need no longer be the case for simple theories without additional set-theoretical assumptions. A weaker version of saturation of cardinality κ > ℵ 0 is the notion of κ-prime models over a given set of parameters. The existence and uniqueness of a κ-prime model over all subsets has been shown by Shelah for superstable theories [17,Theorem IV.4.18]. The second author [4] showed the existence and uniqueness of a κ-prime model over certain subsets for the unstable theory ACFA 0 . In this work, we will show in Theorem 5.9 the simplicity of the group of automorphisms fixing pointwise all non-generic elements over ∅ for a class of uncountable models, encompassing both uncountably saturated models (if they exist) as well as certain κ-prime models, for suitable theories of fields with operators.

Preliminaries
In order to provide a unifying approach to the study of κ-prime models for several theories of fields with operators, we will work with a first-order complete theory T in a language L, which we will assume for the sake of the presentation to be countable, though this is not an actual obstacle as long as we take an uncountable cardinal κ ≥ |L| + . Furthermore, we impose that the L-theory T is simple. We work inside a κ-saturated model U of our theory T with κ uncountable.
Fix a definable group G without parameters. Recall that an element g in G is where the symbol | ⌣ denotes nonforking independence in the sense of the simple theory T .
The inverse of the generic element g is again generic, so we have the independence g ·h | ⌣ A∪{h} whenever g | ⌣A h. The product of two generic independent elements over A is again generic and independent of each factor over A. A type tp(g/A) in G is generic if g is generic over A. Every non-forking extension of a generic type is again generic.
Whenever the underlying group is stable and connected, there exists a unique generic type. This remarkable feature was also shown for difference closed fields in [5,Proposition 2.10]. Since we will be mostly concerned with the group of automorphisms of certain fields with operators, we will impose the following condition: Property 1. The universe U admits a group structure definable over L without parameters. Furthermore, there is a unique generic type over every subset A of U, given by the nonforking extension of the unique generic type p over ∅: every two generic elements over A have the same type over A.
Since every non-forking extension of a generic type is again generic, Property (1) implies in particular that the generic type p over ∅ is stationary: Any two realizations of the generic type which are both independent over the same subset A of U have the same type over A.
Recall that a stationary type q over ∅ is regular if it is orthogonal to every forking extension: if a realizes the unique nonforking extension of q to B and the realization c of q forks with B, then a | ⌣B c. The presence of a regular type q based over the empty set determines a closure operator [11] on the set of realisations of q. This closure operator was later adapted in [19,Section 3.5] to other collections of types. Whilst the general definition of the closure for an arbitrary family of types is somewhat difficult to grasp, it can be simplified in our context, as we will only consider the closure, within the ambient model U, taken with respect to the generic type p of the underlying group. Since every element of the group can be written as the product of two generic elements, every type is thus analyzable with respect to the generic type. In our concrete case, the closure with respect to the (possibly non-regular) generic type p over ∅ can be described as follows: Definition 2.1. Given a subset A, an element b belongs to the closure cl(A) of A if for every subset C containing A and every generic element g over C, we have that g remains generic over C ∪ {b}.
Notice that in particular, every element algebraic (in the model-theoretic sense) over A belongs to cl(A). However, in most examples of fields with operators we are interested in, the closure of a set will be rather big, and in particular cl(∅), taken in our κ-saturated model U, will have cardinality ≥ κ. For example, in the case of differentially closed fields of characteristic 0, every single constant element a (that is, with δ(a) = 0) belongs to cl(∅).
This notion of closure, despite being very general, already satisfies some desired properties: In a stable theory with weak elimination of imaginaries, types over real algebraically closed subsets of small cardinality are stationary. As shown in [6, Theorem 5.3 & Corollary B.11] for difference closed fields, types of the form tp(a/B) are stationary whenever B = cl(B) ∩ acl(B, a), so this motivates the following property. Property 2. Types over relatively cl-closed subsets are stationary: Given subsets A 1 , A 2 and B of size strictly less than κ with a common algebraically closed subset C such that acl(A 1 ) ∩ cl(C) = C = acl(A 2 ) ∩ cl(C), The following property will be fundamental in order to extend partial automorphisms to arbitrary subsets in terms of a small chain of extensions obtained by adding successively generic elements.
Property 3. For every κ-saturated model U of T and every subset A of U of size strictly less than κ, each element b of U is contained in cl(A ∪ C), where C is a subset consisting of a sequence of independent realizations of the generic type over A. Remark 2.3. As the ambient theory T is simple, if Property (3) holds, then we can choose C countable.
by the local character of nonforking independence in the simple countable theory T . By Remark 2.2(b), we conclude that b belongs to cl(A ∪ C 0 ).

Remark 2.4.
If the generic type is regular, then an element is generic over A if and only if it does not belong to cl(A). In particular, Property (3) always holds with C a singleton. Moreover, if tp(g/B) is never isolated, whenever B ⊃ A is arbitrary and g is generic over B, then cl(A) is an elementary substructure of U. (This is always the case for a stable theory when the generic type over ∅ is non-isolated, by the Open Mapping theorem.) Proof. Clearly, a generic element over A cannot lie in cl(A) by Remark 2.2 (a). For the other direction, assume b is not generic and choose g generic over A, b. So write b = g · h, where both g and h = g −1 · b are generic over A ∪ {b}. Now, the realization h of the generic type is not independent from g over A (since b is not generic). By definition 2.1, the element h belongs to cl(A ∪ {g}): Indeed, we need only show that h is independent from every generic element g ′ over any subset C containing A ∪ {g}. Note that h forks with C and thus h is independent from the generic element g ′ over C, by regularity of the generic type. We conclude that b belongs to cl(A ∪ {g}) as well. The Remark 2.2 (b) yields that b belongs to cl(A), as desired.
For the second statement, assume that the generic type is regular but not isolated. That cl(A) is an elementary substructure is a straight-forward application of Tarski's test: consider a consistent formula ϕ(x, b 1 , . . . , b n ), where each b i belongs to cl(A), and choose a realization g of ϕ(x, b 1 , . . . , b n ) in U. If g already belongs to cl(A), then we are done. Otherwise, it is generic over A and thus remains so over B = A ∪ {b 1 , . . . , b n }. As the generic type tp(g/B) is not isolated, there is some c realizing ϕ(x, b 1 , . . . , b n ) whose type differs from tp(g/B). By Property (1), the element c is not generic over B and thus it belongs to cl(B) ⊆ cl(A), as desired.
The following remark is an immediate consequence of the atomicity and the saturation of the model U.
Remark 2.6. Every κ-saturated model U of a theory T satisfying Property (4) is weakly homogenous with respect to the family of subsets of the form with |A i |, |B| < κ for 1 ≤ i ≤ n: given an element c in U and a partial elementary  (2) and (4) with respect to the closure operator defined in Definition 2.1.
Proof. It was already noticed right before stating Property (2) that it always holds for stable theories with weak elimination of imaginaries. Thus, we need only show that Property (4) holds. Let c be a finite tuple of U. In order to show that tp(c/K) is κ-isolated, we must find some subset E of K of size strictly less than κ such that tp(c/E) ⊢ tp(c/K). The local character of forking yields a subset D of K of cardinality bounded by |L| < κ such that c | ⌣D K. Set now E = acl(A 1 , · · · , A n , B, D).
Since D ⊆ E, it follows that c | ⌣E K. We now show that tp(c/E) ⊢ tp(c/K), or equivalently, that tp(c/E) ⊢ tp(c/E, η) for all finite tuples η in K. Notice that tp(c/E) is stationary, by weak elimination of imaginaries, so we need only show that for every realization d in is invariant under all automorphisms of U fixing A. We deduce that η ′ lies in K, as desired.
Remark 2.8. Every stable connected group with weak elimination of imaginaries whose generic type is regular satisfies Properties (1)- (4). If the group is superstable of Lascar rank ω α , then the closure operator equals In particular, both the theory of algebraically closed fields of arbitrary characteristic in the language of rings L Ring as well as the theory of differentially closed fields of characteristic 0 with respect to m commuting derivations in the language L Rings ∪ {δ 1 , . . . , δ m } satisfy Properties (1)-(4).
Notice that, whenever the superstable group has Lascar rank ω α , with α = 0, then the closure of any (small) subset B of the κ-saturated model U has size at least κ.
In contrast to stable theories, types over models in simple theories need not be stationary. However, under some mild conditions it can be the case that the non-forking extension is unique. This was explored in [5,Proposition 4.9] for the theory ACFA 0 (or rather ACF 0 A) of difference closed fields in characteristic 0, using the concept of superficial co-stability, see Lemma 2.9 below. Before showing that properties (1) -(4) hold for ACFA 0 , let us recall some notation and results from [5]: if A is a subset of the κ-saturated difference closed field U, then A σ denotes the inversive difference field generated by A, that is, the smallest difference subfield containing A such that the restriction of the automorphism to A σ is surjective. To avoid any possible confusion, as there will be several powers σ k of the generic automorphism σ around, we will denote by acl σ the model-theoretic algebraic closure of A, that is, the smallest inversive difference field containing A and which is algebraically closed as a field. The type tp σ (b/A) is uniquely determined by the isomorphism type over A of the difference field acl σ (A ∪ {d}) (see [5,Corollary 1.5]). Every completion of ACFA 0 is simple and nonforking independence can be described in algebraic terms: Given two inversive difference fields A and B with a common difference subfield C = acl σ (C), we have that A | ⌣C B if and only if A and B are algebraically independent over C [5, Section 1.9].
By [5,Corollary 1.13], the field (U, σ k ), is again a difference closed field for k = 0 in Z. In particular, we will denote by Note that the closure operator in (U, σ) can be described algebraically as A combination of the results obtained in [5] yields an explicit description of the failure of uniqueness of the non-forking extension to a fixed set of parameters. For the proof of 2.10, we will only need an adaptation of [5, Proposition 4.9 & Lemma 2.8], which we will state in the following Lemma: for every k = 0 in Z and every finite tuple η ′ realizing the quantifier-free type qftp σ k (η). Then the type tp σ (c/E) implies tp σ (c/E ∪ {η}).
Proof. Assume that c, η and E are as in the statement. We can apply [5,Proposition 4.9] and obtain that tp σ (c/E) and tp σ (η/E) are superficially co-stable: for every d | ⌣E η realizing tp σ (c/E), setting K = acl σ (E ∪ {η}) and considering the unique extension σ F of σ ↾aclσ(E,d) ⊗σ ↾K to the compositum field F of acl σ (E, d) and K, we have that F has no proper finite Galois extension invariant under σ F . Moreover, setting k = 1 in our assumption, we deduce from the previous description of the non-forking independence in the simple theory ACFA that c is independent of every realisation η ′ in U of tp σ (η/E), or equivalently, that every realisation d of tp σ (c/E) is independent from η over E.
A realization d of tp σ (c/E) yields an E-isomorphism of difference fields between acl σ (E ∪ {d}) and acl σ (E ∪ {c}) mapping d to c. We need to show that this isomorphism of difference fields extends to an K-isomorphism of difference fields between acl σ (K ∪ {d}) and acl σ (K ∪ {c}).
By superficial co-stability (since every d realizing tp σ (c/E) is automatically independent from K over E), there is a unique extension of the field automorphism σ F to the algebraic closure of F , by [5,Lemma 2.8]. We conclude that there is an isomorphism of difference fields between the algebraic closure of F and the difference field acl σ (K ∪ {c}) over K mapping d to c, so d ≡ K c for every realization d of tp σ (c/E), as desired. We need only show that Property (4) holds in every κ-saturated model U of ACFA 0 . As in the proof of Lemma 2.7, set E = acl σ (A 1 , · · · , A n , B, D), where D is a subset of K = acl σ (cl σ (A 1 ) ∪ · · · ∪ cl σ (A n ) ∪ B) of cardinality bounded by |L| < κ such that c | ⌣D K. Note that c | ⌣E K. The κ-atomicity will follow once we show that tp(c/E) ⊢ tp(c/K), or equivalently, that tp(c/E) ⊢ tp(c/E ∪ {η}) for every finite tuple η in K. We may assume that B = B σ and each A i = acl σ (A i ). Hence K is the field algebraic closure of the (inversive difference) field generated by cl σ (A 1 ) ∪ · · · ∪ cl σ (A n ) ∪ B. Possibly at the cost of enlarging η, we may also assume that η = (η 0 , η 1 , . . . , η n ) with η 0 field algebraic over E ∪ {η i } 1≤i≤n and η i in cl σ (A i ) for 1 ≤ i ≤ n, by the discussion before Lemma 2.9.
Therefore, we need only show by Lemma 2.9 that for every integer k in N and every η ′ in U realizing the quantifier-free type qftp σ k (η/E), the fields E, c σ k and E, η ′ σ k are algebraically independent over E. Note that cl σ (A i ) = cl σ k (A i ), so η i belongs to cl σ k (A i ) for 1 ≤ i ≤ n. The isomorphism of difference fields in the structure (U, σ k ) over E maps the tuple η to η ′ , with , since each of these properties is quantifier-free definable in the language of difference rings. Hence, the whole tuple η ′ is field algebraic over the field generated by E ∪ 1≤i≤n cl σ (A i ) ⊆ K. The difference field K is algebraically closed, so E,η ′ σ k ⊆ K. Since E was chosen so that c | ⌣E K, we deduce the desired independence between E, c σ k and E, η ′ σ k over E.  (1), (2) and (4) with respect to the closure operator given by the unique generic type. We do not know whether this theory satisfies Property (3). Note that the unique generic type is not regular if e = 0: Indeed, given a generic element g over a small set of parameters A, the sequence {λ n (g)} 0≤n<p e is a sequence of independent realisations of the generic type. However, the element g is no longer generic over λ 0 (g), whilst λ 1 (g) remains generic over λ 0 (g). Clearly g and λ 1 (g) fork over λ 0 (g), since λ 1 (g) is definable over g.
Analogously, the theory D-CF 0 of D-closed fields of characteristic 0 equipped with n free derivations in the language L = L Rings ∪ {δ 1 , . . . , δ n } introduced in [16] is stable and eliminates imaginaries [16, Corollary 5.7 and Theorem 5.12], so it satisfies Properties (1), (2) and (4) with respect to the closure operator given by unique generic type. The same holds for any collection of n free operators if the associated endomorphisms are trivial. We do not know whether the theory D-CF 0 satisfies Property (3). Note that the unique generic type is not regular, whenever the number of free derivations is at least 2, by exactly the same argument as for separably closed fields.

Tame models
For this section, we assume that the cardinal κ is uncountable and that the countable simple L-theory T satisfies Properties (1)- (3).
In order to prove Theorem 5.9, we will need to restrict our focus to a particular class of κ-saturated models, which we will denote by tame models. and f (K) = acl(cl(f (A 1 )) ∪ · · · ∪ cl(f (A n )) ∪ f (B)), such that all A i 's and B have cardinality strictly less than κ, extends to a global automorphism of U.
By Property (3) and saturation, it suffices to consider in the above condition (a) a subset A enumerating an independent sequence of realizations of the generic type. In particular, the model U does not contain an independent sequence of realizations of the generic type of length κ + : Indeed, every generic element in U must fork with A, which is witnessed by a finite tuple in A. There are κ many finite tuples in A, but each tuple can only divide (by simplicity of T ) with countably many elements in the independent sequence, which contradicts that the sequence had length κ + . If the generic type is regular, condition (a) is equivalent to (a') The model U does not contain an independent sequence of realizations of the generic type of length κ + . Indeed, choose a subset A enumerating a maximal independent sequence in U of realizations of the generic type. The subset A has size at least κ, by saturation, but it cannot exceed κ, by condition (a'), so it has size exactly κ. Notice now that no element in U lies outside the closure of A, by regularity of the generic type and Remark 2.4. Given a subset K = acl(cl(A 1 ) ∪ · · · ∪ cl(A n ) ∪ B) as above and a sequence (c i ) i<λ with λ < κ, the set acl(K, {c i } i<λ ) is again of the form as in Definition 3.1 (b). Thus, the following remark follows immediately from Property (1).

Remark 3.2.
Consider a κ-tame model U of the theory T and a partial elementary map f : K → U as above. Given two sequences (c i ) i<λ and (d i ) i<λ of realizations of the unique generic type which are respectively independent over K and over f (K), with λ < κ, there is an extension of f to a global automorphism mapping the sequence (c i ) i<λ to the sequence (d i ) i<λ . In particular, the global automorphism maps cl(K, Using a standard chain argument of length κ, it is therefore possible to extend f to a global automorphism of U mapping an sequence of length κ of independent realizations over K of the generic type to sequence of length κ of independent realizations over f (K) of the generic type. Lemma 3.3. Consider a κ-tame model U of a theory T , consider cl-closed subsets X, Y 1 and Y 2 of U with X ⊆ Y 1 ∩ Y 2 , each cl-generated by subsets of size strictly less than κ. Given elementary automorphisms of Y i for i = 1, 2 which agree on X such that g 1 (X) = g 2 (X) = X. If then there is a global automorphism of U which extends both g 1 and g 2 .
Proof. Set Y 1 = cl(A 1 ) for some subset A 1 of cardinality stricly than κ. Note that Y 1 = cl(g i (A 1 )), since g 1 is an elementary map of Y 1 onto itself. By tameness, there exists a global automorphism g 2 of U extending g 2 . After enumeration of Y 1 , the subset Y ′ 1 = g 2 −1 (g 1 (Y 1 )) has the same type as Y 1 over X, since g 1 and g 2 (and thus ), so we obtain by tameness a global automorphism h of U extending h. By construction, the automorphism g 2 • h extends both g 1 and g 2 .
Remark 3.4. If the countable simple theory T satisfies Properties (1)-(4) and has a saturated model U of cardinality κ, then U is κ-tame.
Indeed, the condition Definition 3.1 (a) holds trivially, since the model U has cardinality κ.
The only thing left to show is the strong extension property for elementary maps f : K → U as above. A straight-forward Back-&-Forth argument shows that f extends to a global automorphism using Remark 2.6.
The existence of saturated models of cardinality exactly κ follows whenever the countable theory T is κ-stable [10,17]. For a general theory T , if κ = λ + for some cardinal λ ≥ |T | with λ + = 2 λ , or if κ is regular and strongly inacessible, then there are saturated models of cardinality κ. However, these set-theoretic assumptions go beyond ZFC.

Existence and uniqueness of κ-prime models
Shelah introduced the notion of κ-prime models [17, Chapter IV] (see also [13, Chapitre VI]) for an arbitrary stable theory, generalizing Morley's notion of prime models for an ω-stable theory. Shelah showed the existence and uniqueness (up to isomorphism) of κ-prime models over arbitrary subsets A (even if A has size strictly larger than κ) whenever the countable theory is superstable. They are exactly the κ-saturated κ-atomic models over A which contain no (non-constant) A-indiscernible sequence of length κ + . There are also results for stable theories, but with some restrictions on the cardinal κ.  Proof. Let us first show that U is the closure of a set of size κ. Since the generic type is regular, we need only consider the equivalent condition (a') in Definition 3.1. Clearly, the κ-saturated model U does not contain (non-constant) independent sequences of the generic type of length κ + , since such a sequence is indiscernible by Property (1).
We now show the strong extension property for elementary maps f : ; where all subsets |A i |'s and B of U have cardinality strictly less than κ. By Property (4), the model U is κ-atomic over K as well as over f (K) and does not contain any non-constant indiscernible sequence of length > κ over K nor over f (K). Hence, by uniqueness of κ-prime models, the partial isomorphism f : K → f (K) extends to an automorphism of U.
Since U embeds into M over cl M (∅), it follows immediately that cl U (∅) = cl M (∅).
Notation. From now on, we will say that U is κ-prime over its closure cl(∅) if U is κ-saturated and κ-prime over the subset A = cl U (∅).
The aforementioned characterizations of κ-prime models in terms of κ-atomicity and non-existence of non-constant indiscernible sequences of length κ + yields immediately the following result. 2.10 has a κ-tame model for any κ > ℵ 0 , namely, the κ-prime model U over the corresponding closure cl(∅).

Automorphisms of κ-tame models
Henceforth, we suppose that the countable simple L-theory T satisfies Properties (1)-(3) and has a κ-tame model U.
Notation. We will denote by S the collection of subsets of U of the form cl(A), where A enumerates a sequence of independent realizations of the generic type of length strictly less than κ. Note that each member of S is an algebraically closed substructure of U.
where A enumerates an infinite countable sequence of independent realizations of the generic type over X.

Remark 3.2 yields immediately the following result:
Lemma 5.2. Given an elementary automorphism f of X in S and two acceptable extensions Y 1 and Y 2 of X, there is an extension of f to an elementary partial map sending Y 1 to Y 2 . Thus, any two acceptable extensions Y 1 and Y 2 of X are conjugate by an automorphism of U fixing X pointwise. Remark 5.3. Given X in S, an element b in U as well as a countable collection F of global automorphisms of U which each leave X setwise invariant, there exists an acceptable extension Y of X in S which contains b and is setwise stable under each automorphism of F .
Proof. Without loss of generality, we may assume that F is a group. By induction, successively applying Property (3), we construct a increasing sequence (A n ) n∈N of sets such that: • the set A 0 generates X with respect to the closure operator cl.
• the elements in A n+1 \ A n enumerate a countable independent sequence of realizations of the generic type over A n (and thus over cl(A n )); • the element b belongs to cl(A 1 ); • for every n in N, every τ in F and every a in A n , the element τ (a) belongs to cl(A n+1 ). Set now Y = cl n∈N A n , which has all the desired properties.
We recall now [2, Definition 2.14] (or a slightly modified version thereof, see also [20]). Definition 5.4. An L-automorphism τ of a U is unbounded if, whenever A has cardinality strictly less than κ, there is a generic element b in U over A (so b does not belong to cl(A)) such that τ (b) does not lie in cl (A, b).
The L-automorphism τ is bounded if it is not unbounded.
Recall that Frobenius is bounded, whenever the underlying field has positive characteristic. Similarly, given a κ-saturated model (K, σ) of ACFA, every power σ k of the generic automorphism σ is bounded. It is immediate to see that the collection of bounded automorphisms of an arbitrary model forms a normal subgroup of the automorphism group of U. Lemma 5.5. Let τ be an unbounded automorphism of U fixing the closure cl(∅) in U pointwise. If X in S is stable setwise under the action of τ , then there is an acceptable extension Y of X such that Y and τ (Y ) are independent over X.
Notice that τ (Y ) is again an acceptable extension of X.
Proof. It suffices by Remark 2.2 (b) to inductively build a independent sequence (a n ) n∈N of realizations of the generic type over X such that for any n ∈ N we have a n | ⌣ X a 0 , τ (a 0 ), . . . , a n−1 , τ (a n−1 ) and τ (a n ) | ⌣ X a 0 , τ (a 0 ), . . . , a n−1 , τ (a n−1 ), a n .
Suppose the sequence has been constructed for j < n. The subset X ′ = cl(X, a j , τ (a j ) : j < n) lies in S yet τ is unbounded over the small subset X ′ ∪ τ (X ′ ). Hence, we find an element a n in U generic over X ′ such that τ (a n ) does not lie in cl(X ′ , τ (X ′ ), a n ). In particular, the element a n is independent from a 0 , τ (a 0 ), . . . , a n−1 , τ (a n−1 ) over X. Similarly, the element τ (a n ) is independent from a 0 , τ (a 0 ), . . . , a n−1 , τ (a n−1 ), a n over X, as desired.
It follows from the above proof that every unbounded automorphism fixing pointwise the closure cl(∅) moves maximally, according to the terminology of [18], the unique generic type p over X in S, by Property (1).
Motivated by [2,20], we will now introduce the last property of interest for our purposes.
Property 5. The only bounded automorphism of U fixing pointwise the closure cl(∅) is the identity.
A key fact in the description of bounded automorphisms of fields with operators is that a non-generic definable additive subgroup of G a is given by a system of linear equations on (words on) the operators. A straight-forward adaptation of [2, Théorème, pp. 957-958] yields the following: We will now reproduce verbatim in our context Lascar's proof of the simplicity of the group of field automorphisms of C fixing Q alg [15]. More precisely, we will show the simplicity of the group of automorphisms Aut(U/ cl(∅)), modulo the subgroup of bounded automorphisms.
We fix an unbounded automorphism τ of U fixing cl(∅) pointwise. Denote by Whenever X in S is stable under the action of τ , we denote by ) ↾X for f and g elementary automorphisms of X.
We will show in 5.9 that every automorphism ν of U fixing cl(∅) can be written as for suitable automorphisms f 1 , . . . , f 4 . This will in particular show that the group of automorphisms of U fixing cl(∅) is simple modulo the normal subgroup of bounded automorphisms fixing cl(∅). This process will be done by a chain of approximations of ν to smaller substructures of dimension strictly less than κ.
If the partial isomorphism g extends the partial isomorphism f , we will denote it by f ⊂ g. For the back-and-forth process to describe ν as a product of conjugates of τ and τ −1 , we will need the following central result.
Proposition 5.7. Given X in S stable under the action of τ , two elementary automorphisms f and g of X and an acceptable extension Y ⊇ X in S equipped with an elementary automorphism h of Y extending Ψ X (f, g), there exist two automorphisms f ′ and g ′ of U extending f and g, respectively, such that Proof. By Lemma 5.5, there exists an acceptable extension Y 1 of X such that Y 1 and Y 2 = τ (Y 1 ) are independent over X. By Lemma 5.2, there is an automorphism Again by Lemma 5.2, choose an elementary automorphism g 2 extending g which maps Y 2 to itself. The elementary automorphism We include a diagram to facilitate the arrow-chasing: Lemma 3.3 yields now a common extension g ′ to U of the elementary automorphisms g 1 of Y 1 and g 2 of Y 2 . We need only check now that Ψ(f ′ , g ′ ) extends h when restricted to Y , or equivalently, that f ′−1 • Ψ(f ′ , g ′ ) • f ′ extends h 2 . By the definition of Ψ(f ′ , g ′ ), we have that 2 on Y 2 , as desired. The previous Proposition 5.7 contains all the ingredients to tackle the successor stage of the back-and-forth construction required in the proof of Theorem 5.9 Proposition 5.8. Let ν be an automorphism of U and X in S which is stable under the action of both τ and ν such that for some elementary automorphisms f i of X, with i = 1, . . . , 4.
For every element a in U, there are: • an acceptable extension Y ⊇ X containing the element a which is stable under both τ and ν; Similarly, we find an acceptable extension Y 2 of Y 1 stable under the action of τ , ν, f 1,1 and f 2,1 . Then, we denote by f 1,2 and f 2,2 the restrictions of f 1,1 and f 2,1 to Y 2 .
Notice that ν −1 ↾Y2 •Ψ Y2 (f 1,2 , f 2,2 ) ⊃ Ψ Y1 (f 3,1 , f 4,1 ). Iterating the above argument countably many times, we construct an increasing chain Y n of acceptable extensions (setting Y 0 = X) and compatible elementary extensions f 1,2n and f 2,2n of f 1 = f 1,0 and f 2 = f 2,0 to Y 2n as well as f 3,2n+1 and f 4,2n+1 of f 3 and f 4 to Y 2n+1 such that for all n in N, By construction of the chain, the subset Y = n∈N Y n lies in S and is an acceptable extension of X. For 1 ≤ i ≤ 4, denote f ′ i the corresponding elementary automorphism to Y given by the f i,k 's. By construction, we have that We have now all the ingredients to prove the simplicity, up to bounded automorphisms, of the automorphism group of U which fix pointwise cl(∅). Theorem 5.9. Consider a κ-tame model U of a simple countable theory T satisfying Properties (1)-(3). Fix an unbounded automorphim τ of U fixing cl(∅) pointwise. Every automorphism ν of U fixing cl(∅) pointwise can be written as the product of four conjugates of τ and τ −1 .
In particular, the group Aut(U/ cl(∅)) of automorphisms of U fixing cl(∅) pointwise is simple modulo the normal subgroup of all bounded automorphisms fixing cl(∅) pointwise.
For the beginning of the recursion, set X 0 = cl(∅) and f i,0 = Id X0 . Assume now that X β has already been constructed for β < α. If α is a limit ordinal, the union X α = β<α X β with f i,α = β<α f i,β , for 1 ≤ i ≤ 4 is cl-generated by an independent sequence of length |α| and ν restricted to X α satisfies the above identity. If α is the successor of β, Proposition 5.8 applied to X β yields an acceptable extension X β+1 of X β containing a β and elementary extensions f i,β+1 , as desired.
Finally, the union α<κ X α is cl-closed, so it must equal U. By construction, the automorphism ν equals a product of four conjugates of τ and τ −1 globally on U, since the automorphisms at every step are compatible.
Corollary 5.10. For each of the following theory of fields with operators: • algebraically closed field with the closure operator given by the field algebraic closure acl; • differentially closed field in characteristic 0 with finitely many commuting derivations with the closure operator given by the elements which are not differentially transcendental; • difference closed field in characteristic 0 with the closure operator given by the elements of transformal transcendence degree 0; the group of automorphisms of every uncountable model saturated in its cardinality (if it exists) fixing pointwise cl(∅) is simple.
More generally, given κ > ℵ 0 , the group of automorphisms of the κ-prime model over its closure cl(∅) fixing pointwise cl(∅) is simple.
Proof. By Remark, 2.8, Proposition 2.10 and Remark 5.6, each of these theories satisfies Properties (1)- (5). Furthermore, these theories either have uncountable saturated or κ-prime models, which are in both cases κ-tame by Remark 3.4 & Corollary 4.4. The triviality of the subgroup of bounded automorphisms of such models in Remark 5.6 and Theorem 5.9 yields the desired result. Question 2. Let K be a saturated uncountable separably closed fields of positive characteristic p and finite degree of imperfection e. Is the automorphism group of K fixing pointwise cl(∅) simple?