Regular singular differential equations and free proalgebraic groups

We determine the differential Galois group of the family of all regular singular differential equations on the Riemann sphere. It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$.


Introduction
The following table depicts the beautiful analogy between classical Galois theory and differential Galois theory over the rational function field C(x).Excellent introductions to these topics can be found in [Sza09] and [vdPS03].

Classical Galois theory Differential Galois theory 1
Univariate polynomials over C(x) Linear differential equations over C(x) 2 Galois extension of C(x) and their Galois groups Picard-Vessiot extensions of C(x) and their differential Galois groups 3 A finite Galois extensions L of C(x) corresponds to a ramified cover p : X → P 1 (C).If S is the finite set of branch points and x 0 ∈ C is not in S, then the Galois group of L/C(x) can be identified with the image of π 1 (P 1 (C) S, x 0 ) under its action on p −1 (x 0 ).
Schlesinger's density theorem: Let S be a finite subset of P 1 (C) and x 0 ∈ C not in S.
The differential Galois group of a regular singular linear differential equation with singularities in S, can be identified with the Zariski closure of the image of π 1 (P 1 (C) S, x 0 ) under its action on the local solution space at x 0 .4 For x 0 ∈ C and S a finite subset of P 1 (C) not containing x 0 , the Galois G group of the maximal algebraic extension of C(x) with ramification only over S, is the profinite completion of π 1 (P 1 (C) S, x 0 ), i.e., G is the free profinite group on a set of cardinality |S| − 1.
For x 0 ∈ C and S a finite subset of P 1 (C) not containing x 0 , the differential Galois group G of the family of all regular singular differential equations with singularities inside S, is the proalgebraic completion of π 1 (P 1 (C) S, x 0 ), i.e., G is the free proalgebraic group on a set of cardinality |S| − 1. 5 Douady's Theorem: The absolute Galois group of C(x) is the free profinite group on a set of cardinality |C|.?
The main goal of this article is to fill in the above question mark.On the face of it, it may seem that the appropriate differential analog of Douady's theorem is "The absolute differential Galois group of C(x) is the free proalgebraic group on a set of cardinality |C|".This is in fact a true statement ([BHHW21a]).However, Douady arrived at 5 via 3 and 4. In this sense, an appropriate differential analog of Douady's theorem should only be concerned with regular singular differential equations.Our main result is the following differential analog of Douady's theorem.
Theorem A (Theorem 4.4).The differential Galois group of the family of all regular singular differential equations over C(x) is the free proalgebraic group on a set of cardinality |C|.
At first glance, it might seem that 5 should follow from 4 rather immediately.However, in general, the projective limit of free profinite groups need not be a free profinite group ([RZ10, Ex. 9.1.14]).The question, when a projective limit of free profinite groups is itself free has attracted some attention but does not seem to be fully understood ([RZ10, Thm.3.5.15and Open Question 9.5.2]).
To get from 4 to 5 in the proof of Douady's theorem (see [Sza09,Sec. 3.4] or [Dou64] for the original reference) one uses a compactness argument and that π 1 (P 1 (C) S, x 0 ) has more or less canonical generators.Another ingredient of the proof is that in the free profinite group on a set with r elements, any subset of r topological generators is a basis.As we will show (Example 2.3), the corresponding statement fails for free proalgebraic groups.Therefore, genuinely new ideas are needed in the differential case.In particular, we will use a characterization of free proalgebraic groups in terms of embedding problems.
For a finite subset X of C, it is an immediate consequence of the Riemann-Hilbert correspondence, that the differential Galois group of the family of all regular singular differential equations with singularities in X ∪ {∞}, is the free proalgebraic group on a set of cardinality |X|.We generalize this result from finite subsets of C to arbitrary subsets of C. Indeed, the case X = C is exactly Theorem A.
We note that there is also an analogy between differential Galois theory over C(x) and classical Galois theory over k(x), with k an algebraically closed field of characteristic p > 0, such that regular singular points correspond to tamely ramified points, while irregular singular points correspond to wildly ramified points ([vdPS03,Section 11.6]).Based on this analogy, our result may seem more surprising, because the Galois group of the maximal tamely ramified extension of k(x) with branch locus in a fixed subset S of P 1 (k) is not a free profinite group.
We conclude the introduction with an outline of the article.In Section 2 we recall the definition of free proalgebraic groups and the required results concerning differential Galois theory and the Riemann-Hilbert correspondence.We then study projetive systems of abstract free groups in Section 3. Finally, in the last section the previous results are applied to prove Theorem A.
The author is grateful to David Harbater and Michael Singer for helpful comments.

Preliminaries and Notation
In this preparatory section we recall the basic definitions and results concerning regular singular differential equations and differential Galois theory.We also review the definition of free proalgebraic groups.
We use "=" or " " to denote canonical isomorphisms and " ∼ =" to denote isomorphisms.(As the implied isomorphism in Theorem A is not canonical, it seems worthwhile to make this distinction.)2.1.Free proalgebraic groups.Throughout this article we work over the field C of complex numbers.We use the term "algebraic group" in lieu of "affine group scheme of finite type over C".Similarly, a "proalgebraic group" is an "affine group scheme over C".By a closed subgroup of a proalgebraic group, we mean a closed subgroup scheme.Following [Mil17,Def. 5.5] a morphism G → H of proalgebraic groups is called a quotient map if it is faithfully flat.We use G H to indicate quotient maps.
We begin by recalling the definition of free proalgebraic groups from [Wib20].Let Γ be a proalgebraic group and let X be a set.A map ϕ : X → Γ(C) converges to 1 if almost all elements of X map to 1 in any algebraic quotient of Γ, i.e., for every algebraic group G and every quotient map φ : Γ G, all but finitely many elements of X map to 1 under The following definition is the special case of [Wib20, Def.2.18], where C is the formation of all algebraic groups and R = k = C. Definition 2.1.Let X be a set.A proalgebraic group Γ(X) together with a map ι : X → Γ(X)(C) converging to one is called a free proalgebraic group on X if ι satisfies the following universal property.For every proalgebraic group G and every map ϕ : X → G converging to 1, there exists a unique morphism φ : Γ(X) → G of proalgebraic groups such that As the pair (ι, Γ(X)) is unique up to a unique isomorphism, we will usually speak of the free proalgebraic group Γ(X) on X.
Remark 2.2.To verify that Γ(X) is the free proalgebraic group on X, it suffices to verify the universal property of Definition 2.1 for G algebraic ([Wib20, Rem.2.19]).
For a proalgebraic group G and a subset X of G(C), we denote the smallest closed subgroup H of G such that X ⊆ H(C) with X .In other words, X is the closed subgroup of G generated by X.By [Wib20, Thm.2.17] we have Γ(X) = X .
We next explain why the proof of Douady's theorem (as presented in [Dou64] or [Sza09, Sec.3.4]) does not have a direct differential analog.This proof uses a result ([Dou64, Prop.1] or [Sza09,Lem. 3.4.11])attributed to Serre by Douady, stating that, any set of n elements that topologically generates a free profinite group of rank n is a basis.Equivalently, a surjective endomorphism of a free profinite group of finite rank is an isomorphism.The proof of this result uses a counting argument and therefore does not apply in our context, where finite groups are replaced by algebraic groups.In fact, as shown in the following example, Serre's result does not hold in our context.
Example 2.3.Let X = { * } be a set with one element * .We will show that not every generator of Γ(X) is a basis, i.e., there exists a quotient map Γ(X) → Γ(X) that is not an isomorphism.
The free proalgebraic on one element is of the form Γ(X) = G a × D(C × ).See [Wib20, Ex. 2.22] or [Sau16,Cor. 16.26].Here G a is the additive group and, as in [DG70, Ch.IV, §1, Sec.1], for any abelian group M , D(M ) denotes the diagonalizable proalgebraic group with character group M , i.e., D(M )(T ) = Hom(M, T × ) for any C-algebra T .The map ι : As an abelian group, C × is isomorphic to (Q/Z) ⊕ V , where V is a Q-vector space of dimension |C|.In particular, there exists an injective endomorphism ψ : C × → C × that is not an isomorphism.
Dualizing ψ, we find a quotient map D(C × ) → D(C × ) that is not an isomorphism.This trivially extends to a quotient map Γ(X) → Γ(X) that is not an isomorphism.
We will need the notion of proalgebraic completion of an abstract group.See e.g., [BLMM02].Note that the proalgebraic completion is sometimes also referred to as the proalgebraic hull (e.g., in [Sau16]) or as the Hochschild-Mostow group (in honor of [HM57]).
Definition 2.4.Let F be an (abstract) group.The proalgebraic completion F alg of F is a proalgebraic group equipped with a morphism F → F alg (C) of groups satisfying the following universal property: If G is a proalgebraic and F → G(C) is a morphism of groups, then there exists a unique morphism φ : F alg → G of proalgebraic groups such that For X a finite (!) set and F (X) the (abstract) free group on X, it follows from the universal properties that F (X) alg Γ(X).The proalgebraic completion F alg of F can be constructed as the fundamental group of the neutral tannakian category of all finite dimensional C-linear representations of F .

Differential Galois theory.
Introductions to this topic can be found in [Mag94,vdPS03], [CH11] and [Sau16].We recall the basic definitions and results, introducing our notation for the subsequent sections along the way.
We fix a differential field K with derivation δ : K → K.We assume that the field of constants K δ = {a ∈ K| δ(a) = a} of K is the field C of complex numbers.We are mainly interested in the case when K = C(x) is the rational function field in one variable x and δ = d dx .We consider a family F = (δ(y) = A i y) i∈I of linear differential equations indexed by some set I, where A i ∈ K ni×ni is a square matrix for every i ∈ I.
and L is generated as a field extension of K be all entries of all Y i 's.
The K-subalgebra R of L generated by all of entries of all Y i 's and 1 det(Yi) 's is a K-δ-subalgebra of L and called a Picard-Vessiot ring for F.
For a given family F, a Picard-Vessiot extension exists and is unique up to a K-δ-isomorphism.The differential Galois group G(L/K) of the Picard-Vessiot extension L/K, or of the family F, is the functor T Aut(R ⊗ C T /K ⊗ C T ), from the category of C-algebras to the category groups, where T is considered as a constant differential ring and the automorphisms are required to commute with the derivation.The functor G(L/K) can be represented by a C-algebra, i.e., G(L/K) is a proalgebraic group.
A Picard-Vessiot extension L/K is of finite type if it is the Picard-Vessiot extension for a single differential equation.This is the case if and only of G(L/K) is algebraic.
Since L is the field of fractions of R, any g ∈ G(L/K)(C) extends uniquely to a K-δ-automorphism of L. For a closed subgroup H of G = G(L/K) we set Theorem 2.6 (The differential Galois correspondence).Let L/K be a Picard-Vessiot extension.The assignment M → G(L/M ) defines an inclusion reversing bijection between the set of intermediate differential fields of L/K and the set of closed subgroups of G(L/K).The inverse is given by H → L H .
If M corresponds to H under this bijection, then M/K is Picard-Vessiot if and only if H is normal in G(L/K).Moreover, if this is the case, the restriction morphism An alternative definition of the differential Galois group of a family of linear differential equations can be given via the tannakian formalism ( [DM82,Del90]).As a first step, one has to define the "category of differential equations".This is formalized through the notion of a differential module.A differential module over K is a finite dimensional K-vector space M equipped with an additive map ∂ : M → M such that ∂(am) = δ(a)m + a∂(m) for all a ∈ K and m ∈ M .A morphism of differential modules over K is a K-linear map that commutes with ∂.
To a linear differential equation δ(y) = Ay with A ∈ K n×n , one associates the differential module M A by setting M A = K n and ∂(ξ) = δ(ξ) − Aξ for all ξ ∈ K n .Conversely, if (M, ∂) is a differential module with basis e = (e 1 , . . ., e n ), we can write ∂(e) = e(−A) for a unique matrix Via the choice of the basis e, we can thus associate to M the differential equation δ(y) = Ay.A different choice of basis leads to a Gauge equivalent differential equation.
For a differential module (M, ∂) over K one sets The category of all differential modules over K is a neutral Tannakian category over C. If F = (δ(y) = A i y) i∈I is a family of differential equations over K and M = (M Ai ) i∈I is the corresponding family of differential modules, then the differential Galois group of F is isomorphic to the fundamental group of the neutral Tannakian category M generated by M.
We now consider the global picture.Let (M, ∂) be differential module over C(x).Here, as throughout the paper, the rational function field C(x) is considered as a differential field via the derivation δ = d dx .Let P 1 (C) = C ∪ {∞} denote the Riemann sphere.For every point p ∈ P 1 (C) we have a "local" differential field C(x Note that (C(x), d dx ) is a differential subfield of (C((x − p)), d d(x−p) ) for p ∈ C. Thus, we obtain a differential module M p = M ⊗ C(x) C(x) p over C(x) p for every p ∈ C. For p = ∞ it is not true that (C(x), d dx ) is a differential subfield of (C((x −1 )), d dx −1 ).However, (C(x), −x 2 d dx ) is a differential subfield of (C((x −1 )), d dx −1 ).So we can base change the differential module (M, −x 2 ∂) over (C(x), −x 2 d dx ) to a differential module M ∞ over C(x) ∞ .This awkwardness at infinity is one of the reasons why some authors prefer to work with connections, rather than differential modules.This way one can avoid the a priori choice of a derivation on C(x).However, since the module Ω C(x)/C of differentials of C(x) over C is one dimensional, these two approaches are equivalent.
A point p ∈ P 1 (C) is a singularity of M if the differential module M p over C((t)) is not regular.A point p ∈ P 1 (C) is regular singular for M if the differential module M p over C((t)) is regular singular.Finally, M is called regular singular if every point p ∈ P 1 (C) is regular singular for M .
Consider a differential equation δ(y) = Ay with A ∈ C(x) n×n .A point p ∈ P 1 (C) is a singularity of δ(y) = Ay if it is a singularity of the associated differential module M A .Note that this definition is at odds with the common terminology, referring to the poles of A as the singularities of δ(y) = Ay.If p ∈ C is a singularity of δ(y) = Ay, then p must be a pole of A. However, the converse is not true.A pole of A that is not a singularity of δ(y) = Ay is sometimes called an apparent singularity.The differential equation δ(y) = Ay is called regular singular if the associated differential module M A is regular singular.Fix a proper subset X of C ⊆ P 1 (C).As X is assumed to be a proper subset of C, we can choose a "base point" x 0 ∈ C with x 0 / ∈ X.Let RegSing(C(x), X) denote the category of all regular singular differential modules over C(x) with singularities contained in X ∪ {∞}.This is a Tannakian category over C. A fibre functor ω X,x0 : RegSing(C(x), X) → Vec C , with values in the category Vec C of finite dimensional C-vector spaces, is given by where M x0 is the differential field of germs of meromorphic functions at x 0 .We denote with Aut ⊗ (ω X,x0 ) the proalgebraic group of tensor automorphisms of ω X,x0 .Now assume that X is finite.Let π 1 (C X, x 0 ) be the topological fundamental group of the Riemann sphere with the points X ∪ {∞} removed, with base point x 0 .Then the local solution space (M ⊗ C(x) M x0 ) ∂ is naturally equipped with the monodromy action of π 1 (C X, x 0 ).We denote with Rep(π 1 (C X, x 0 )) the category of finite dimensional C-linear representations of π 1 (C X, x 0 ).The following theorem is sometimes referred to as the (global) Riemann-Hilbert correspondence.The essential surjectivity of the functor in the theorem is also known as the solution of the (weak form of the) Riemann-Hilbert problem.
Theorem 2.7 (Riemann-Hilbert correspondence).Let X ⊆ C be finite.Then the functor is an equivalence of Tannakian categories.
From Theorem 2.7 we immediately obtain: Corollary 2.8.For X ⊆ C finite we have Aut ⊗ (ω X,x0 ) π 1 (C X, x 0 ) alg .In particular, the differential Galois group of the family of all regular singular differential equations over C(x) with singularities in X ∪ {∞} is isomorphic to the free proalgebraic group on X.
Proof.As π 1 (C X, x 0 ) alg is the proalgebraic group of tensor automorphisms of the forgetful functor Rep(π 1 (C X, x 0 )) → Vec C , Theorem 2.7 yields an isomorphism π 1 (C X, x 0 ) alg → Aut ⊗ (ω X,x0 ).The last statement follows because the differential Galois group of the family of all regular singular differential equations with singularities in X ∪ {∞} is isomorphic to Aut ⊗ (ω X,x0 ) and the group π 1 (C X, x 0 ) is free on |X| generators.
We stress the fact that the morphism of groups π 1 (C X, x 0 ) → Aut ⊗ (ω X,x0 )(C) is canonical.If X is a subset of the finite set X ⊆ C, then RegSing(C(x), X ) is a subcategory of RegSing(C(x), X) and ω X ,x0 is the restriction of ω X,x0 to RegSing(C(x), X ).We thus have a morphism Aut ⊗ (ω X,x0 ) → Aut ⊗ (ω X ,x0 ) of proalgebraic groups.As C X ⊆ C X , we also have a morphism π 1 (C X, x 0 ) → π 1 (C X , x 0 ) of groups.The diagram commutes and so also the diagram commutes.
The family of isomorphisms π 1 (C X, x 0 ) alg Aut ⊗ (ω X,x0 ), one for every finite subset X of C not containing x 0 , can thus be seen as defining an isomorphism between two projective systems of proalgebraic groups.The projective limit on the right hand side is lim ), whereas the projective limit lim ← −X π 1 (C X, x 0 ) alg on the left hand side is a projective limit of free proalgebraic groups.The following section provides the necessary tools to show that this limit itself is free.

Projective systems of free groups
Let X be a set and consider the directed set Y of all finite subsets of X ordered by inclusion.For Y ∈ Y let F (Y ) denote the (abstract) free group on Y and for Y ⊆ Y , define a map ϕ Y,Y : and the induced map F (X) → lim ← −Y ∈Y F (Y ) is injective.However, this map need not be surjective.Intuitively, surjectivity fails because F (X) only contains words of finite length, while lim ← −Y ∈Y F (Y ) may contain words of infinite length.For example, if but not in the image of F (X).In the nomenclature of [Hig52] this limit is an unrestricted free product.For more on projective limits of abstracts free groups see [CK12] and [EN13].However, when working with free profinite groups instead of abstract free groups, the above construction leads to a free profinite group.See [RZ10, Cor.3.3.10b)] or [Sza09,Lem. 3.4.10].As we shall now explain, also in the case of free proalgebraic groups, the above construction leads to a free proalgebraic group.
As above, let X be a set and let Y be the directed set of all finite subsets of X.For Y ⊆ Y , by the universal property of Γ(Y ) (Definition 2.1), the map ϕ Y,Y : ) is a projective system of free proalgebraic groups on finite sets.The projective limit is Γ(X).
G to an algebraic group G factors through some Γ(Y ).
To verify the universal property of ι, according to Remark 2.2, it suffices to consider a map ϕ : X → G(C) converging to 1, with G an algebraic group.Then Y 0 = {x ∈ X| ϕ(x) = 1} is finite.Define a morphism ψ : Γ(Y 0 ) → G by ψ(y) = ϕ(y) for y ∈ Y 0 and let φ be the composition We now specialize to the case that X is a proper subset of C. In this case, besides the projective system ((F (Y )) Y ∈Y , (ϕ Y ,Y ) Y ⊇Y ) from the beginning of this section, we can associate another projective system of finite rank (abstract) free groups to X as follows.Fix x 0 ∈ C with x 0 / ∈ X.For Y = {y 1 , . . ., y n } ∈ Y, it is well known (and follows from Van Kampen's theorem) that the fundamental group π 1 (C Y, x 0 ) of C Y with base point x 0 is isomorphic to the free group on n generators.A free set of generators is given by choosing, for each i = 1, . . ., n, a loop based at x 0 that passes once around y i counterclockwise and does not enclose any other points of Y . For ) is a projective system of groups.
Lemma 3.2.The projective systems commutes for every Y ⊇ Y .
Proof.For Y ∈ Y and y ∈ Y , we define a canonical generator at y to be an element of π 1 (C Y, x 0 ) that is the homotopy class of a loop based at x 0 that passes once counterclockwise around y, not enclosing any other points of Y .The following graphics depict two canonical generators at y 1 .
Note that π 1 (C Y, x 0 ) contains only finitely many canonical generators at y. Thus the set

Main result
Throughout Section 4 we will use the following notation.We fix a Picard-Vessiot extension L/C(x) for the family all regular singular differential equations over C(x).For a subset X of C we denote with L X ⊆ L the Picard-Vessiot extension of C(x) for the family of all regular singular differential equations over C(x) with singularities in X ∪ {∞}.So L = L C .We also set Γ X = G(L X /C(x)).With this notation, our goal is to show that Γ X ∼ = Γ(X) for every subset X of C.
We first tackle the case of proper subsets of C. The following proposition generalizes Corollary 2.8 from finite subsets of C to arbitrary proper subsets of C. Proposition 4.1.Let X be a proper subset of C. Then the differential Galois group Γ X of the family of all regular singular differential equations over C(x) with singularities contained in X ∪ {∞}, is isomorphic to the free proalgebraic group Γ(X) on X.
Proof.Fix a base point x 0 ∈ C with x 0 / ∈ X.As Γ X is isomorphic to Aut ⊗ (ω X,x0 ), it suffices to show that Aut ⊗ (ω X,x0 ) is isomorphic to Γ(X).(Note that the isomorphism between Γ X and Aut ⊗ (ω X,x0 ) is arguably not canonical.It depends on an isomorphism between the fibre functor ω X,x0 and the fibre functor defined by L X .) As in Section 3, we consider the directed set Y of all finite subsets of X.Because the category RegSing(C(x), X) is the union of the subcategories RegSing(C(x), Y ), where Y runs through all elements of Y, it is clear that lim alg between the corresponding projective limits.The latter limit is, by Lemma 3.1, isomorphic to Γ(X).
In summary, we have Note that Proposition 4.1 and its proof does not apply to the case X = C of prime interest, because for X = C we cannot choose a base point x 0 / ∈ C X. To accomplish the case X = C, we will use a characterization of free proalgebraic groups in terms embedding problems.
To this end, we need to recall some definitions from [Wib20].Let Γ be a proalgebraic group.An embedding problem for Γ consists of two quotient maps α : G H and β : Γ H of proalgebraic groups.The embedding problem is algebraic if G (and therefore also H) is an algebraic group.The embedding problem is trivial if α is an isomorphism.A solution of the embedding problem is a quotient map φ : Γ is a quotient map.Here i∈I (G H) denotes the fibre product of |I|-copies of G over H.The rank rank(Γ) of a proalgebraic group Γ is defined as the smallest cardinal κ such that Γ can be written as a projective limit of algebraic groups over an directed set of cardinality κ.See [Wib20, Prop.3.1] for other characterizations of the rank.
The following theorem provides a characterization of free proalgebraic groups in terms of algebraic embedding problems.Theorem 4.2.Let Γ be a proalgebraic group with rank(Γ) ≤ |C|.Then Γ is isomorphic to the free proalgebraic group on a set of cardinality |C| if and only if every non-trivial algebraic embedding problem for Γ has |C| independent solutions.Proof.First assume that Γ is isomorphic to Γ(X), where X is a set of cardinality |C|.Since rank(Γ(X)) = |X| = |C| by [Wib20, Cor.3.12], we find rank(Γ) = |C|.Thus the claim follows from Theorem 3.42 (applied with C the formation of all algebraic groups) paired with Definition 3.25 of [Wib20].
Conversely, assume that every non-trivial algebraic embedding problem for Γ has |C| independent solutions.We claim that rank(Γ) = |C|.Consider the embedding problem α : G a → 1, β : Γ → 1 for Γ, where G a is the additive group.By assumption, there exist solutions (φ x ) x∈C such that the induced morphism Γ → G The following lemma is our crutch to go from proper subsets of C to all of C. Lemma 4.3.Let Γ be a proalgebraic group with rank(Γ) ≤ |C| and let X be a set of cardinality |C|.Assume that every quotient map Γ H to an algebraic group H can be factored as Γ Γ(X) H. Then Γ is isomorphic to Γ(X).We are now prepared to prove our main result.Let H be an algebraic group and β : Γ H a quotient map.By the differential Galois correspondence (Theorem 2.6) the extension L ker(β) /C(x) is a Picard-Vessiot extensions with differential Galois group H. Since H is algebraic, L ker(β) /C(x) is a Picard-Vessiot extension of finite type.Therefore, there exists a finite family F of regular singular differential equations over C(x) such that the Picard-Vessiot extension L F of C(x) for F (inside L) contains L ker (β) .Let Y ⊆ C be the finite set of (finite) singularities of the differential equations contained in F. Let X be a proper subset of C with |X| = C and Y ⊆ X (e.g., X = C {x 0 } with x 0 / ∈ Y ).Then

Proof
and so, the quotient map β : Γ H factors as β : Γ Γ X H.As Γ X ∼ = Γ(X) by Proposition 4.1, we see that the condition of Lemma 4.3 is satisfied.
Combining Proposition 4.1 and Theorem 4.4 we obtain: Corollary 4.5.Let X be a subset of C. Then the differential Galois group of the family of all regular singular differential equations with singularities in X ∪ {∞} is isomorphic to the free proalgebraic group on X.
4.1.Open Questions.It seems natural to wonder if Corollary 4.5 remains valid when C is replaced with an algebraically closed field of characteristic zero.Our proof uses transcendental tools, such as the fundamental group, and therefore does not generalize.Even when X is finite, it seems to be unknown if Corollary 4.5 is true for an algebraically closed field of characteristic zero in place of C.

2. 3 .
Regular singular differential equations.For background on regular singular differential equations and the Riemann-Hilbert correspondence see [Sau16, Part 3], [vdPS03, Chapters 5 and 6], [MS16, Part I] and [ABC20, Part III].We first treat the local definitions.Consider a differential module (M, ∂) over the field C((t)) of formal Laurent series in t equipped with the usual derivation δ = d dt .Let C[[t]] ⊆ C((t)) be the differential subring of formal power series.
is a canonical generator at y for every y ∈ Y } is finite.The map ψ Y,Y : π 1 (C Y , x 0 ) → π 1 (C Y, x 0 ) maps a canonical generator at y ∈ Y ⊇ Y either to a canonical generator at y (if y ∈ Y ) or to 1 if y / ∈ Y .For Y, Y ∈ Y with Y ⊆ Y , we can define a map Ψ Y,Y : B Y → B Y , by Ψ Y,Y ((g y ) y ∈Y ) = (ψ Y,Y (g y )) y∈Y .Then (B Y ) Y ∈Y , (Ψ Y,Y ) Y ⊇Y is a projective system of finite sets.Thus the corresponding projective limit is non-empty ([RZ10, Prop.1.1.4]).Let ((g y ) y∈Y ) Y ∈Y be an element of lim ← −Y ∈Y B Y .For Y ∈ Y define α Y : F (Y ) → π 1 (C Y, x 0 ) by α Y (y) = g y for y ∈ Y .Then α Y is an isomorphism and by construction diagram (2) commutes.

Theorem 4. 4 .
The differential Galois group of the family of all regular singular differential equations over C(x) is isomorphic to the free proalgebraic group on a set of cardinality |C|.Proof.Let Γ = Γ C = G(L/C(x)) be the differential Galois group of the family of all regular singular differential equations.By [BHHW21b, Lem.3.3], the rank of Γ is the smallest cardinal number κ such that L/C(x) is a Picard-Vessiot extension for a family of differential equations of cardinality κ.Since the family of all differential equations over C(x) has cardinality |C|, we have rank(Γ) ≤ |C|.