Vik S Kulikov
email: kulikov@mi.ras.ru

V M Kharlamov
email: kharlam@math.unistra.fr

COVERING SEMIGROUPS

We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application, we give new conditions on the ramification type that are sufficient for irreducibility of the Hurwitz spaces, suggest some bounds on the number of irreducibility components under certain more general conditions, and show that the number of irreducible components coincides with the number of topological classes of the coverings if the number of brunch points is big enough.

Introduction

Let f : E → F be a finite morphism between complex non-singular irreducible projective curves. Denote by C(E) and C(F ) the fields of rational functions on E and F , respectively. The morphism f defines a finite extension f * : C(F ) ֒→ C(E) of the field C(F ) (reciprocally, the field extension defines the covering f uniquely up to isomorphisms of coverings over a fixed base). We denote by G the Galois group of the Galois closure of this extension and call it the Galois group of f . Let us fix a point q ∈ F that is not a branch point of f and order the points of E lying over q. We call the morphism f with a fixed ordering of the points of f -1 (q) a marked covering.

Consider the fundamental group π 1 (F \ B, q) of the complement of the branch set B ⊂ F of a marked covering f of degree d = deg f . Then, the ordering of the points of f -1 (q) defines a homomorphism f * : π 1 (F \ B, q) → S d of π 1 (F \ B, q) to the symmetric group S d . Due to irreducibility of E, the image imf * ⊂ S d acts transitively on f -1 (q) and is isomorphic to G, so that we can identify imf * and G and thus fix this embedding G ֒→ S d .

The movement along a standard simple loops γ around branch points b ∈ B the local monodromy f * (γ) ∈ G of f at b. The homotopy class of this standard loop, and hence the local monodromy, are defined by b uniquely only up to conjugation, in π 1 (F \ B, q) and G, respectively. We denote by O ⊂ G the union of the conjugacy classes of all the local monodromies of f and call the pair (G, O) the equipped Galois group associated with f . The collection τ = (τ 1 C 1 , . . . , τ m C m ), where C 1 , . . . , C m list

The first author was partially supported by grants of NSh-4713.2010.1, RFBR 11-01-00185, and by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023. The second author was partially funded by the ANR-09-BLAN-0039-01 grant of Agence Nationale de la Recherche, and is a member of FRG Collaborative Research "Mirror Symmetry & Tropical Geometry" (Award No. 0854989).

all the conjugacy classes included in O and τ i counts the number of branch points of f with the local monodromies belonging to C i , is called the monodromy type of f . The degree d marked coverings of F with Galois group G and monodromy type τ form a so called Hurwitz space HUR d,G,τ (F ) (for precise definitions see subsection 2.7).

In the case F = P 1 , G = S d and O is the set of transpositions, the famous Clebsch -Hurwitz Theorem [START_REF] Clebsch | Zür Theorie der Riemann'schen Fläche[END_REF], [START_REF] Hurwitz | Ueber Riemann'she Flächen mit gegebenen Verweigugspunkten[END_REF] states that HUR d,S d ,τ (P 1 ) consists of a single irreducible component if τ = (nO) with even n 2(d -1) and it is empty otherwise. Generalizations of Clebsch -Hurwitz Theorem were obtained in [START_REF] Berstein | On the classification of generic branched coverings of surfaces[END_REF], [START_REF] Wajnryb | Orbits of Hurwitz action for coverings of a sphere with two special fibres[END_REF], [START_REF] Protopopov | Topological classification of branched coverings of the twodimensional sphere[END_REF], [START_REF] Fried | The inverse Galois problem and rational points on moduli space[END_REF], and [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF] - [START_REF] Vik | Factorizations in finite groups[END_REF]. In particular, Clebsch -Hurwitz Theorem was extended to the following cases: in [START_REF] Berstein | On the classification of generic branched coverings of surfaces[END_REF], if all but one local monodromies are transpositions; in [START_REF] Wajnryb | Orbits of Hurwitz action for coverings of a sphere with two special fibres[END_REF], if all but two local monodromies are transpositions; in [START_REF] Protopopov | Topological classification of branched coverings of the twodimensional sphere[END_REF], if all local monodromies are either transpositions or cyclic permutations of length three; and in [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF], if there are at least 3(d -1) transpositions among the local monodromies.

In [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF], it is proved that for an equipped group (S d , O) such that the first conjugacy class C 1 of O contains an odd permutation leaving fixed at least two elements, the Hurwitz space HUR d,S d ,τ (P 1 ) is irreducible if τ 1 is big enough. On the other hand, the example in [START_REF] Wajnryb | Orbits of Hurwitz action for coverings of a sphere with two special fibres[END_REF] shows that HUR 8,S 8 ,τ (P 1 ) consists of at least two irreducible components if τ = (1C 1 , 1C 2 , 1C 3 ), where C 1 is the conjugacy class of the permutation (1, 2) [START_REF] Clebsch | Zür Theorie der Riemann'schen Fläche[END_REF][START_REF] Fadell | Configuration spaces[END_REF][START_REF] Fox | Covering Spaces with Singularities[END_REF], C 2 is the conjugacy class of (1, 2, 3)(4, 5, 6, 7), and C 3 is the conjugacy class of [START_REF] Berstein | On the classification of generic branched coverings of surfaces[END_REF][START_REF] Fulton | Hurwitz schemes and irreducibility of moduli of algebraic curves[END_REF][START_REF] Clebsch | Zür Theorie der Riemann'schen Fläche[END_REF][START_REF] Fadell | Configuration spaces[END_REF][START_REF] Fox | Covering Spaces with Singularities[END_REF][START_REF] Fox | The braid groups[END_REF][START_REF] Fried | The inverse Galois problem and rational points on moduli space[END_REF]. Articles [START_REF] Fried | The inverse Galois problem and rational points on moduli space[END_REF] and [START_REF] Vik | Factorizations in finite groups[END_REF] are devoted to partial generalizations of Clebsch -Hurwitz Theorem to the case of arbitrary group G. In particular, in [START_REF] Vik | Factorizations in finite groups[END_REF], it was proved that for a fixed equipped finite group (G, O) the number of irreducible components of HUR d,G,τ (P 1 ) (if it is non-empty) does not depend on τ if all τ i are big enough.

For higher genus, the irreducibility of HUR d,S d ,τ (F ) is proved in [START_REF] Graber | A note on Hurwitz schemes of covers of a positive genus curve[END_REF] under hypothesis that n ≥ 2d and all local monodromies are transpositions. After that, this result was improved, first, in [START_REF] Kanev | Irreducibility of Hurwitz spaces[END_REF] where the hypothesis n ≥ 2d was replaced by n ≥ 2d -2, and next, in [START_REF] Vetro | Irreducibility of Hurwitz spaces for coverings with one special fibre[END_REF], where the second hypothesis was replaced by assumption that all but one local monodromies are transpositions. Let us notice that the irreducibility of the quotient of HUR d,S d ,τ (F ) by the action of the mapping class group of F (considered as a real surface) was proved in [START_REF] Berstein | On the classification of generic branched coverings of surfaces[END_REF] under a weaker hypothesis n > d 2 . One of the aims of this article is to extend results of [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF] - [START_REF] Vik | Factorizations in finite groups[END_REF] from F = P 1 to the case of F of arbitrary genus. The approach used there for counting the number of irreducible components of HUR d,G,τ (P 1 ) is based on a systematic work with semigroups over groups; in particular, factorization semigroups S(G, O) with factors belonging to O (cf., subsections 1.1 and 1.2 below) play the crucial role in this study, especially since subsets of elements of type τ of subsemigroup S(G, O) G 1 ⊂ S(G, O) are in a canonical bijection with the sets of irreducible components of the Hurwitz space HUR d,G,τ (P 1 ).

In the present paper, to treat the coverings of projective curves (or, the same, real surfaces) of arbitrary genus we generalize the notion of factorization semigroups to that of semigroups of marked coverings. One can consider different levels of the equivalence relations of coverings and so we introduce, respectively, different species of semigroups of marked coverings. The equivalence relation of the level that is most appropriate to construction of Hurwitz spaces is based essentially on moving of branch points, while that the level most appropriate to topological classification of coverings (like in [START_REF] Berstein | On the classification of generic branched coverings of surfaces[END_REF], for example) includes, in addition, the action on the base of coverings by the whole mapping class group. In particular, considering the coverings up to moving of branch points we introduce a semigroup GS d (G, O) of marked degree d coverings with Galois group G and set of local monodromies O ⊂ G. If we consider the same coverings up to the action of the modular group, then we obtain another semigroup, which we denote by GW S d (G, O). They are related by a natural epimorphism Φ :

GS d (G, O) → GW S d (G, O) of semigroups.
Similar to genus 0 case, certain specific subsemigroups of these two semigroups are in a canonical bijection with the set of irreducible components of the Hurwitz space HUR d,G, (F ) and, respectively, the set of topological classes of marked degree d ramified coverings of F with Galois groups G.

By definition, the monodromy type of an element s = (f : E → F ) belonging to one of these semigroups is the collection τ (s) = (τ 1 C 1 , . . . , τ m C m ) of local monodromies of f . The monodromy type behaves additively and gives a homomorphism from semigroups of coverings to the semigroup Z m 0 . Therefore, for any constant T ∈ N, there appear well defined subsemigroups In [START_REF] Vik | Factorizations in finite groups[END_REF], there was defined an ambiguity index for each equipped finite group (G, O) (see subsection 1.3).

GS d,T (G, O) = {s ∈ GS d (G, O) | τ i (s) T for i = 1, . . . ,

Theorem 2. For each equipped finite group

(G, O), O = C 1 ⊔ • • • ⊔ C m ,
d,G,τ (F ) is equal to a (G,O) if τ i T for all i = 1, . . . , m. If the elements of O k = C 1 ⊔ • • • ⊔ C k for some k < m generate the group G, then there is a constant T ′ such that the number of irreducible components of HUR m d,G,τ (F ) is less or equal to a (G,O k ) if τ i T ′ for all i = 1, . . . , k.
Theorem 3. Let C be the conjugacy class of an odd permutation σ ∈ S d such that σ leaves fixed at least two elements. Then there is a constant N C such that for any projective irreducible non-singular curve F the Hurwitz space

HUR d,S d ,τ (F ) is irreducible if C enters in τ with a factor ≥ N C .
The article consists of two sections. Section 1 is devoted to the algebraic part of the proof. In subsections 1.1 -1.3 we fix notation and recall necessary definitions and results from [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF] - [START_REF] Vik | Factorizations in finite groups[END_REF]. In subsection 1.4 we introduce a notion of admissible subgroups of the automorphism groups of free groups, which is necessary for the next subsection where we define the algebraic coverings semigroups. The remaining subsections of Section 1 contain the proofs of the algebraic part of main results. Section 2 starts from two preliminary subsections where we introduce such auxiliary notions like monodromy encoding of ramified coverings and skeletons of surfaces. In 2.3 -2.6 we introduce a series of geometric covering semigroups and prove comparison statements between algebraic and geometric covering semigroups. In the final subsections, we relate elements of the geometric coverings semigroups with irreducible components of Hurwitz spaces and complete the proofs of main theorems.

Semigroups over groups

1.1. Definition of semigroups over groups. Here, we recall basic definitions and some properties of semigroups over groups with a special emphasis to factorization semigroups (for more details, see [START_REF] Vik | Hurwitz curves[END_REF] - [START_REF] Vik | Factorizations in finite groups[END_REF]).

A collection (S, G, α, ρ), where S is a semigroup, G is a group, and α : S → G, ρ : G → Aut(S) are homomorphisms, is called a semigroup S over a group G if for all s 1 , s 2 ∈ S we have

s 1 • s 2 = ρ(α(s 1 ))(s 2 ) • s 1 = s 2 • λ(α(s 2 ))(s 1 ), (1)
where λ(g) = ρ(g -1 ). Let (S 1 , G 1 , α 1 , ρ 1 ) and (S 2 , G 2 , α 2 , ρ 2 ) be two semigroups over groups G 1 and G 2 , respectively. A pair (h 1 , h 2 ) of homomorphisms h 1 :

S 1 → S 2 and h 2 : G 1 → G 2 is called a homomorphism of semigroups over groups if (i) h 2 • α 1 = α 2 • h 1 , (ii) ρ 2 (h 2 (g))(h 1 (s)) = h 1 (ρ 1 (g)(s)) for all s ∈ S 1 and all g ∈ G 1 . In particular, if G 1 = G 2 = G, then a homomorphism of semigroups ϕ : S 1 → S 2 is said to be defined over G if α 1 (s) = α 2 (ϕ(s)) and ρ 2 (g)(ϕ(s)) = ϕ(ρ 1 (g)(s)) for all s ∈ S 1 and g ∈ G.
1.2. Factorization semigroups. One of the main examples of semigroups over groups is given by, so called, factorization semigroups. To define them, consider an equipped group (G, O), that is, G is a group and O is a subset of G invariant under the inner automorphisms. Here and further on, we assume that:

(i) 1 ∈ O; (ii) O consists of a finite number of conjugacy classes C i of G, O = C 1 ⊔ • • • ⊔ C m ;
(iii) the (linear) ordering of these conjugacy classes is fixed.

By homomorphisms of equipped groups

(G 1 , O 1 ) and (G 2 , O 2 ) we understand homo- morphisms f : G 1 → G 2 such that f (O 1 ) ⊂ O 2 .
The factorization semigroup with factors in O is, by definition, the semigroup S(G, O) generated by alphabet X O = {x g | g ∈ O} and subject to relations

x g 1 • x g 2 = x g 2 • x g g 2 1 , g 1 , g 2 ∈ O, (2)
where g g 2 1 denotes g -1 2 g 1 g 2 . The homomorphism α : S(G, O) → G, given by α(x g ) = g for each x g ∈ X O , is called the product homomorphism. The simultaneous conjugation

x a ∈ X O → x gag -1 ∈ X O defines a homomorphism G → Aut(S(G, O)
), which we denote by ρ. It is easy to see that under such a choice, (S(G, O), G, α, ρ) is a semigroup over G.

Note that there is a well defined length homomorphism of semigroups,

l : S(G, O) → Z 0 = {a ∈ Z | a 0} that is defined by l(x g 1 • . . . • x gn ) = n. Put ρ S = ρ • α, λ S = λ • α, where as above λ(g) = ρ(g -1 ). Claim 1. ([15]) For all s 1 , s 2 ∈ S(G, O) we have s 1 • s 2 = s 2 • λ S (s 2 )(s 1 ) = ρ S (s 1 )(s 2 ) • s 1 .
To each s = x g 1 • . . . • x gn ∈ S(G, O), we associate a subgroup G s of G generated by the images α(x g 1 ) = g 1 , . . . , α(x gn ) = g n of the factors x g 1 , . . . , x gn and denote by G O the subgroup of G generated by the elements of O.

Claim 2. ( [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF]) The subgroup G s of G is well defined, that is, it does not depend on a presentation of s as a product of generators x g i ∈ X O .

For subgroups H and K of a group G, we put

S(G, O) H = {s ∈ S(G, O) | G s = H}, S(G, O) K = {s ∈ S(G, O) | α(s) ∈ K}, and S(G, O) H K = S(G, O) K ∩ S(G, O) H .
(3) if α(s • x g ) belongs to the center Z(G s•xg ) of G s•xg , then s • x g = x g • s, (4) if α(s) = 1, then s • s ′ = s ′ • s for any s ′ ∈ S(G, O).
Claim 3. ( [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF]) For any equipped group (G, O) the semigroup S(G, O) 1 is contained in the center of the semigroup S(G, O) and, in particular, it is commutative.

Note that if g ∈ O is an element of order n, then x n g ∈ S(G, O) 1 . Lemma 1. ([12]) Let s ∈ S(G, O) Z(G O ) and s 1 ∈ S(G, O) G O , where Z(G O ) is the center of G O . Then s • s 1 = ρ(g)(s) • s 1 (3) for all g ∈ G O .
In particular, if s ∈ S(G, O) G , C ⊂ O is a conjugacy class of G, and g n 1 belongs to the center Z(G) of G for certain g 1 ∈ C, then for any g 2 ∈ C we have

x n g 1 • s = x n g 2 • s. (4
O 1 → O 2 that induces an isomorphism of C-graphs between Γ (G 1 ,O 1 ) and Γ (G 2 ,O 2 ) .
To each

C-graph Γ = Γ (G,O) one associates a C-group G Γ = ( G, O) equivalent to (G, O).
Denoting by g → g the bijection O → Õ, we can describe G as the group defined by generators g ∈ Õ and the relations { g -1 3 g 1 g 3 = g 2 if and only if there is an edge e vg 1 ,vg 2 ,g 3 ∈ Γ} (equivalently, g -1 3 g 1 g 3 = g 2 if and only if the relation g -1 3 g 1 g 3 = g 2 holds in G). These generators are called C-generators. They are in one-to-one correspondence with the vertices of Γ due to the composed bijection g → g → v g . Furhermore, two C-generators g 1 and g 2 belong to the same connected component of Γ (G,O) if and only if they are conjugate. The set O of C-generators of the C-group G satisfies the assumptions (i)-(iv). By adding the commutativity relations one shows that the abelianization

H 1 ( G, Z) = G/[ G, G] of G is isomorphic to Z m .
Moreover, due to a fixed ordering of the conjugacy classes { C 1 , . . . , C m } it comes with a natural basis, where in terms of the abelianization homomorphism ab : G → H 1 ( G, Z) the i-th element of the basis is given by ab( g) In [START_REF] Vik | Factorizations in finite groups[END_REF] the following theorems are proved. T for all i = 1, . . . , m there exist a (G,O) elements s 1 , . . . , s a (G,O) ∈ S(G, O) G such that [START_REF] Berstein | On the classification of generic branched coverings of surfaces[END_REF]

with g ∈ C i , 1 ≤ i ≤ m. The homomorphism τ = ab • β -1 * : S(G, O) → Z m 0 is called the type homomor- phism, the image τ (s) = (τ 1 (s), . . . , τ m (s)) ∈ Z m 0 is called the type of s ∈ S(G,
s i = s j for 1 i < j a (G,O) ; (2) τ (s i ) = τ (s 1 ) for 1 i a (G,O) ; (3) α G (s i ) = α G (s 1 ) for 1 i a (G,O) ; (4) if s ∈ S(G, O) G , τ (s) = τ (s 1 ) and α G (s) = α G (s 1 ), then s = s i for some i, 1 i a (G,O) . (5) if s ∈ S(G, O) G and α G (s) = α G (s 1 ), then s = s 1 .
Theorem 5. Let G be a finite group and O ′ ⊂ O be two its equipments such that the elements of

O ′ = C 1 ⊔• • •⊔C k generate the group G. Then there is a constant T = T O ′ such that if for an element s 1 ∈ S(G, O) G the ith type τ i (s 1 ) T for all i = 1, . . . , k, then there are not more than a (G,O ′ ) elements s 1 , . . . , s n ∈ S(G, O) G such that (i) s i = s j for 1 i < j n; (ii) τ (s i ) = τ (s 1 ) for 1 i n; (iii) α G (s i ) = α G (s 1 ) for 1 i n, where a (G,O ′ ) is the ambiguity index of (G, O ′ ).
Theorem 5 is exactly Theorem 7 from [START_REF] Vik | Factorizations in finite groups[END_REF]. Theorem 4, items (1)-( 4), is Theorem 6 from [START_REF] Vik | Factorizations in finite groups[END_REF], while the item ( 5) is a direct consequence of ( [START_REF] Vik | Factorizations in finite groups[END_REF], Theorems 5 and 6) and the following straightforward remark.

G (s i ) ∈ G. Namely, for i = j the element α G (s i )α G (s j ) -1 is a non-trivial element of ker β (G,O) ∩ [ G, G].
1.4. Admissible subgroups of Aut(F n+2p ). In this subsection, in order to introduce a notion of algebraic covering semigroups, we pick out some class of subgroups of the automorphism groups of free groups, called admissible automorphism groups.

Let F n+2p be a free group freely generated by n+2p elements. Let G = {γ 1 , . . . , γ n } ⊂ F n+2p , L = {λ 1 , . . . , λ p } ⊂ F n+2p , and M = {µ 1 , . . . , µ p } ⊂ F n+2p be three ordered sets such that the elements of B = G ∪ L ∪ M generate the group F n+2p .

Let n = (n 1 , . . . , n p+1 ) be an ordered non-negative partition of the number n, that is, an ordered (p + 1)-tuple of non-negative integers whose sum is equal to n:

n = n 1 + • • • + n p+1 , n i ∈ Z 0 .
We put k i = i j=1 n j . Each partition n defines its own ordering on B:

γ 1 , . . . , γ k 1 , λ 1 , µ 1 , . . . , γ k i-1 +1 , . . . , γ k i , λ i , µ i , γ k i +1 , . . . , γ k i+1 , . . . , λ p , µ p , γ kp+1 . . . , γ n
(here the set {γ k i-1 +1 , . . . , γ k i } is empty if n i = 0). Denote by B n the set B with the ordering defined by partition n and call it a frame of F n+2p . In particular, B (n,0,...,0) = {γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p }. The element

∂B n = γ 1 . . . γ k 1 [λ 1 , µ 1 ] . . . γ k i-1 +1 . . . γ k i [λ i , µ i ]γ k i +1 . . . γ k i+1 . . . [λ p , µ p ]γ kp+1 . . . γ n of F n+2p is called the boundary of B n .
Given a set B ′ = G ′ ∪ L ′ ∪ M ′ as above and two adjacent partitions, n ′ = (. . . , n i-1 , n i , n i+1 , n i+2 , . . . ) and n ′′ = (. . . , n i-1 , n i -1, n i+1 + 1, n i+2 , . . . ), we define an elementary frame change h n ′ ,n ′′ that results both in change of the generating set and the ordering. Namely, we put

h n ′ ,n ′′ (B ′ n ′ ) = B ′′ n ′′ , where B ′′ = G ′′ ∪ L ′′ ∪ M ′′ with λ ′′ j = λ ′ j and µ ′′ j = µ ′ j for all j = 1, . . . , p, while γ ′′ j = γ ′ j for j = k i = n 1 + • • • + n i and γ ′′ k i = ([λ ′ i , µ ′ i ]) -1 γ ′ k i [λ ′ i , µ ′ i ]. The inverse change h -1 n ′ ,n ′′ = h n ′′
,n ′ also will be called an elementary frame change. Two frames of F n+2p are said to be strongly equivalent if one of them can be obtained from the other one by a finite sequence of elementary frame changes. Any composition of elementary changes transforming a frame B ′ n ′ into a frame B ′′ n ′′ will also be denoted by h n ′ ,n ′′ . The proof of the following properties is straightforward.

Claim 4. Let B ′

n and B ′′ n be two frames strongly equivalent to a frame B (n,0,...,0) . Then

B ′ n = B ′′ n .
Claim 5. Let B ′ n ′ and B ′′ n ′′ be two strongly equivalent frames. Then ∂B ′ n ′ = ∂B ′′ n ′′ . The group Aut(F n+2p ) naturally acts on the set of frames. This action respects the partition. Given h ∈ Aut(F n+2p ) and a frame B n , we put B ′ = h(B) and h(B n ) = B ′ n . As usually, the orbit of B n under the action of a subgroup H of Aut(F n+2p ) is denoted by HB n . The following Lemma is obvious.

Lemma 2. Let H be a subgroup of Aut(F n+2p ) and B ′ n ′ , B ′′ n ′′ two strongly equivalent frames. Then (i) for any h ∈ H, the frames h(B ′ n ′ ) and h(B ′′ n ′′ ) are strongly equivalent and h(B ′ n ′ ) = h -1 n ′ ,n ′′ (h(B ′′ n ′′ )), (ii) the map h n ′ ,n ′′ : HB ′ n ′ → HB ′′ n ′′ is one-to-one.
Let us fix a frame

B 1 = B (n,0,...,0) = {γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p } and, for each i with 2 ≤ i ≤ p + 1, put B i = h n,n ′ B 1
where n = (n, 0, . . . , 0) and n ′ = (n -1, 0, . . . , 0, 1, 0, . . . , 0) with 1 on the i-th place.

We specify several auxiliary automorphisms of F n+2p . The automorphism σ i with i = 1, . . . , n -1 is defined by its action in the frame B 1 as follows:

σ i (λ j ) = λ j for j = 1, . . . , p, σ i (µ j ) = µ j for j = 1, . . . , p, σ i (γ j ) = γ j for j = i, i + 1, σ i (γ i ) = γ i+1 , σ i (γ i+1 ) = γ γ i+1 i .
The automorphism ξ i,λ with i = 1, . . . , p is defined by its action in the frame B i as follows:

ξ i,λ (λ j,i ) = λ j,i for j = i, ξ i,λ (µ j,i ) = µ j,i for j = 1, . . . , p, ξ i,λ (γ j,i ) = γ j,i for j = n, ξ i,λ (γ n,i ) = γ c 1,i n,i , where c 1,i = λ i,i µ -1 i,i λ -1 i,i γ -1 n,i , ξ i,λ (λ i,i ) = γ n,i λ i,i .
The automorphism ξ i,µ with i = 1, . . . , p is defined by its action in the frame B i as follows:

ξ i,µ (µ j,i ) = µ j,i for j = i, ξ i,µ (λ j,i ) = λ j,i for j = 1, . . . , p, ξ i,µ (γ j,i ) = γ j,i for j = n, ξ i,µ (γ n,i ) = γ c 2,i n,i , where c 2,i = µ i,i λ -1 i,i µ -1 i,i γ n,i , ξ i,µ (µ i,i ) = γ -1 n,i µ i,i .
The automorphism ζ i with i = 1, . . . , p is defined by its action in the frame B i as follows:

ζ i (λ j,i ) = λ j,i for j = i, ζ i (µ j,i ) = µ j,i for j = i, ζ i (γ j,i ) = γ j,i for j = n, ζ i (γ n,i ) = c 3,i , ζ i (λ i,i ) = λ c 3,i i,i , ζ i (µ i,i ) = µ c 3,i i,i , where c 3,i = γ [λ i,i ,µ i,i ] n,i .
For 0 p 1 p, denote by Br n,p 1 the subgroup of the group Aut(F n+2p ) generated by the elements σ 1 , . . . , σ n-1 , ξ

= γ 1 . . . γ n [λ 1 , µ 1 ] . . . [λ p , µ p ] ∈ F n+2p is fixed under the action of Br n,p .
Proof. Obviously, ∂B 1 is fixed under the actions of σ i for i = 1, . . . , n -1 and the actions of ξ 1,λ , ξ 1,µ , ζ 1 , as well as ∂B i with i 2 is fixed under the actions of the automorphisms ξ i,λ , ξ i,µ , ζ i . Now, the statement follows from Claim 5. Let (. . . , γ ′ i , γ ′ i+1 , . . . ) be a part of a frame B ′ n that we assume to be strongly equivalent to B 1 . Denote by σ i,n an automorphism of

F n+2p such that σ i,n (γ ′ i ) = γ ′ i+1 , σ i,n (γ ′ i+1 ) = (γ ′ i ) γ ′ i+1 ,
h ∈ S n such that h(γ i ) is conjugate to γ σ h (i) ; 3) for each h ∈ H n,p it holds the relation h(γ 1 . . . γ n [λ 1 , µ 1 ] . . . [λ p , µ p ]) = γ 1 . . . γ n [λ 1 , µ 1 ] . . . [λ p , µ p ] (here γ 1 . . . γ n = 1 if n = 0). Let us fix a frame B 1 = {γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p } of F n+2p and let f : F n+2p → G be a homomorphism to an equipped group (G, O) such that f (γ i ) ∈ O (we call such an f an equipped homomorphism to (G, O)). Put g i = f (γ i ) for 1 ≤ i ≤ n and a j = f (λ j ), b j = f (µ j ) for 1 ≤ j ≤ p.
To each frame B n strongly equivalent to B 1 , we associate a word

W f,B n in the alphabet Z = Z (G,O) = X O ∪ Y G , where X O = {x g | g ∈ O} is the alphabet we used already in subsection 1.2 and Y G = {y a,b | (a, b) ∈ G 2 }. We put W f,B 1 = x g 1 . . . x gn y a 1 ,b 1 . . . y ap,bp
and then construct the words W f,B n iteratively by elementary moves: in notation used in the definition of an elementary frame change

h n ′ ,n ′′ (B ′ n ′ ) = B ′′ n ′′ , where n ′ = (. . . , n i-1 , n i , n i+1 , n i+2 , . . . ) and n ′′ = (. . . , n i-1 , n i -1, n i+1 + 1, n i+2 , . . . ) are two adjacent partitions, the elementary move W f,B ′ n ′ → W f,B ′′ n ′′ consists in the replacement of two adjacent letters x g ′ k i y a ′ i ,b ′ i in W f,B ′ n ′ by y a ′ i ,b ′ i x ([a i ,b i ]) -1 g ′ k i [a ′ i ,b ′ i ]
(as in the definition of elementary frame changes,

k i = n 1 + • • • + n i ).
Denote by W f (G, O) the set of words which can be obtained from W f,B 1 by finite sequences of elementary moves and put

W n,p (G, O) = f W f (G, O)
, where the union is taken over all equipped homomorphisms f : F n+2p → (G, O). We say that two words are qf -equivalent if they belong to the same set W f (G, O).

Every admissible group H n,p acts on W n,p (G, O). Namely, we put

h(W f,B n ) = W f,B ′ n , B ′ n = h(B n ). In particular, if W f,B n is obtained from W f,B 1 by a finite sequence of elementary moves, then h(W f,B n ) is obtained from h(W f,B 1
) by the same sequence of elementary moves.

Let h be an element of an admissible group H n,p . We have h(γ i ) = γ w i σ h (i) , h(λ i ) = u i , and h(µ i ) = v i , where w i , u i , v i are some elements of F n+2p . Denote by the same letters the words w i = w i (γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p ), u i = u i (γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p ), and v i = v i (γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p ) in letters γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p and their inverses representing these elements in F n+2p . Consider elements g 1 , . . . , g n , a 1 , b 1 , . . . , a p , b p of an equipped group (G, O), where g 1 , . . . , g n ∈ O, and let us substitute g j for γ j , a j for λ j , b j for µ j into the words w i , u i , v i and denote the corresponding elements of G by

w i = w i (g 1 , . . . g n , a 1 , b 1 , . . . , a p , b p ), u i = u i (g 1 , . . . g n , a 1 , b 1 , . . . , a p , b p ), and v i = v i (g 1 , . . . g n , a 1 , b 1 , . . . , a p , b p ). Denote by g 1 , . . . , g n , a 1 , b 1 , . . . , a p , b p a sub- group of G generated by the elements g 1 , . . . , g n , a 1 , b 1 , . . . , a p , b p ∈ G.
Claim 7. In notations and assumptions used above, we have All the covering semigroups considered below are factor semigroups of F S(G, O). In particular, this is the case of what we call the quasi-free algebraic covering semigroup qF S(G, O) that we define as a semigroup generated by the alphabet Z and subject to relations

g 1 , . . . g n , a 1 , b 1 , . . . , a p , b p = g w 1 σ h (1) , . . . , g wn σ h (n) , u 1 , v 1 , • • • , u p , v p . Proof. It suffices to note that the subgroup g 1 , . . . g n , a 1 , b 1 , . . . , a p , b p of G is the image in G of the group F n+2p under the homomorphism f : F n+2p → G given by f (γ i ) = g i , f (λ j ) = a j ,
x g • y a,b = y a,b • x g [a,b] , g ∈ O, a, b ∈ G (5)
(in other words, the elements of qF S(G, O) are the sets of qf -equivalent words (see subsection 1.4)).

We follow notation of Subsection 1.4. Let H = {H n,p } {n 0,p 0} be a collection of automorphism groups that satisfy conditions 2), 3) from the definition of admissible automorphism groups. We associate with each h ∈ H n,p a set R h of relations

x g 1 • . . . • x gn • y a 1 ,b 1 • . . . • y ap,bp = x g w 1 σ h (1) • . . . • x g wn σ h (n) • y u 1 ,v 1 • . . . . • y u p ,vp (6)
taken over all (g 1 , . . .

x g 1 • x g 2 = x g 2 • x g g 2 1 ( 7
)
for any x g 1 , x g 2 ∈ X O , and

x g • y a,b = y a,b • x g [a,b] , (8)
x g • y a,b = x g c 1 • y ga,b , c 1 = ab -1 a -1 g -1 , (9)
y a,b • x g = y a,g -1 b • x g c 2 , c 2 = ba -1 b -1 g, (10)
x g • y a,b = x g [a,b] • y a g [a,b] ,b g [a,b] (11)
for any x g ∈ X O and any y a,b ∈ Y G .

Proof. Follows from Claims 5, 6 and Lemmas 2, 3.

Since every admissible automorphism group H n,p contains the group Br n,p , for any collection H of admissible automorphism groups there is a natural epimorphism r H-equiv : S(G, O) → S H-equiv (G, O) of semigroups.

The semigroup S(S d , S d \{1}) that we denote by V S d will be called a strong algebraic versal degree d covering semigroup. Note that an embedding i :

G ֒→ S d of a group G into S d induces the semigroup embedding of S(G, O) into V S d . Claim 8. The map α : Z → G given by α(x g ) = g for x g ∈ X O and α(y a,b ) = [a, b] for y a,b ∈ Y G defines a homomorphism S H-equiv (G, O) → G.
Proof. Straightforward inspection of relations ( 5) and [START_REF] Fox | The braid groups[END_REF] shows that for each of these relations the product of the images of the left-side factors is equal in G to the product of the right-side factors.

Further on we denote this homomorphism S H-equiv (G, O) → G by α G,H-equiv , or simply α G , and call it the product homomorphism.

The action ρ of the group G on the set Z, given by

x g 1 ∈ X O → ρ(g)(x g 1 ) = x gg 1 g -1 ∈ X O , y a,b ∈ Y G → ρ(g)(y a,b ) = y gag -1 ,gbg -1 ∈ Y G , defines a homomorphism ρ S : G → Aut(S(G, O))
and homomorphisms ρ H-equiv : G → Aut(S H-equiv (G, O)). Obviously, these actions are compatible with the homomorphism r H-equiv .

If it does not lead to a confusion, we replace the notation S H-equiv (G, O) by S(G, O) and then denote the both homomorphisms ρ S and ρ H-equiv simply by ρ. The action ρ(g) on S(G, O) is called the simultaneous conjugation by g ∈ G. Put λ(g) = ρ(g -1 ) and

λ S = λ • α G , ρ S = ρ • α G .
Whatever is an admissible covering semigroup S H-equiv (G, O) = S(G, O), the collection (S(G, O), G, α G , ρ) is a semigroup over the group G and the embedding i : X O ֒→ Z defines an embedding i * : S(G, O) ֒→ S(G, O), which is a semigroup homomorphism over G. Note also that epimorphisms r H-equiv : S(G, O) → S H-equiv (G, O) are also semigroup homomorphisms over G.

Using relations ( 7) -( 11), any element s ∈ S(G, O) can be written in a so called reduced form, s = s 1 • s 2 , where s 1 ∈ S(G, O) and s 2 = y a 1 ,b 1 • . . . • y ap,bp for some a 1 , b 1 , . . . , a p , b p ∈ G. We put τ (s) = τ (s 1 ) and g(s) = p and call them type of s and genus of s, respectively. It is easy to see that the type and the genus of s ∈ S(G, O) are well defined, that is, τ (s) and g(s) do not depend on the reduction of s to a reduced form s = s 1 • s 2 .

Let As in the case of factorization semigroups, for subgroups H 1 and H 2 of a group G, we put S(G, O) Proof. Obvious. 1.6. Solvability of some equations in strong covering semigroups. In this subsection, we will assume that in S(G, O) there is the unity, 1 ∈ S(G, O) (we add it into S(G, O)).

s 1 • s 2 with s 1 = x g 1 • . . . • x
H 1 = {s ∈ S(G, O) | G s = H 1 }, S(G, O) H 2 = {s ∈ S(G, O) | α(s) ∈ H 2 }, and
S(G, O) H 2 H 1 = S(G, O) H 2 ∩ S(G, O) H 1 . Let G Γ = ( G,
Let s 1 , s 2 , s 3 ∈ S(G, O). We say that an equation

s 1 = s 2 • z • s 3 (12) is solvable in S(G, O) ⊂ S(G, O) if there is an element s ∈ S(G, O) such that s 1 = s 2 • s • s 3 .
Note that s is a solution of equation ( 12) if and only if the following holds: if we write s, s 1 , s 2 , s 3 as products of generators of S(G, O), then there is a finite sequence of elementary transformations transforming the factorization of s 1 into the factorization of s 2 • s • s 3 , here an elementary transformation means a change of some pair of two neighboring factors into another one according to the one of relations ( 7) -( 11) (reading either from the left to the right or from the right to the left).

Consider four elements s 1 , . . . , s 4 of S(G, O) and let us fix their presentations as products of generators of S(G, O). Let S be a subset of S(G, O) the elements of which have a fixed type. We say that the equations

s 1 • s • s 2 = s 3 • z • s 4 , (13)
where s ∈ S, are universally solvable if, first, there is a solution s ∈ S of (13) for any s and, second, there is a finite sequence of elementary transformations which satisfy the following property: for any presentation of s as the product of generators of S(G, O) there is a presentation of a solution s as the product of generators such that this sequence of elementary transformations transforms the factorization of s 1 • s • s 2 into the one of s 3 • s • s 4 . The element s together with its factorization mentioned above will be called the universal solution (for s and its factorization) of equation ( 13).

y a,b • s = y a,(α(s)) -1 b • z, (14)
s • y a,b = z • y a,(α(s) -1 ) [a,b] b ( (15)
) 16
is universally solvable in S(G, O).

Proof. Let s = x g 1 • . . . • x gn . If n = 1
, then a universal solvability of equation ( 14) follows from relation [START_REF] Hurwitz | Ueber Riemann'she Flächen mit gegebenen Verweigugspunkten[END_REF]. Assume that equation ( 14) is universally solvable for any s of length n -1 and let us write an element s of length n in the form: s = s 1 • x gn . We have

s • y a,b = s 1 • x gn • y a,b = s 1 • z 1 • y gna,b = ρ S (s 1 )(z 1 ) • s 1 • y gna,b ,
where z 1 is an universal solution of equation

x gn • y a,b = z • y gna,b .
By assumption, for some z 2 we have:

s 1 • y gna,b = z 2 • y α(s 1 )ga,b . Therefore s • y a,b = ρ S (s 1 )(z 1 ) • s 1 • y gna,b = ρ S (s 1 )(z 1 ) • z 2 • y α(s 1 )gna,b = (ρ S (s 1 )(z 1 ) • z 2 ) • y α(s)a,b ,
that is, equation ( 14) is universally solvable always. The proof of universal solvability of equation ( 15) is similar to one for equation [START_REF] Vik | Factorizations in finite groups[END_REF]. Only we must use relation [START_REF] Kanev | Irreducibility of Hurwitz spaces[END_REF] instead of relation [START_REF] Hurwitz | Ueber Riemann'she Flächen mit gegebenen Verweigugspunkten[END_REF].

To prove the universal solvability of equation ( 16), note that α(λ([a, b])(s)) = (α(s)) [a,b] . Therefore, by relation (8), we have

s • y a,b = y a,b • λ([a, b])(s) = y a,(α(s) -1 ) [a,b] b • z 1 , where z 1 is a universal solution of equation y a,b • λ([a, b])(s) = y a,(α(s) -1 ) [a,b] b • z. Now to
prove the universal solvability of equation ( 16), it suffices several times to use relation [START_REF] Graber | A note on Hurwitz schemes of covers of a positive genus curve[END_REF].

The following proposition is a generalization of Main Lemma 2.1 in [START_REF] Kanev | Irreducibility of Hurwitz spaces[END_REF]. and for any s g -1 ∈ S(G, O) such that α(s g -1 ) = g -1 the following equation

x g 1 • . . . • x gn • x g • s g -1 • y a 1 ,b 1 • . . . • y a k ,b k = x g 1 • . . . • x gn • x g h • z • y a 1 ,b 1 • . . . • y a k ,b k is solvable in S(G, O). Proof. For s 1 , s 2 ∈ S(G, O) denote by G(s 1 , s 2 ) the subset of G such that h ∈ G(s 1 , s 2 )
if and only if the equations

s 1 • x g • s g -1 • s 2 = s 1 • x g h • z • s 2 (17)
are universally solvable in S(G, O) for each g ∈ O ⊂ G and any s g -1 ∈ S(G, O) of fixed type and such that α(s g -1 ) = g -1 . Note that if z 1 is a solution of equation ( 17), then α(z 1 ) = (g h ) -1 .

Claim 11. Let O ⊂ G be a finite set. Then for any

s 1 , s 2 ∈ S(G, O) the set G(s 1 , s 2 ) is a subgroup of G. Proof. Let us show that if h ∈ G(s 1 , s 2 ), then h -1 ∈ G(s 1 , s 2 ).
Let z 1 be an universal solution of equation ( 17). If we apply the inverse sequence of the sequence of elementary transformations giving the universe resolution of equation ( 17), then it is easy to see that s g -1 is the universal solution of the equation

s 1 • x g 1 • z 1 • s 2 = s 1 • x g h 1 1 • z • s 2 ,
where g 1 = g h . For any h ∈ G the conjugation of the elements of S(G, O) by h is a bijection, and for each g ∈ G the set of different factorizations of elements s ∈ S(G, O) of fixed type and such that α

(s) = g -1 is finite. Therefore h -1 ∈ G(s 1 , s 2 ) if h ∈ G(s 1 , s 2 ).
Let us show that if h 1 , h 2 ∈ G(s 1 , s 2 ), then h 1 h 2 ∈ G(s 1 , s 2 ). It is easy to see that if z 1 is an universal solution of the equation

s 1 • x g • s g -1 • s 2 = s 1 • x g h 1 • z • s 2
and if z 2 is an universal solution of the equation

s 1 • x g h 1 • z 1 • s 2 = s 1 • x (g h 1 ) h 2 • z • s 2 ,
then z 2 is an universal solution of the equation

s 1 • x g • s g -1 • s 2 = s 1 • x g h 1 h 2 • z • s 2 . Claim 12. For any s 1 = s ′ 1 • s ′′ 1 , s 2 = s ′ 2 • s ′′ 2 ∈ S(G, O), we have G(s ′′ 1 , s ′ 2 ) ⊂ G(s 1 , s 2 ). Proof. Obvious. Claim 13. For any h 1 , . . . , h n ∈ O, we have h 1 , . . . , h n ⊂ G(x h 1 • . . . . • x hn , 1).
Proof. It easily follows from Claim 11 and from the following equalities:

x h • x g • s g -1 = x g (h -1 ) • ρ(h)(s g -1 ) • x h = x h • x g (h -1 ) • ρ(h)(s g -1 ), since x g (h -1 ) • ρ(h)(s g -1 ) • x h = ρ S (x g (h -1 ) • ρ(h)(s g -1 ))(x h ) • x g (h -1 ) • ρ(h)(s g -1 ) and α(x g (h -1 ) • ρ(h)(s g -1 )) = 1. Claim 14. For any a, b ∈ G, we have ab -1 a -1 ∈ G(1, y a,b ).
Proof. By Claim 10, we have

x g • s g -1 • y a,b = x g • z 1 • y g -1 a,b = ρ(g)(z 1 ) • x g • y g -1 a,b ,
where z 1 is an universal solution of equation s g -1 • y a,b = z • y g -1 a,b . By relation [START_REF] Hurwitz | Ueber Riemann'she Flächen mit gegebenen Verweigugspunkten[END_REF],

x g • y g -1 a,b = x gg -1 aba -1 ggg -1 ab -1 a -1 • y a,b = x g ab -1 a -1 • y a,b . Therefore x g • s g -1 • y a,b = ρ(g)(z 1 ) • x g • y g -1 a,b = ρ(g)(z 1 ) • x g ab -1 a -1 • y a,b , that is, ab -1 a -1 ∈ G(1, y a,b ). Claim 15. For any a, b ∈ G, we have aba -1 b -1 a -1 ∈ G(1, y a,b ).
Proof. Applying ln(x g • s g -1 ) times relation [START_REF] Graber | A note on Hurwitz schemes of covers of a positive genus curve[END_REF] and after that applying ln(s g -1 ) times relation [START_REF] Fried | The inverse Galois problem and rational points on moduli space[END_REF], we have

x g • s g -1 • y a,b = y a,b • x g [a,b] • z 1 = y a,b • z 2 • x g [a,b] ,
where

z 1 = λ([a, b])(s g -1 ) and z 2 = ρ(g [a,b] )(z 1 ). It is easy to see that α(z 2 ) = (g [a,b] ) -1 .
By Claim 10 and relation [START_REF] Kanev | Irreducibility of Hurwitz spaces[END_REF],

y a,b • z 2 • x g [a,b] = z 3 • y a,g [a,b] b • x g [a,b] = z 3 • y a,b • x g [a,b]ba -1 b -1 , where z 3 is an universal solution of equation y a,b • z 2 = z • y a,g [a,b] b .
Applying relation [START_REF] Graber | A note on Hurwitz schemes of covers of a positive genus curve[END_REF], we obtain

z 3 • y a,b • x g [a,b]ba -1 b -1 = z 3 • x g [a,b]ba -1 b -1 [b,a] • y a,b = z 3 • x g aba -1 b -1 a -1 • y a,b
and to complete the proof it suffices to use the relation Proof. By Claims 14 and 15, the elements ab -1 a -1 and aba

z 3 • x g aba -1 b -1 a -1 = x g aba -1 b -1 a -1 • λ(g aba -1 b -1 a -1 )(z 3 ).
-1 b -1 a -1 belong to G(1, y a,b ). It follows from Claim 11 that (ab -1 a -1 )(aba -1 b -1 a -1 ) = (ab) -1 ∈ G(1,
a 1 , b 1 , . . . , a k , b k ⊂ G(1, y a 1 ,b 1 • . . . • y a k ,b k ).
Proof. Let us use induction on k. In the case k = 1, it is Claim 16. Assume that for some k -1 Claim is true. Therefore, by Claim 12,

a 1 , b 1 , . . . , a k-1 , b k-1 ⊂ G(1, y a 1 ,b 1 • . . . • y a k ,b k ).
Denote by

u t = [a 1 , b 1 ] . . . [a t , b t ].
We have

x g • s g -1 • y a 1 ,b 1 • . . . • y a k-1 ,b k-1 • y a k ,b k = y a 1 ,b 1 • . . . • y a k-1 ,b k-1 • x g u k-1 • λ(u k-1 )(s g -1 ) • y a k ,b k .
Denote by z 1 = λ(u k-1 )(s g -1 ) and let z c (see Claim 16) be an universal solution of equation

x g u k-1 • z 1 • y a k ,b k = x g u k-1 c • z c • y a k ,b k , where c = a k or b k . We have x g • s g -1 • y a 1 ,b 1 • . . . • y a k ,b k = y a 1 ,b 1 • . . . • y a k-1 ,b k-1 • x g u k-1 c • z c • y a k ,b k = x g c u k-1 • λ(u k-1 )(z c ) • y a 1 ,b 1 • . . . • y a k-1 ,b k-1 • y a k ,b k . Now, since by assumption, u k-1 ∈ G(1, y a 1 ,b 1 • . . . • y a k ,b k ), we obtain that the element c ∈ G(1, y a 1 ,b 1 • . . . • y a k ,b k ).
Now the proof of Proposition 4 easily follows from Claims 11 -17.

Proposition 5. Let (G, O) be an equipped finite group such that the elements of

O = C 1 ⊔ • • • ⊔ C m generate the group G.
Denote by n i the number of elements of the conjugacy class C i and by p i the order of elements of C i . Let s 1 ∈ S(G, O) be such that τ i (s 1 ) > n i p i for all i, 1 i m, and

s 2 = y a 1 ,b 1 • . . . • y a k ,b k be such that s 1 • s 2 ∈ S(G, O) G . Then the equation s 1 • s 2 = z • (y 1,1 ) k is solvable in S(G, O).
Proof. Let a 1 = g -1 1 . . . g -1 n , where g 1 , . . . , g n ∈ O, and let

s 1 = x h 1 • . . . . • x h N . Let g 1 ∈ C i . Since τ i (s 1 ) > n i p i ,
among the factors x h 1 , . . . , x h N there are at least p i + 1 factors with the same h j ∈ C i . Moving p i of these factors to the right (using relation ( 7)), we obtain that

s 1 • s 2 = s ′ 1 • x h j • (x h j ) p i -1 • s 2 for some s ′ 1 ∈ S(G, O) such that s ′ 1 • s 2 ∈ S(G, O) G . Applying Proposition 4, we have s ′ 1 • x h j • (x h j ) p i -1 • s 2 = s ′ 1 • s ′ • x g 1 • s 2 for some s ′ ∈ S(G, O) such that τ (s ′ ) = τ ((x h j ) p i -1 ) and α(s ′ ) = g -1
1 . By relation ( 9), we have

s ′ 1 • s ′ • x g 1 • s 2 = s ′ 1 • s ′ • x g 1 • y a 1 ,b 1 • . . . • y a k ,b k = s ′ 1 • s ′ • x g ′ 1 • y a ′ 1 ,b 1 • . . . • y a k ,b k , where g ′ 1 is an element conjugate to g 1 and a ′ 1 = g -1 2 . . . g -1 n . Note that s 1 = s ′ 1 • s ′ • x g ′ 1 and s 2 = y a ′ 1 ,b 1 • . . . • y a k ,b
k satisfy all conditions of Proposition 5. Therefore, by induction on n, we obtain that

s 1 • s 2 = s 1 • (y 1,b 1 • y a 2 ,b 2 • . . . • y a k ,b k ) for some s 1 which together with s 2 = y 1,b 1 • y a 2 ,b 2 • . . . • y a k ,b k satisfies all conditions of Proposition 5.
The same arguments (only instead of relation ( 9) we must use relation ( 10)) give that In [START_REF] Vik | Factorizations in finite groups[END_REF], this problem was solved in the case of p = 0 and t = (t 1 , . . . , t m ) such that all t i are big enough (see Theorems 4 and 5). In this subsection we generalize these results to the case of arbitrary genus, namely, we prove

s 1 • s 2 = s 1 • (y 1,1 • y a 2 ,b 2 • . . . • y a k ,b k ) for some
s i = s j for 1 i < j a (G,O) ; (2) τ (s i ) = τ (s 1 ) for 1 i a (G,O) ; (3) g(s i ) = g(s 1 ) for 1 i a (G,O) ; (4) α G (s i ) = α G (s 1 ) for 1 i a (G,O) ; (5) if s ∈ S(G, O) G is such that τ (s) = τ (s 1 ), g(s) = g(s 1
), and α G (s) = α G (s 1 ), then s = s i for some i, 1 i a (G,O) .

Proof. Let p be the genus of s 1 and T be a constant the existence of which is claimed in Theorem 4. Without loss of generality, we can assume that T > max 1 i m n i p i , where n i is the number of elements of the conjugacy class C i and p i is the order of elements of C i . By Proposition 5 (and since r H-equiv is an epimorphism), the element s 1 can be written in the form s 1 = s 1 • (y

i = s i • (y 1,1 ) p , 1 i a (G,O)
. By Proposition 5 and Theorem 4, they satisfy conditions (2) -( 5) of Theorem 6. Let us show that they also satisfy condition (1) of Theorem 6. Assume that for some i = j we have s i = s j , that is, if we write s i and s j as products of generators x g , g ∈ O, then there is a finite sequens Tr of elementary transformations (see subsection 1.6) transforming the factorization of s i into the factorization of s j . By Claim 9, the selected factorizations alow us to lift the element s i into S( G, O) = S H-equiv ( G, O) (that is, to consider an element s i ∈ S( G, O) with the same factorization as the one of s i ), where G Γ = ( G, O) is the C-group equivalent to (G, O). Let us apply the same sequence Tr of elementary transformations to the element s i . As a result we obtain an element

s j = s j • (y a 1 ,b 1 • . . . • y ap,bp ) = s i such that a l , b l ∈ ker β (G,O) ⊂ Z( G) for 1 l p.
But it is impossible, since in this case we have α G (y a l ,b l ) = 1 for 1 l p and therefore by Remark 1, we must have α G ( s i ) = α G ( s j ).

Theorem 7. Let G be a finite group and O ′ ⊂ O be two its equipments such that the elements of

O ′ = C 1 ⊔ • • • ⊔ C k generate the group G. Then there is a constant T = T O ′ such that if for an element s 1 ∈ S(G, O) G its i-th type τ i (s 1 )
T for all i = 1, . . . , k, then there are not more than

a (G,O ′ ) elements s 1 , . . . , s n ∈ S(G, O) G such that for 1 i < j n (i) s i = s j ; (ii) τ (s i ) = τ (s 1 ); (iii) α G (s i ) = α G (s 1 ), (iv) g(s i ) = g(s 1 ), where a (G,O ′ ) is the ambiguity index of (G, O ′ ).
Proof. It is similar to the proof of Theorem 6.

Corollary 1. Let C 1 be a conjugacy class of an odd permutation σ 1 ∈ S d such that σ 1 leaves fixed at least two elements. Then in the case when

C 1 is contained in an equipment O = C 1 ⊔ • • • ⊔ C m of S d ,
there is a constant T = T C 1 such that for any σ ∈ S d , any fixed integer p 0, and any t = (t 1 , . . . , t m ) ∈ Z m 0 such that t 1 T the equation α S d (z) = σ (18) has in each covering semigroup S(S d , O) at most one solution s satisfying conditions g(s) = p and τ (s) = t. Under assumption t 1 T , the existence of solution of equation [START_REF] Wajnryb | Orbits of Hurwitz action for coverings of a sphere with two special fibres[END_REF] does not depend on p and depends only on t.

Proof. It follows from Theorem 7 and the main result of [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF].

Let S(G, O) be an admissible covering semigroup over an equipped finite group

(G, O), O = C 1 ⊔ • • • ⊔ C m , and let T = (t 1 , . . . , t m ) ∈ Z o . Denote by S(G, O) T = {s ∈ S(G, O) | τ i (s) t i } a subsemigroup of S(G, O). By Theorem 6, we have Corollary 2. For any equipped finite group (G, O), O = C 1 ⊔ • • • ⊔ C m , there is a constant T = (t 1 , . . . , t m ) ∈ Z o such that the restrictions r H-equiv : S(G, O) G T → S H-equiv (G, O) G T of the epimorphisms r H-equiv : S(G, O) → S H-equiv (G, O) to the subsemigroup S(G, O) G
T are isomorphisms for any set R H of admissible relations.

Geometric semigroups of coverings

2.1. Monodromy encoding of ramified coverings. To describe ramified coverings of a given connected manifold M (in most cases, M will be a connected compact oriented surface with one hole and a base point on the boundary) we use a traditional monodromy encoding of non-ramified coverings and the unicity of manifold ramified completions.

Namely, given a non-ramified, possibly disconnected, degree d covering π : M → M over a connected manifold M with a base point q ∈ M, the lifts of a loop at q form a set of d paths in M starting each at a different point of π -1 (q) and, thus, they give rise to a permutation of the set π -1 (q). This permutation depends only on the homotopy type of the loop and in this way one obtains an encoding of the covering by a homomorphism from π 1 (M, q) to the permutation group of π -1 (q). In particular, if the covering π is equipped with a marking ν :

I d = [1, . . . , d] → π -1 (q) it gives a well defined homomorphism π 1 (M, q) → S d .
The foliowing Proposition is well known and straightforward.

Proposition 6. Two non-ramified marked coverings over the same based space (M, q) are isomorphic as marked coverings if and only if they define the same homomorphism π 1 (M, q) → S d . If an isomorphism exists, it is unique; equivalently, marked coverings have no non-trivial automorphisms. Each homomorphism π 1 (M, q) → S d corresponds to a certain non-ramified marked degree d covering of M. The covering space is connected if and only if the action of π 1 (M, q) on π -1 (q) is transitive. The orbits of the action correspond canonically to connected components of the covering space.

Manifold completions of non-ramified coverings by ramified ones are most transparent in low dimensions.

The following result is also well known and straightforward; it can be found, for example, in [START_REF] Fox | Covering Spaces with Singularities[END_REF]. Proposition 7. Let (M, q) be a based two-dimensional manifold and B = {P 1 , . . . , P n } a finite subset of M disjoint from q and ∂M. Then each marked non-ramified covering of (U, q) with U = M \ B has one and only one ramified completion M → M.

2.2.

Surfaces with a hole and their skeletons. Our main building blocks are connected compact oriented surfaces (2-dimensional manifolds) with one hole and one marked point on the boundary. We equip them, in addition, with a semi-skeleton, skeleton, or caudate skeleton.

We define a semi-skeleton of a connected compact oriented 2-dimensional manifold (with one hole) to be the union of disjoint embedded bouquets of two oriented circles with a property that the complement of the union is homeomorphic to a punctured disc. Clearly, the number of connected components of a semi-skeleton is equal to the genus of the surface. We distinguish the two circles of a bouquet that represents a connected component of a semi-skeleton by means of intersection index, namely, we speak on λ-and µ-circles in a bouquet C λ ∨ C µ by respecting the convention that C λ • C µ = -1 (and not 1).

Given such a triple (F, q, S ∞ ), where S ∞ is a semi-skeleton of a surface F with a fixed point q ∈ ∂F , we can represent it by an open-eyes plane diagram, that is to draw a disc with a marked point on its boundary and p = g(F ) holes inside the disc, and trace the standard 4-gone identification scheme on the boundary of each hole, see the left drawing on Figure 1; the orientation of F should be induced by the counter-clock wise orientation of the disc. When it happens to be more convenient and transparent, we use also another, equivalent, presentation and draw a disc with "pince-nez", that is p pairs of holes with "bridges", see the right drawing on Figure 1 (there, the λ-circles are a and c, while b and d are the µ-circles).

Open-eyes plane diagrams of a given triple (F, q, S ∞ ) are defined up to orientation preserving (stratified) homeomorphisms of the disc respecting the marked point and the orientation of the boundary identification strata. Converse statement is also true, an open-eyes plane diagram defines the triple (F, q, S ∞ ) up to orientation preserving homeomorphism (of triples). A similar statement is also true for diagrams with pincenez, but as to the former one, one should take also into account the possibility for each handle to replace its λ-pince-nez presentation by the µ-pince-nez one, and vice versa.

A skeleton of a genus p connected compact oriented 2-dimensional manifold with one hole is, by definition, a semi-skeleton enhanced by a system of pathes that join the marked point q ∈ ∂F with the components of the semi-skeleton, the pathes are called strings, they are taken disjoint and each of the p strings is chosen in such a way that in the disc model with p holes the string riches its hole at the vertex with outgoing λ-and µ-edges. Now, let us assume that F is equipped with a finite subset B ⊂ F \ ∂F (later on, such a subset B is appearing as the branch locus of a finite cover). In such a case, by a caudate skeleton we understand a triple (F, q, S cdt ) where S cdt is a skeleton disjoint from B and extended by a system of tails, that is a collection of n = |B| simple paths connecting the points of B one-by-one with q, the tails being chosen disjoint from each other and from the skeleton. In particular, S cdt is homeomorphic to the wedge sum of a skeleton with n intervals.

The above notion of open-eyes plane diagrams extends to triples (F, q, S), where S is either a skeleton or caudate skeleton of F . These diagrams consist of p = g(F ) holes or pince-nez in a disc, a marked point on the boundary of the disc, and a system of strings, which is enhanced by a system of tails in the case of caudate skeletons, see Figure 2. Open-eyes plane diagrams of surfaces with a skeleton or, respectively, caudate skeleton are defined up to isotopies; and conversely the triple (F, q, S) is defined up to orientation preserving homeomorphisms of triples by its open-eyes plane diagram. In the case of skeletons, and up to isotopy, for a given genus p there is one and only one open-eyes plane diagram. We denote this diagram by ∆ p and write (Π p , q, Σ p ) to denote the triple that this diagram defines.

If p = 0, then ∆ 0 = Π 0 is just the standard disc and Σ 0 is reduced to q (the marked point on the boundary of the disc).

If p > 0, then the skeleton Σ p contains a non-trivial semi-skeleton, which we denote by Σ ∞ p . A choice of a skeleton induces in a natural way an ordering on the set of the components of the semi-skeleton. Namely, we fix the ordering induced by the counterclockwise order on the strings of the skeleton, see Figure 3, and denote the strings of Σ p by T 1 , . . . , T p following this, counter-clockwise, order. Note that an ordering of the components of Σ ∞ p is equivalent to a choice of the skeleton up to isotopy.

a 2 b 2 a -1 2 b -1 2 a 2 b 2 a 1 b 1 a -1 1 b -1 1 b 1 a 1 t 1 t 2 t 1 t 2
Note also that a skeleton S being given, one has a canonical choice of geometric free generators λ 1 , µ 1 , . . . , λ p , µ p of the fundamental group π 1 (F, q), where λ i , 1 ≤ i ≤ p, (respectively, µ i ) are represented by the loops T i ⋆C λ,i ⋆T -1 i (respectively, T i ⋆C µ,i ⋆T -1 i ); the numbering respects the above ordering of the strings.

For given genus p ≥ 1 and number n ≥ 1 of tails, the number of isotopy classes of plane diagrams of surfaces with caudate skeletons is greater than 1 and equal to the binomial coefficient C n p+n . Indeed, similar to the case of skeletons, a choice of a caudate skeleton induces a counter-clock wise ordering on the the set of tails and strings (or, equivalently, on the set that consist of points of B and the connected components of the semi-skeleton). Conversely, the counter clock-wise ordering of the set of tails and strings determines the diagram up to isotopy.

Thus, a caudate skeleton S cut being given, one gets not only a canonical choice of geometric free generators λ 1 , µ 1 , . . . , λ p , µ p of the fundamental group π 1 (F, q), but also an extension of it to a set of geometric free generators of π 1 (F \ B, q) by a sequence γ 1 , . . . , γ n represented by the loops Γ i ⋆ C i ⋆ Γ -1 i , where C i denotes a small loop around a point b i of B and Γ i a portion of the tail going to b i . Note that this whole set of generators of π 1 (F \ B, q) is equipped with counter clock-wise ordering.

2.3.

Free semigroups of marked coverings. We continuer to consider connected compact oriented 2-dimensional manifolds F with one hole and a marked point q ∈ ∂F and turn to a study of their ramified finite degree coverings f : E → F . Let us recall

a 3 b 3 a 3 b 3 a 1 b 1 a 1 b 1 a 2 b 2 a 2 b 2 T 1 T 2 T 3
q once more that we allow disconnected covering spaces, but forbid ramifications at the boundary of F . Our aim is to organize such coverings of a fixed degree d in a semigroup.

To achieve this goal we equip each covering with a marking, that is a numbering ν by 1, . . . , d of the elements of f -1 (q), and consider the coverings up to certain natural equivalence relations. Different choices of the equivalence relations lead to different semigroups.

We start from introducing the free geometric degree d covering semigroup, which we denote by GFS d . To build such a semigroup, we equip the base F of each marked covering (f : E → F, ν, q) with a caudate skeleton S cdt whose tails end at the branch points of the covering. The elements of GFS d are the triples (f : E → F, ν, S cdt ) considered up to homeomorphisms of coverings respecting all the ingredients; more precisely, two triples (f 1 :

E 1 → F 1 , ν 1 , S cdt 1 ) and (f 2 : E 2 → F 2 , ν 2 , S cdt 2 ) are equivalent if there are homeomorphisms φ : E 1 → E 2 and ψ : F 1 → F 2 such that f 2 • φ = ψ • f 1 , φ • ν 1 = ν 2 ,
and ψ(S cdt 1 ) = S cdt 2 . The semigroup structure on GFS d is defined in a similar way that was used in [START_REF] Vik | Factorization semigroups and irreducible components of Hurwitz space[END_REF] in the case of genus zero. Namely, the product h = f • g of two elements of GFS d represented by marked ramified coverings (f :

E 1 → F 1 , ν 1 , S cdt 1
) and (g :

E 2 → F 2 , ν 2 , S cdt
2 ) (by abuse of notation we denote by the same symbols both the elements of GFS d and the underlying coverings) is given by the marked ramified covering (h : E → F, ν, S cdt 1 ∪ S cdt 2 ), where F and h : E → F are obtained, first downstairs, by gluing F 1 with F 2 along an arc of ∂F 2 issued from q 2 in the counter-clockwise direction and an arc of ∂F 1 issued from q 1 in the clockwise direction, (see Figure 4) and, second upstairs, by a gluing of f and g that preserves the markings over q = q 1 = q 2 . This operation respects the equivalence relation.

We equip GFS d with a map α : GFS d → S d that evaluates the boundary monodromy (taken in the direction of boundary orientation). As it follows from the gluing procedure, this map is a homomorphism. Furthermore, the symmetric group S d naturally acts on GFS d by renumbering the points of the fibre f -1 (q). Thus, GFS d becomes in a canonical way a semigroup over S d .

For each g ∈ S d denote by X g ∈ GFS d the element represented by a ramified covering f : E → Π 0 with one branch point, a marked point q ∈ ∂Π 0 , and the monodromy α(X g ) equal to g. Such an element is defined uniquely, as it follows, for example, from Proposition 7.

Next, consider the torus Π 1 with a hole, a marked point q ∈ ∂Π 1 and a skeleton Σ 1 that includes the semi-skeleton Σ Proof. Follows from the following isotopy unicity: for a surface with a given caudate skeleton, its open-eyes caudate skeleton plane diagram (Σ cdt p , q) ⊂ (∆ p , q) is unique up to isotopies in (∆ p , q).

As a set, the semigroup GFS d splits in a disjoint union of subsets, (GFS d ) n,p , that correspond to coverings with a given number n of branch points over surfaces of given genus p, or, saying in another way, to words with n letters X g and p letters Y a,b .

X g • Y a,b = Y a,b • X g [a,b] (19) and X g 1 • X g 2 = X g 2 • X g g 2 1 ( 20
)
for any g 1 , g 2 , a, b ∈ S d . These are the defining relations of GVS d .

Proof. For a surface with a given skeleton and branch locus, an enhancing of its open-eyes skeleton plane diagram (Σ p , q) ⊂ (∆ p , q) to an open-eyes caudate skeleton diagram (Σ cdt p , q) ⊂ (∆ p , q) with a given ordering of strings and tails is unique up to isotopies of (Σ p ∪ B, q) in (∆ p , q). The relation (19) reflects the elementary change of ordering (see Figure 5), and (20) the standard Artin-Hurwitz half-twist of two points of B in ∆ p \ Σ p (cf., [START_REF] Vik | Factorizations in finite groups[END_REF]).

Similar to GFS d , the semigroup GVS d splits as a set in a disjoint union of subsets, (GVS d ) n,p , that correspond to coverings with a given number n of branch points over surfaces of given genus p. We equip the base of each covering with a skeleton and consider the triples (f 1 :

E 1 → F 1 , ν 1 , S 1 ) and (f 2 : E 2 → F 2 , ν 2 , S 2 )
, where S 1 and S 2 are skeletons of F 1 and F 2 , respectively, up to the equivalence generated by two binary relations: first, up to isotopy of coverings with fixed base, marking, and skeleton, but moving branch points (which can move, in particular, through the skeleton); second, up to homeomorphisms respecting all the ingredients, that is up to homeomorphisms φ : E 1 → E 2 and ψ :

F 1 → F 2 such that f 2 • φ = ψ • f 1 , φ • ν 1 = ν 2 , and ψ(S ∞ 1 ) = S ∞ 2 .
The only, but major, difference with respect to the previous very strong covering semigroups is that we authorize the branch points to cross the skeleton.

Taking into account this additional equivalence relation, we obtain another semigroup over S d , which we denote by GS d and call the strong versal geometric degree d covering semigroup. The quotient map GVS d → GS d is a homomorphism of semigroups over S d and, set theoretically, it splits in quotient maps (GVS d ) n,p → (GS d ) n,p .

For each n and p, let us fix the surface F (of genus p) and its skeleton S, and place the branch locus B (of cardinality n) to be disjoint from S. Then the braid group on n strands, that is the group Br n (F, ∂F ) of isotopy classes of orientation preserving identical on the boundary self-homeomorphisms of (F, B), becomes to act naturally on (GVS d ) n,p , and, as it follows also directly from the definitions, the fibers of the quotient map (GVS d ) n,p → (GS d ) n,p are the orbits of this action.

Let us recall that, on the other hand, the braid group Br n (F, ∂F ) can be canonically identified with the fundamental group π 1 (F (n) \ ∆), where F (n) is the symmetric product of n copies of F and ∆ is the discriminant locus, that is, the set of those n-tuples that contain fewer than n distinct points. More precisely, we start from fixing a set B = {P 1 , . . . , P n } ⊂ F \ ∂F consisting of n distinct points and treat (2) ψ 1 (t), . . . , ψ n (t) are distinct points of F \ ∂F for each t ∈ [0, 1]; and multiplication is given by concatenation of paths.

By a λ-(respectively, µ-) move we understand a geometric braid whose all but one strands are constant and the remaining one follows a path I ⋆ C λ ⋆ I -1 (respectively, I ⋆ C µ ⋆ I -1 ) where I is a simple path in the complement of S ∪ (B \ {b}) joining a point b ∈ B with the vertex of a bouquet C λ ∨ C µ ⊂ S, see Figure 6. The standard Artin-Hurwitz (half-twist) geometric braids exchanging two points of B will be called H-moves.

The following proposition is well known. In fact, it follows easily, for example, from the exact sequences 2.6. Admissible geometric covering semigroups. Again, to define new geometric covering semigroups, we start from the very strong semigroup of degree d marked coverings, GVS d , and add supplementary relations between the triples (f : E → F, ν, S cdt ) representing the elements of GVS d . For that purpose, it is convenient to introduce some auxiliary category F .

The objects of F are the triples (F, q, S cdt ) where F is a connected compact oriented surface with one hole, q is a fixed point on its boundary, and S cut is a caudate skeleton of F ; we denote the set of ends of the tails of S cdt by B, its cardinality by n, and the genus of F by p. The morphisms of F are the preserving orientation homeomorphisms of triples (F, q, B). 1We denote by Homeo the whole set of morphisms of F ; by Homeo n,p the subset of Homeo consisting of the morphisms between the triples (F, q, S cdt ) with given |B| = n and g(F ) = p; and by H(F, q, S cdt ) the group consisting of the self-homeomorphisms (F, q, B) → (F, q, B). 2 We say that a collection of subsets H n,p ⊂ Homeo n,p , n 0, p 0, is geometrically admissible, if it contains the isotopies of B in F \ ∂F and for each two triples (F, q, S cdt ) and (F, q, S cdt ) with the same n and p there is a morphism (F, q, S cdt ) → (F, q, S cdt ) belonging to H n,p .

For each n and p, we fix the triple (F, q, S cdt ). The caudate skeleton S cdt defines a frame in the free group π 1 (F \ B, q), while each element ψ ∈ H(F, q, S cdt ) defines an automorphism of this free group F n+2p = π 1 (F \ B, q). Let us denote by H n,p the subgroups of Aut(F n+2p ), F n+2p = π 1 (F \ B, q), representing H n,p ∩ H(F, q, S cut ).

As follows directly from the definitions, H n,p are admissible subgroups of Aut(F n+2p ), if the collection H n,p ⊂ Homeo n,p is geometrically admissible. For example, one get geometrically admissible collections by considering the homeomorphisms preserving λ-circles up to isotopy (respectively, µ-circles), or taking the whole sets, H n,p = Homeo n,p . In fact, one can easily go other way round and, starting from a collection of admissible subgroups H n,p of Aut(F n+2p ), build a geometrically admissible collection by attributing to Hn,p ∩ H(F, q, S cdt ) all the elements in H(F, q, S cdt ) that act in F n+2p = π 1 (F \ B, q) as elements of H n,p (with respect to the frame defined by S cdt ).

Let H = { H n,p } n 0,p 0 be a geometrically admissible collection. Assume in addition that this collection of homeomorphisms is closed under the boundary connected sum of the triples (F, q, B) (see definition of the product in the semigroup GFS d ). We say that two triples (f 1 : E 1 → F 1 , ν 1 , S cdt 1 ) and (f 2 : E 2 → F 2 , ν 2 , S cdt 2 ) are H-equivalent if there are homeomorphisms φ : E 1 → E 2 and ψ : (F 1 , q 1 , B 1 ) → (F 2 , q 2 , B 2 ) such that f 2 • φ = ψ • f 1 , φ • ν 1 = ν 2 , and ψ ∈ H n,p . By means of such an additional equivalence relation, we obtain a semigroup over the group S d taking the quotient

  m} and GW S d,T (G, O) = {s ∈ GW S d (G, O) | τ i (s) T for i = 1, . . . , m}. The main results are as follows. Theorem 1. For any equipped finite group (G, O) such that the elements of O generate the group G, there is a constant T ∈ N such that the restriction of Φ to GS d,T (G, O) is an isomorphism of GS d,T (G, O) and GW S d,T (G, O).

  The one-to-one map β (G,O) : O → O given by β (G,O) ( g) = g defines an epimorphism β = β (G,O) : ( G, O) → (G, O) of equipped groups and an isomorphism β * : S( G, O) → S(G, O) of semigroups. By Claim 8 in [14], ker β is a subgroup of the center Z( G) of the C-group G.

Theorem 4 .

 4 Let (G, O), O = C 1 ⊔ • • • ⊔ C m , be an equipped finite group and ( G, O) = G Γ with Γ = Γ (G,O) the C-group equivalent to (G, O). Then there is a constant T ∈ N such that for any element s 1 ∈ S(G, O) G with τ i (s 1 )

Remark 1 .

 1 The elements s 1 , . . . , s a (G,O) ∈ S(G, O) G , whose existence is claimed by Theorem 4, are distinguished by their valuers α

  and f (µ j ) = b j . 1.5. Definition of covering semigroups. Let (G, O) be an equipped group. Denote by F S(G, O) the free semigroup over the alphabet Z = Z (G,O) = X O ∪ Y G introduced in subsection 1.4 and call F S(G, O) free covering semigroup over the equipped group (G, O).

Proposition 3 .

 3 , g n ) ∈ O n and all (a 1 , b 1 , . . . , a p , b p ) ∈ G 2p . Denote by factor semigroup qF S(G, O)/R H . In particular, the semigroup S(G, O) = qF S(G, O)/{R B } is called the strong covering semigroup, where B = {Br n,p } {n 0,p 0} . For a collection H = {H n,p } {n 0,p 0} of admissible automorphism groups, we denote the semigroup qF S(G, O)/R H by S H-equiv (G, O) and call it an admissible covering semigroup. The strong covering semigroup S(G, O) is isomorphic to the semigroup generated by the alphabet Z = Z (G,O) = X O ∪ Y G and subject to relations

  gn and s 2 = y a,b • . . . • y a k ,b k be a reduced form of an element s = s 1 • s 2 ∈ S(G, O). As it follows from Claim 7, the subgroup of G generated by g 1 , . . . , g n , a 1 , b 1 , . . . , a k , b k does not depend on the choice of a reduced form of s. In what follows, we denote this subgroup by G s .

  O) be the C-group equivalent to (G, O) (see subsection 1.3). For any set R H of admissible relations, the epimorphism β = β (G,O) : ( G, O) → (G, O) of equipped groups defines an epimorphism β * = β (G,O) * : S( G, O) = S H-equiv ( G, O) → S(G, O) = S H-equiv (G, O) over groups. Claim 9. The restriction of β * to the subsemigroup S( G, O) ⊂ S( G, O) coincides with the isomorphism of semigroups S( G, O) and S(G, O) ⊂ S(G, O) (defined in subsection 1.3).

Claim 10 .

 10 For any s ∈ S(G, O) of any fixed type and any a, b ∈ G each of the following equations s • y a,b = z • y α(s)a,b ,

Proposition 4 .

 4 Let O ⊂ G be a finite set. Then for any h ∈ g 1 , . . . , g n , a 1 , b 1 , . . . , a k , b k

Claim 16 .

 16 For any a, b ∈ G, we have a, b ∈ G(1, y a,b ).

  y a,b ). Therefore ab ∈ G(1, y a,b ) and hence a = (ab -1 a -1 )(ab) ∈ G(1, y a,b ). Applying one more Claim 11 to a and ab, we obtain that b ∈ G(1, y a,b ). Claim 17. For any a 1 , b 1 , . . . , a k , b k ∈ G, we have

1 . 7 .

 17 s 1 which together with s 2 = y a 2 ,b 2 • . . . • y a k ,b k satisfies all conditions of Proposition 5. Now to complete the proof of Proposition 5, it suffices to note that y 1,1 commutes with any element s ∈ S(G, O) (relation (8)) and to use induction on k. On the number of solutions of equation α(z) = g. Let (G, O) be an equipped group such that the elements of O generate the group G, O = C 1 ⊔ • • • ⊔ C m decomposition into a disjoint union of conjugacy classes of G. In this subsection, we investigate the following problem: for fixed type t ∈ Z m 0 , fixed genus p and given element h ∈ G to estimate the number of solutions in an admissible covering semigroup S(G, O) G of the equation α G (z) = h under the restrictions τ (z) = t and g(z) = p.

Theorem 6 .

 6 Let S(G, O) = S H-equiv (G, O) be an admissible covering semigroup over an equipped finite group (G, O), O = C 1 ⊔ • • • ⊔ C m . Then there is a constant T 1 ∈ N such that if for an element s 1 ∈ S(G, O) G the i-th type τ i (s 1 ) T 1 for all i = 1, . . . , m, then there are a (G,O) elements s 1 , . . . , s a (G,O) ∈ S(G, O) G such that (1)

Figure 1 .

 1 Figure 1. Plane diagrams of a genus-2 surface with its semi-skeleton.

Figure 2 .

 2 Figure 2. Plane diagrams of a genus-2 surface with its caudate skeleton.

Figure 3 .

 3 Figure 3. Strings ordered counter-clock wise.

Figure 4 .

 4 Figure 4. Plane diagram of a semigroup product.

1 . 8 .

 18 ∞ 1 and the string T 1 . Pick a pair a, b ∈ S d and denote by Y a,b ∈ GFS d the element represented by a non-ramified covering f : E → Π 1 with monodromy a along the loop T 1 ⋆ C λ ⋆ T -1 1 and b along the loop T 1 ⋆ C µ ⋆ T -1 1 . Such an element is also defined uniquely, as it follows, for example, from Proposition 6. Note that α(Y a,b ) = aba -1 b -Proposition The semigroup GFS d is a free semigroup over the group S d , its set of free generators is formed by X g , g ∈ S d \ {1}, and Y a,b , a, b ∈ S d .

2. 4 .Proposition 9 .

 49 Very strong semigroup of marked coverings. As a next step, we replace caudate skeletons by skeletons, that is forget the tails going to the branch points, and thus construct another semigroup over S d replacing everywhere in the above construction of GFS d the caudate skeletons by skeletons. We call this new semigroup the very strong semigroup of degree d marked coverings and denote it by GVS d . The forgetful map GFS d → GVS d consisting in replacing a caudate skeleton by the skeleton is a well defined homomorphism of semigroups over S d . Let us denote by Kh: us the same symbols X g , Y a,b the images in GVS d of the above free generators X g , Y a,b ∈ GFS d . The elements X g , g ∈ S d \ {1}, and Y a,b , a, b ∈ S d , form a set of generators of the semigroup GVS d . They satisfy the relations

Figure 5 . 2 . 5 .

 525 Figure 5. Plane diagram of an elementary change of ordering.

Figure 6 .

 6 Figure 6. Simple path joining a branch point with a bouquet vertex.

1 → π 1 1 and 1 →

 111 (F \ B ′ ) → P Br n (F, ∂F ) → P Br n-1 (F, ∂F ) → P Br n (F, ∂F ) → Br n (F, ∂F ) → S n → 1,where the second sequence is the definition of the pure braid groups, P Br n , and B ′ denotes B \ {P 1 }. Proposition 10. ([START_REF] Fadell | Configuration spaces[END_REF][START_REF] Fox | The braid groups[END_REF]) The braid group Br n (F, ∂F ) is generated by H-, λ-, and µ-moves.

Figure 7 .= ga g c 1 X

 71 Figure 7. Moving a branch point through an λ-cut in an XY -product

Figure 8 . 2 c 2 =

 822 Figure 8. Moving a branch point through a µ-cut in a Y X-product

  such that the elements of O generate the group G, there is a constant T such that for any projective irreducible non-singular curve F the number of irreducible components of each nonempty Hurwitz space HUR

  ) C-graph was associated with each equipped group. By definition, the C-graph Γ (G,O) of an equipped group (G, O) is a directed labeled graph. Its vertices are labeled by elements of O, and this labeling is a bijection between O and the set of vertices, V = {v g | g ∈ O}. Each edge of Γ (G,O) also is labeled by an element of O.Namely, two vertices v g 1 and v g 2 , g 1 , g 2 ∈ O, are connected by an edge e vg 1 ,vg 2 ,g with label g ∈ O if and only if g -1 g 1 g = g 2 . (A C-graph may contain loops, and several edges, but with distinct labels, may connect the same pair of vertices in the same direction.)Obviously, the conjugacy classesC i ⊂ O, 1 ≤ i ≤ m,are in one-to-one correspondence with the connected components Γ i of the C-graph Γ (G,O) ; more precisely, v g ∈ Γ i if and only if g ∈ C i . Two equipped groups (G 1 , O 1 ) and (G 2 , O 2 ) are called equivalent if their C-graphs, Γ (G 1 ,O 1 ) and Γ (G 2 ,O 2 ) , are isomorphic as C-graphs; in other words, if there is a bijection

	Proposition 2. ([12]) The elements of S(G, O) G 1 are fixed under the conjugation action of G.
	1.3. Factorization semigroups over equivalent equipped groups. Here, we
	introduce an additional assumption with regard to O in an equipped group (G, O):
	we assume that
	(iv) the elements of O generate the group G.
	In [14], a

  O), and the ith coordinate τ i (s) of τ (s) is called the ith type of s. A C-group G is called C-finite if the number of vertices of the graph Γ ( G, O) is finite. By Proposition 3 in [14], the commutant [ G, G] of a C-finite group G is a finite group. The order a (G,O) = | ker β ∩ [ G, G]| of the group ker β ∩ [ G, G] is called the ambiguity index of an equipped finite group (G, O). If O ′ ⊂ O are two equipments of a finite group G such that the elements of O ′ generate the group G, then by Corollary 2 in [14], we have a (G,O) a (G,O ′ ) .

  1,λ , . . . , ξ p 1 ,λ , ξ 1,µ , . . . , ξ p 1 ,µ , ζ 1 , . . . , ζ p 1 . Obviously, for p 1 p the group Br n,p 1 is a subgroup of Br n,p and the groups Br n,p 1 ⊂ Aut(F n+2p 1 ) and Br n,p 1 ⊂ Br n,p ⊂ Aut(F n+2p ) are naturally isomorphic. The groups Br n,p will be called algebraic braid groups. Claim 6. The boundary ∂B 1

  Lemma 3. Let B ′n be strongly equivalent to B 1 . Then the automorphisms σ i,n , ξ i,n,λ , ξ i,n,µ , and ζ n,i of F n+2p belong to Br n,p .Proof. Follows from Claim 5 and Lemma 2 by straightforward induction on p.We say that a subgroup H n,p of Aut(F n+2p ) is admissible if:1) Br n,p ⊆ H n,p ; 2) for each h ∈ H n,p there is a permutation σ

and σ i,n leaves fixed all the other elements of B ′ n . Similarly, if (. . . , γ ′ j , λ ′ i , µ ′ i , . . . ...) is a part of a frame B ′ n , then denote by ξ i,n,λ , ξ i,n,µ , and ζ n,i the automorphisms of F n+2p that leave fixed all the elements of B ′ n except γ ′ j , λ ′ i , µ ′ i and act on γ ′ j , λ ′ i , µ ′ i by the same formulas as ξ i,λ , ξ i,µ , and ζ n,i act on the elements γ n , λ i , µ i of the frame B i (we just replace n by j).

Note that morphisms are not supposed to respect skeletons.

Note that H(F, q, S cut ) is in a canonical bijection with the set of self-morphisms (F, q, S cdt ) → (F, q, S cdt ).

The following claim is an immediate consequence of the above Proposition.

for any g 1 , g 2 ∈ S d ; and Under pulling the branch point through λ-or µ-cuts as it is shown in Figures 7 and8, we obtain the the relations (23) and (24). In particular, the elements g ′ appearing in the relations 7 and8) can be found, for example, from the identities expressing the unchanged monodromy along the hole: gaba -1 b -1 = g ′ a ′ b(a ′ ) -1 b -1 , a ′ = ga under the move across the λ-cut, as is shown in Figure 7, and gaba

under the µ-cut, as is shown in Figure 8.

2.7. Construction of Hurwitz spaces of marked coverings. Here, we adapt Fulton's construction of Hurwitz spaces, see [START_REF] Fulton | Hurwitz schemes and irreducibility of moduli of algebraic curves[END_REF], to the case of marked coverings. Let D ⊂ F be an open disc in a projective irreducible non-singular algebraic curve F . We put F = F \ D, choose a point q ∈ ∂F , and fix an n-point set B ⊂ F \ ∂F .

Let us recall that for any surface F its braid group on n strands, Br n (F, ∂F ), can be seen as fundamental group π 1 (F (n) \ ∆), where F (n) is the symmetric product of n copies of F and ∆ is the discriminant locus.

Due to our assumptions, the fundamental group π 1 (F \ B, q) is isomorphic to the free group F n+2p where p is the genus of F , and in such a way the braid group Br n (F, ∂F ), which acts naturally (the right action) on π 1 (F \ B, q) ≃ F n+2p , becomes anti-isomorphic to the algebraic braid group Br n,p introduced in Subsection 1.4 (this is usually called Artin presentation theorem and follows from a comparison of the actions of the generators of these groups on F n+2p ≃ π 1 (F \ B, q)).

To detail these identifications, let us pick a caudate skeleton S cdt of F the set of ends of whose tails is B. In notation of subsection 2.2, the choice of S cdt defines the set {λ 1 , µ 1 , . . . , λ p , µ p } of free generators of the group π 1 (F, q) and loops γ i around the points of B, so that γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p are free generators of

The set {γ 1 , . . . , γ n , λ 1 , µ 1 , . . . , λ p , µ p } is a frame of the free group π 1 (F \ B, q) ≃ F n+2p , which in accordance with notation of Subsection 1.4 we denote by B 1 . As to the standard generators σ 1 , . . . , σ n-1 , ξ 1,λ , . . . , ξ p,λ , ξ 1,µ , . . . , ξ p,µ , ζ 1 , . . . , ζ p of the algebraic braid group Br n,p , they turn in terms of the geometric braid group Br n (F, ∂F ) into H-moves, λ-and µ-moves, and braiding of a point around a handle (see Subsection 2.5). This anti-isomorphism Br n (F, ∂F ) → Br n,p defines a right action of Br n (F, ∂F ) on the set of words W B 1 (G, O) = f W f,B 1 , where the union is taken over all equipped homomorphisms f :

1 is invariant under the action of Br n (F, ∂F ). Therefore this action defines homomorphisms ω = ω n,p,(G,O) :

As it follows from Proposition 6, the homomorphisms ω and ω G

respectively. Furthermore, there is a canonical embedding j :

Moreover, the both coverings are marked (by words of

\ ∆ represented by B, and j respects the markings.

According to the usual construction of covering spaces by means of the groupoid of homotopy classes of paths, the covering space HUR (G,O),n (F ) as a set is the set of pairs (B ′ , f ′ ), where B ′ ∈ F (n) 0 \ ∆ and f ′ : π 1 (F \ B ′ , q) → G are epimorphisms such that the conjugacy classes of their values at the loops around the points of B ′ belong to O and f ′ (∂F ) = 1. We call HUR (G,O),n (F ) the Hurwitz space of marked n-branched coverings of F with equipped Galois group (G, O). This construction being functorial, a choice of an embedding i : G ֒→ S d provides an embedding of HUR (G,O),n (F ) into HUR d,n = HUR (S d ,S d \{1}),n (F ), which we call the Hurwitz space of marked n-branched degree d coverings of F .

The advantage of considering marked coverings is that the Hurwitz spaces HUR d,n come then with a universal family of coverings, F d,n → HUR d,n . Such a family can be obtained as manifold completion (see Subsection 2.1) of the unramified covering of

, that is the covering defined by the homomorphism π 1 (U, (q, B))

that sends the images of elements ς ∈ π 1 (F \ B, q) to permutations (x, w) → (α(ς)x, w) and the images of elements χ ∈ π 1 (F (n) \ ∆) to permutations (k, w) → (k, ω(χ)w).

i=1 D i = ∅ and q ∈ ∂D i for all i define an unramified covering θ n (G, O) G 1 : HUR (G,O),n (F , q) → (F \ q) (n) \ ∆ the covering space of which is called the Hurwitz space of marked (at a point q ∈ F ) coverings of a projective algebraic curve F with equipped Galois group (G, O) and branched at n points.