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Covering semigroups. (English. Russian original) Zbl 1286.14045

Izv. Math. 77, No. 3, 594-626 (2013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 3, 163-198 (2013).
The article under review provides a sufficient condition for the irreducibility, as well as a bound on the number of the irreducible components, of the Hurwitz space of the marked covering of a curve with fixed Galois group and monodromy type. This is done by means of a semigroup structure on the set of irreducible components of the aforementioned Hurwitz space.
In order to formulate some results of the article, let us note that a finite morphisms \(f: E \rightarrow F\) of smooth irreducible complex projective curves is determined up to isomorphism by the finite extensions \(f^* : {\mathbb C}(F) \hookrightarrow {\mathbb C}(E)\) of the corresponding function fields. The Galois group \(G\) of the Galois closure of this extension is called the Galois group of \(f\). If \(B \subset F\) is the branch locus of \(f\) and \(q \in F \setminus B\), then any ordering of \(f^{-1} (q)\) turns \(f\) into a marked covering and determines a homomorphism \(f_* : \pi _1 (F \setminus B, q) \rightarrow \mathcal{S}_d\) in the symmetric group of degree \(d = \deg f\) with \(G \simeq \text{im} f_*\). The homotopy class of a standard simple loop around \(b \in B\) in \(\pi _1 (F \setminus B, q)\) is determined by \(b\) up to a conjugation. Let \(O = C_1 \coprod \ldots \coprod C_m\) be the union of the conjugacy classes of the local monodromies of \(f\) and \(\tau _i\), and let \(1 \leq i \leq m\) be the number of branch points of \(f\) with local monodromies from \(C_i\). Denote by \(\tau = ( \tau _1 C_1, \ldots , \tau _m C_m)\) the monodromy type of \(f\) and write \(HUR _{d,G, \tau} (F)\) for the Hurwitz space of the marked coverings of \(F\) of degree \(d\) with Galois group \(G \subset \mathcal{S}_d\) and monodromy type \(\tau\). Let us assume that the elements of \(O\) generate \(G\) and consider the group \(\widetilde{G}\), whose generating set \(\widetilde{O}\) is isomorphic to \(O\) and whose relations are given by \(\widetilde{g_3}^{-1} \widetilde{g_1} \widetilde{g_3} = \widetilde{g_2}\) for those \(\widetilde{g_i} \in \widetilde{O}\) whose counterparts \(g_i \in O\) satisfy \(g_3^{-1} g_1 g_3 = g_2\). Then there is an epimorphism \(\beta : ( \widetilde{G}, \widetilde{O}) \rightarrow (G, O)\) and the abelianization \(H_1 ( \widetilde{G}, {\mathbb Z}) = \widetilde{G} / [ \widetilde{G}, \widetilde{G}]\) is isomorphic to \(({\mathbb Z}^m, +)\). If \(O \simeq \widetilde{O}\) are finite sets, the commutator subgroup \([ \widetilde{G}, \widetilde{G}]\) of \(\widetilde{G}\) is finite and the order \(a_{(G,O)} := | \ker \beta \cap [ \widetilde{G}, \widetilde{G}]|\) of the intersection \(\ker \beta \cap [ \widetilde{G}, \widetilde{G}]\) is called the ambiguity index of the equipped group \((G, O)\). The authors show that if \(O = C_1 \coprod \ldots \coprod C_m\) is a finite generating set of \(G\), then there is a constant \(T \in {\mathbb N}\) such that for any smooth irreducible complex projective curve \(F\) and any \(\tau _1 \geq T, \ldots \tau _m \geq T\) with non-empty Hurwitz space \(HUR _{d,G,\tau} (F)\), this Hurwitz space has \(a_{(G,O)}\) irreducible components. Moreover, if \(O_k = C_1 \coprod \ldots \coprod C_k\) generates \(G\) for some \(k < m\), then there is a constant \(T' \in {\mathbb N}\) such that for any smooth irreducible complex projective curve \(F\) and any \(\tau _1 \geq T', \ldots , \tau _k \geq T'\) the corresponding Hurwitz space has at most \(a_{(G,O)}\) irreducible components.
Let \(C\) be the conjugacy class of an odd permutation \(\sigma \in \mathcal{S}_d\), which leaves fixed at least two elements. Then there is a constant \(N_C \in {\mathbb N}\), such that for any smooth irreducible complex projective curve \(F\) and any monodromy type \(\tau\), in which \(C\) is of multiplicity at least \(N_C\), the Hurwitz space \(HUR _{d, \mathcal{S}_d, \tau} (F)\) is irreducible.
Let us consider a connected compact oriented real surface \(F\) with one hole and one marked point \(q \in \partial F\) on the boundary. A semi-skeleton \(S^{\infty}\) of \(F\) is a disjoint union of embedded bouquets of two oriented circles, whose complement is homeomorphic to a punctured disc. A skeleton \(S\) is a semi-skeleton \(S^{\infty}\), endowed with a system of paths, joining \(q \in \partial F\) with the connected components of \(S^{\infty}\). A caudate skeleton \(S^{\text{cdt}}\) is a skeleton that is disjoint from the branch locus \(B\) and endowed with \(n = |B|\) simple paths, connecting the points of \(B\) with \(q\) and intersecting each other only at \(q\). Let \(f_i : E_i \rightarrow F_i\) be finite ramified coverings, equipped with enumerations \(\nu _i : [1, \ldots , d_i] \rightarrow f^{-1} (q)\), \(1 \leq i \leq 2\). The product of the marked ramified coverings \((f_1 : E_1 \rightarrow F_1, \nu _1, S_1 ^{\text{cdt}})\) and \((f_2 : E_2 \rightarrow F_2, \nu _2, S_2 ^{\text{cdt}})\) is represented by a marked ramified covering \((f: E \rightarrow F, \nu, S_1 ^{\text{cdt}} \cup S_2 ^{\text{cdt}})\), where \(F\) is obtained by gluing \(F_1\) and \(F_2\) along an arc of \(\partial F_2\), issuing from \(q_2\) in the anticlockwise direction and an arc of \(\partial F_1\), issuing from \(q_1\) in the clockwise direction. The coverings \(f_1 : E_1 \rightarrow F_1\) and \(f_2 : E_2 \rightarrow F_2\) are glued in such a way that to preserve the enumeration of the fibres over \(q = q_1 = q_2\). Let us equip the triples \((f: E \rightarrow F, \nu, S)\) by the equivalence relation \((f_1 : E_1 \rightarrow F_1, \nu _1, S_1) \sim (f_2 : E_2 \rightarrow F_2, \nu _2, S_2)\), generated by the following two binary relations: first, the isotopy of marked coverings with fixed base, marking and skeleton and moving branch points (possibly moving across the skeleton) and, secondly, the homeomorphisms \(\phi : E_1 \rightarrow E_2\), \(\psi : F_2 \rightarrow F_1\), such that \(f_2 \circ \phi = \psi \circ f_1\), \(\phi \circ \nu _1 = \nu _2\), \(\psi (S_1 ^{\infty}) = S_2^{\infty}\). The semigroup of the equivalence classes of \((f: E \rightarrow F, \nu , S)\) under the aforementioned two binary relations is denoted by \(G{\mathbb S}_d (G,O)\) and called the strong versal geometric covering semigroup. Let \(\text{Homeo} _{n,p}\) be the set of the morphisms of \((f: E \rightarrow F, \nu, S^{\text{cdt}})\) with fixed number of branch points \(n = |B|\) and fixed genus \(p = g(F)\). The triples \((f_1 : E_1 \rightarrow F_1, \nu _1, S_1 ^{\text{cdt}})\) and \(f_2 : E_2 \rightarrow F_2, \nu _2, S_2 ^{\text{cdt}})\) are \(\text{Homeo} _{n,p}\)-equivalent if there are homeomorphisms \(\phi : E_1 \rightarrow E_2\) and \(\psi : (F_1, q_1, B_1) \rightarrow (F_2, q_2, B_2)\), such that \(f_2 \circ \phi = \psi \circ f_1\), \(\phi \circ \nu _1 = \nu _2\) and \(\psi \in \text{Homeo} _{n,p}\). The semigroup \(GW {\mathbb S}_d (G,O)\) of the \(\text{Homeo} _{n,p}\)-equivalence classes is called the weak versal geometric covering semigroup of degree \(d\). There is a natural epimorphism \(\Phi : G {\mathbb S}_d (G,O) \rightarrow GW {\mathbb S}_d (G,O)\). For any \(T \in {\mathbb N}\) let us consider the sub-semigroup \[ G{\mathbb S}_{d,T} (G,O) = \{ s \in G{\mathbb S}_d (G,O) \;\;| \;\;\tau _i (s) \geq T, \;\;\forall 1 \leq i \leq m \} \] of \(G{\mathbb S}_d (G,O)\) and the sub-semigroup \[ GW{\mathbb S}_{d,T} (G,O) = \{ s \in GW{\mathbb S}_d (G,O) \;\;| \;\;\tau _i (s) \geq T, \;\;\forall 1 \leq i \leq m \} \] of \(GW{\mathbb S}_d (G,O)\). The article under review proves that for any equipped group \((G,O)\) with finite generating set \(O = C_1 \coprod \ldots \coprod C_m\) of \(G\) there is a constant \(T \in {\mathbb N}\), such that the restriction \(\Phi : G {\mathbb S}_{d,T} (G,O) \rightarrow GW{\mathbb S}_{d,T} (G,O)\) is an isomorphism of semigroups.

MSC:

14H30 Coverings of curves, fundamental group
20M50 Connections of semigroups with homological algebra and category theory
57M05 Fundamental group, presentations, free differential calculus
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