On n -th roots of meromorphic maps

. Let S be a connected Riemann surface and let ϕ : S → (cid:98) C be branched covering map of ﬁnite type. If n ≥ 2, then we describe a simple geometrical necessary and suﬃcient condition for the existence of some n -th root, that is, a meromorphic map ψ : S → (cid:98) C such that ϕ = ψ n .


Introduction
In this paper, S will denote a connected (not necessarily compact or of finite type) Riemann surface and C = C ∪ {∞} will be the Riemann sphere. A holomorphic surjective map ϕ : S → C is a holomorphic branched covering if: (i) it has a finite set B ϕ ⊂ C of branching points, (ii) ϕ : S \ ϕ −1 (B ϕ ) → C \ B ϕ is a holomorphic covering map and (iii) around each point q ∈ B ϕ there is a open disc ∆ q such that ϕ −1 (∆ q ) consists of a collection of pairwise disjoint discs V j such that each of the restrictions ϕ : V j → ∆ q is a finite degree d q,j holomorphic map (i.e., there are biholomorphisms z : V j → D and w : ∆ q → D, where D is the unit disc, such that w • ϕ • z −1 (z) = z dq,j ). For each q ∈ B ϕ , let M q ⊂ {1, 2, . . .} be the set of local degrees of ϕ at the points in the fiber ϕ −1 (p). We say that ϕ is of finite type if the sets M q are finite. This condition permits to define the branch order of q ∈ B ϕ as the minimum common multiple of the values in M q .
Let ϕ : S → C a holomorphic branched covering of finite type. If n ≥ 2, then a meromorphic map ψ : S → C such that ϕ = ψ n is called an n-th root of ϕ (the others n-th roots of ϕ are of the form e 2kπi/n ψ, where k = 0, 1, . . . , n − 1).
The existence of an n-th root of ϕ necessarily implies that: (a) ∞, 0 ∈ B ϕ and (b) the branch orders of both 0 and ∞ are multiples of n. These two conditions are not sufficient for ϕ to have an n-root. For S = C the existence of an n-th root is equivalent for each zero and each pole of the rational map ϕ to have degree a multiple of n (which in particular asserts conditions (a) and (b)). But, for other Riemann surfaces, the above is not sufficient in general.
In [3] there was provided a simple geometrical necessary and sufficient condition for ϕ to have a 2-th root. In this paper, we generalize such a description for every n ≥ 2 (Theorem 2.3).
In the final section we generalize some of the tools in the proof of the main result to the context of Kleinian groups of higher dimension.
Remark 1.1 (A connection to Fuchsian groups). Let K be a finitely generated Fuchsian group, acting on the hyperbolic plane H 2 , such that H 2 /K is an orbifold of genus zero (so its underlying Riemann surface structure is C) and let k 1 , . . . , k r be the orders of its cone points. Let Γ be a subgroup of K and let S be the underlying Riemann surface structure associated to the hyperbolic orbifold H 2 /Γ (if Γ is assumed to be torsion free, then this orbifold has no cone points). It is well known that S is of finite type if and only if Γ is finitely generated (in particular, by taking infinitely generated subgroups we obtain examples of surfaces of infinite type). The inclusion Γ ≤ K induces a holomorphic branched covering of finite type S = H 2 /Γ → C. The local degree of ϕ at each point of S is either 1 (the generic case) or a divisor of some k j (if Γ is torsion free, then the local degree at each point over a cone point of order k j is also k j ). Conversely, the uniformization theorem asserts that, for a connected Riemann surface S of hyperbolic type, each branched covering of finite type ϕ : S → C is obtained in such a way for suitable choices of Γ and K.

Main results
Before stating our main result we need some definitions.

Admissible arcs and n-Z-orientability
Let us recall that a map on an orientable and connected surface X is a 2-cell decomposition of it, induced by the embedding of a connected graph H for which each of its vertices has a finite degree and each face (i.e., the connected components of X \ H) are finite-sided polygons.
Remark 2.1. For instance, if we let ϕ : C → C to be ϕ(z) = z 3 , then it can be seen that the positive real line is an admissible arc. As in this case F δ has exactly three faces (cyclicly permuted by the rotation A(z) = e 2πi/3 z), this cannot be n-Z-orientable for n = 3.
The 2-Z-orientable definition was introduced by Zapponi in [9, 10, 11] (he used the term "orientable") in order to decide if a given Strebel quadratic meromorphic form Q [8] on a closed Riemann surface has an square root (the 2-cell decomposition is obtained from the graph whose vertices are the zeroes of Q and the edges are its non-compact horizontal trajectories).
Remark 2.2. If δ 1 and δ 2 are admissible arcs for ϕ, then there is an orientationpreserving homeomorphism h : C → C fixing the points 0 and ∞ and such that h(δ 1 ) = δ 2 .

n-Z-orientability is a necessary and sufficient condition
The following generalizes the results in [3] done for the case n = 2. Theorem 2.3. Let S be a connected Riemann surface. Let ϕ : S → C be a holomorphic branched covering map of finite type with 0, ∞ ∈ B ϕ and each one with branch order a multiple of n ≥ 2. If δ is an admissible arc for ϕ, then the existence of n-th roots of ϕ is equivalent for the map F δ to be n-Z-orientable.
In terms of Fuchsian groups, Theorem 2.3 asserts the following.
Corollary 2.4. Let K < PSL 2 (R) be a co-compact Fuchsian group acting on the hyperbolic plane H 2 such that H 2 /K has genus zero. Let n ≥ 2 be an integer and assume that there are two cone points p, q ∈ H 2 /K whose cone orders are multiples of n. Let δ ⊂ H 2 /K be a simple arc whose end points are p and q and containing all other cone points in its interior. Let F δ be the map of H 2 induced by the lifting of δ to H 2 . Then the existence of a normal subgroup Γ n , of index n ≥ 2, in K such that K/Γ n ∼ = Z n and H 2 /Γ n has genus zero, is equivalent for F δ to be n-Z-orientable.
In order to see the above, we identify H 2 /K with C and p = 0, q = ∞. Then in Theorem 2.3 we set S = H 2 and take ϕ : H 2 → C a branched regular covering with K as its deck group.

Case of compact Riemann surfaces
If S is a compact Riemann surface, then every non-constant meromorphic map ϕ : S → C is a holomorphic branched covering of finite type. In this way, Theorem 2.3 can be rewritten as follows.
Corollary 2.5. Let S be a compact Riemann surface. Let ϕ : S → C be a nonconstant meromorphic map with 0, ∞ ∈ B ϕ and each one with branch order a multiple of n ≥ 2. If δ is an admissible arc for ϕ, then the existence of n-th roots of ϕ is equivalent for the associated map F δ to be n-Z-orientable.
The compact Riemann surface S can be defined by a complex projective algebraic curve inside P n and the meromorphic map ϕ : S → C can be described by a rational map.
Let us assume S is defined as the zero locus of the homogeneous polynomials P 1 , . . . , P r ∈ C[x 1 , · · · , x n+1 ] and that ϕ corresponds to the quotient ] are homogeneous polynomials of the same degree. If σ ∈ Gal(C), the group of field automorphisms of C, then we set S σ (respectively, ϕ σ ) the projective algebraic curve defined by the polynomials P σ 1 , . . . , P σ r (respectively, Q σ 1 /Q σ 2 ), where P σ j (respectively, Q σ j ) is obtained from P j (respectively, Q j ) by applying σ to all of its coefficients. It can be checked that S σ is again a compact Riemann surface and that ϕ σ : S σ → C is a holomorphic branched covering map of finite type.
Corollary 2.6. Let S be a compact Riemann surface. Let ϕ : S → C be a nonconstant meromorphic map with 0, ∞ ∈ B ϕ and each one with branch order a multiple of n ≥ 2. Then the n-Z-orientability property of a pair (S, ϕ) is a Gal(C)-invariant.
Belyi's theorem [2] asserts that a compact Riemann surface S can be defined by a curve over the field Q of algebraic numbers if and only if there is a nonconstant meromorphic map, called a Belyi map, β : S → C whose branching points are contained inside {∞, 0, 1}. On S there is a 2-cell decomposition D β , called a dessin d'enfant [6], whose underlying graph β −1 ([0, 1]) is bipartite (the black vertices are β −1 (1) and the white ones are β −1 (0)). Corollary 2.5 provides a geometrical condition for the new Belyi map ϕ = β/(β − 1) to have n-square roots. Such a geometrical condition, in terms of the dessin D β , is that its faces can be labeled using numbers in {1, 2, . . . , n} such that around each black vertex (respectively, white vertex), following the counterclockwise orientation, we see a finite consecutive sequence of the tuple (1, 2, . . . , n) (respectively, (n, n − 1, . . . , 2, 1)). This condition provides a Galois invariant for the dessin D β . In [4], for n = 2, it was observed that this is a new Galois invariant on dessins d'enfants. We expect (but we have no explicit evidence) that for each n ≥ 3 it provides a new Galois invariant.

3.1.
Let r ≥ 2 be the cardinality of B ϕ and let δ ⊂ C an admissible arc for ϕ, starting at the branch point p 1 = 0, ending at the branch point p r = ∞. We label the rest of the branch points of ϕ as p 2 , . . . , p r−1 , such that p j is between p j−1 and p j+1 . Let us denote by k j be the branch order of p j . We are assuming that k 1 and k r are both multiples of n ≥ 2. Set X to be either C, C or H 2 depending on if r j=1 (1 − k −1 j ) − 2 is negative, zero or positive, respectively.

3.2.
Let K be a discrete group of isometries of X such that X/K = C and whose cone points are p 1 , . . . , p r , with respective cone orders k 1 , . . . , k r . Let π K : X → C be a regular holomorphic branched covering with K as deck group.
The arc δ defines a fundamental domain P δ for K (see Figure 2 at the end), with 2(r − 1) sides, and set of side pairings A P δ = {C 1 , . . . , C r−1 }, such that The K-translates of P δ produces a 2-cell decomposition T K,P δ of X, that is, a map on X. As k 1 and k r are multiples of n, the following produces a surjective homomorphism θ 0 : K → G = σ ∼ = Z/nZ : C j → σ, j = 1, . . . , r − 1.
As ker(θ 0 ) is the group generated by the K-conjugates of the elements it follows that adjacent faces of the map T K,P δ have different labels.
Let x 1 , . . . , x r ∈ X be the fixed points of the elements C 1 , C −1 1 C 2 , C −1 2 C 3 , . . . , C −1 r−2 C r1 and C r−1 , respectively. Then, π K (x j ) = p j , for j = 1, . . . , r. Remark 3.1. The map T K,P δ is n-Z-orientable. To see this, for each T ∈ K, we label the T -translated of P δ by the element θ 0 (T ) ∈ G. Now, in order to be consequent with our definiton of n-Z-orientability as in the introduction, we make the identification of σ j with the integer j + 1, for j = 0, 1, . . . .n − 1.
The orbifold X/ ker(θ 0 ) can be identified with the Riemann sphere C with exactly n(r − 2) + 2 cone points, these being of orders k 1 /n, k 2 , n . . ., k 2 , . . . , k r−1 , n . . ., k r−1 , k r /n. The pair (K, ker(θ 0 )) induces a Möbius transformation A, of order n, and (by using the above identification) a degree n meromorphic map η : C → C, whose deck group is A ∼ = Z/nZ, branched at the end points of δ, i.e., 0 and ∞. Up to conjugation by a suitable Möbius transformation, we may assume that A(z) = e 2πi/n z and η(z) = z n .
The set η −1 (δ) is a collection of n simple arcs (containing all the cone points of X/ ker(θ 0 ) and whose end points are the two fixed points of A, i.e., 0 and ∞) which are cyclically permuted by A. This provides a n-Z-orientable map F 0 on C and it is induced by the map T K,P δ .
Remark 3.2. Each index normal subgroup Γ 0 of K, such that that K/Γ 0 ∼ = Z/nZ, is given as the kernel of a surjective homomorphism θ : K → Z/nZ. Let us assume that X/Γ 0 has genus zero. In this case, the inclusion Γ 0 ¡ K induces a regular holomorphic branched covering π Γ0 : X → C with Γ 0 as its deck group, such that π K = R • π Γ0 , where R(z) = z n . Let us restrict to those Γ 0 such that π Γ0 (x 1 ) = 0 and π Γ0 (x r ) = ∞. (Note that if each of the k j , where j = 2, . . . , r −1, are not a multiple of n, then this is the only possibility). In this case, for j = 2, . . . , r − 1, π Γ0 (x j ) is a cone point of order k j (since that point is not critical point of R). This asserts that C −1 j−1 C j ∈ Γ 0 . If we set σ = θ(C 1 ), which is a generator of Z/nZ, the previous asserts that θ(C j ) = σ, for every j = 1, . . . , r − 1. In other words, up to post-composing by an automorphism of Z/nZ, we obtain θ 0 , i.e., Γ 0 is uniquely determined.

3.3.
As a consequence of the uniformization theorem, there is a proper subgroup Γ of K such that S is the Riemann surface structure of the orbifold X/Γ, a regular holomorphic (possible branched) covering π Γ : X → S, with deck group Γ, such that π K = ϕ • π Γ (i.e., ϕ is induced by the inclusion Γ < K).
Let us observe that the π Γ -image of the n-Z-orientable map T K,P δ on X is the map F δ (which might or not be n-Z-orientable in principle). Now, the existence of a meromorphic map ψ : S → C, such that ϕ = η • ψ = ψ n is equivalent to have that Γ ≤ ker(θ 0 ) (see Remark 3.2). By lemma 3.3 (whose arguments follow the same ideas as in [3] for n = 2), the previous is equivalent for the n-Z-orientability of F δ . This will provide the desired result. Proof. Let us assume Γ ≤ ker(θ 0 ). The idea is to push-down the (n-Z-orientable) labelling on the faces of T K,P δ to the faces of F δ . Two faces F 1 and F 2 of T K,P δ are projected to the same face if and only if there is some T ∈ Γ such that F 2 = T (F 1 ). As we are assuming Γ ≤ ker(θ 0 ), the induced labelling is well defined. It is not difficult to observe that induced labelling on the map F δ satisfies the condition for being n-Z-orientable.
In the other direction, let us assume we have a labelling for F δ satisfying the n-Z-orientability. By the connectivity of S, we may construct a fundamental (connected) domain Q for Γ by gluing some copies K-translated of P δ (as many as the index of Γ in K). The projection of those copies of P δ , used in the construction of Q, projects under π Γ exactly to the faces of F δ . Now, lift the labelling of the n-Z-orientable map F δ to obtain labelling of these copies of P δ included in Q. Use the group K to translate these labels to the rest of K-translates of P δ . This provides a labelling on T K,P δ satisfying the n-Zorientability property. By Remark 3.2, this can be assumed to be the labelling provided by θ 0 . As the above procedure of pulling-down the labelling from T K,P δ to F δ induces the given labelling, it follows from the first part that Γ ≤ ker(θ 0 ).

A remark: θ-Zapponi-orientability of Kleinian groups
In the previous section we have considered a Fuchsian group K, a fundamental polygon P , the set A P ⊂ K of its side pairings, and a surjective homomorphism θ 0 : K → G = Z/nZ such that ker(θ 0 ) ∩ A P = ∅. The homomorphism θ 0 permitted to label each of the faces of the map T K,P , using as labelling the elements of G, and such that adjacent faces have different labels. This procedure can be generalized for any Kleinian group as follows.
Let K be a discrete group of isometries of X m , where X m is either the mdimensional hyperbolic H m or the m-dimensional Euclidian space E m or the m-dimensional sphere S m . Let P ⊂ X m be a fundamental polyhedron of K and let A P the subset of K consisting of the side-pairings of P . It is well known that A P is a set of generators for K and that a complete set of relations is provided by how the sides of P are glued by these side-pairings (Poincaré Polyhedron Theorem, see [1,5,7]). The K-translates of P provides a n-tessellation T K,P of X m .

(K, P )-admissible homomorphisms
Let θ : K → G, where G is a finite group, be a surjective homomorphism. For each T ∈ K we proceed to label the n-face T (P ) using the element θ(T ) ∈ G. If adjacent faces have different labels, then we say that θ is (K, P )-admissible. Proof. This follows from the fact that, for T 1 , T 2 ∈ K, one has that T 1 (P ) and T 2 (P ) are adjacent if and only if there is some L ∈ A P such that T 2 = T 1 L.
Remark 4.2. For every surjective homomorphism θ : K → G, it is possible to find a fundamental polyhedron P for K such that θ is (K, P )-admissible. (1) If K P is the subgroup of K generated by all the elements of the form AB, where A, B ∈ A P , then either K P = K or has index two in K. If θ : K → G = Z 2 is any homomorphism, then K P ≤ ker(θ). It follows that θ is (K, P )-admissible if and only if K = K P = ker(θ) (in particular, there is at most one (K, P )-admissible homomorphism onto Z 2 ). (2) Let n, r ≥ 2 and let us consider a Fuchsian group, acting in the hyperbolic plane H 2 , with the following presentation Let P be a fundamental domain of K as shown in Figure 2. Its set of sidepairings is A P = {C 1 , . . . , C r−1 }. Let G = σ ∼ = Z/nZ. If k 1 and k r are both multiples of n, then we may consider the surjective homomorphism θ 0 : K → G, defined by θ 0 (C j ) = σ, for every j = 1, . . . , r − 1. As ker(θ 0 ) is the group generated by the conjugates of the elements C n 1 , C −1 1 C 2 , C 1 C −1 2 , . . . , C −1 1 C r−1 , C 1 C −1 r−1 , it follows that θ 0 is (K, P )-admissible. The induced labelling of T K,P by θ 0 satisfies to be n-Z-orientable.

θ-Z-orientable subgroups
Let θ : K → G be a (K, P )-admissible homomorphism. If Γ is a proper subgroup of K, then the tessellation T K,P induces an m-dimensional tessellation T K,P,Γ on the geometric orbifold O Γ = X m /Γ. The labelling on the faces of T K,P , provided by the (K, P )-admissible homomorphism θ, induces a labelling of the faces of the tessellation T K,P,Γ . It is not difficult to see that the adjacent mfaces of this last tessellation have different labels if and only if Γ ≤ ker(θ). If this is the situation, we say that (K, P, Γ) is θ-Z-orientable.
We summarize all the above in the following. Lemma 4.4. Let K be a discrete group of isometries of X m , P ⊂ X m be a fundamental polyhedron for it and A P ⊂ K be the set of side-pairings of P . Let θ : K → G be a (K, P )-admissible homomorphism onto a finite group G (equivalently, ker(θ) ∩ A P = ∅). Then (K, P, Γ) is θ-Z-orientable if and only if Γ ≤ ker(θ).