# Rubén A. Hidalgo

## Departamento de Matemática y Estadística

PUBLICATIONS

1989

 1.-On Γ      Notas Soc. Mat. Chile 8 (1) (1989), 27-36. In this article we provide Schottky uniformizations of closed Riemann surfaces admitting a conformal involution so that the involution is reflected in the uniformization.

1993

 Tôhoku Math. J. 45 (1993), 499 - 503. We extend a result on homology coverings of closed Riemann surfaces due to Maskit to the class of analytically finite ones. We show that if S is an analytically finite hyperbolic Riemann surface, then its conformal structure is determined by the conformal structure of its homology cover. The homology cover of a Riemann surface S is the highest regular covering of S with an Abelian group of covering transformations. In fact, we show that the commutator subgroup of any torsion-free, finitely generated Fuchsian group of the first kind determines it uniquely.

1994

 Ann. Acad. Fenn. 19 (1994), 259-289. We consider a closed Riemann surface S and a group H of conformal automorphisms of S . We seek a Schottky uniformization (Ω, G, π: Ω → S ) of the surface S with the property that every element of H can be lifted to a conformal automorphism of the region Ω. We obtain necessary conditions, called Condition (A), on the set of fixed points of the non-trivial elements of H in order to find a Schottky uniformization as desired. For instance, Condition (A)is trivially satisfied by groups acting freely, groups isomorphic to Z/2Z ⊕....⊕ Z/2Z and dihedral groups. We show that Condition (A) is sufficient when H is a cyclic group.

 4.- The mixed Elliptically Fixed Point property for Kleinian groups.      Ann. Acad. Fenn. 19 (1994), 247-258. We define the mixed elliptically fixed point (MEFP) property for Kleinian groups. Such a property is a set of possibilities for the locations on the Riemann sphere of  the fixed points of elliptic elements in Kleinian groups. We show that the MEFP property is invariant under the operations given by the Maskit-Klein Combination Theorems. As a consequence, we obtain that finitely generated function groups satisfy such a property. We also show that geometrically finite Kleinian groups satisfy the MEFP property. Examples of Kleinian groups where such a property does not hold are provided.

 Glasgow Math. J. 36 (1994), 17 - 32. Let S be a closed Riemann surface and let H be an Abelian group of conformal automorphisms of S. We obtain necessary and sufficient conditions to be satisfied by H in order to find a Schottky group G uniformizing S for which H lifts.

1995

 The Pacific Journal of Math. 169 (1995), 291-310. We study some topological and analytical properties of a Kleinian group G f or which there is an invariant Jordan curve by the action of G. These groups may be infinitely generated.

 7.- (with C. Pommerenke: TU-Berlin, Berlin, Germany)      On simultaneous homomorphisms onto Kleinian groups.      Complex Variables: Theory and Applications 27 (1995), 309-315. Let Γ be a Fuchsian group acting on the upper half-plane H and let G be a Kleinian group with 0, 1, ∞ in its limit set Λ(G). If L denotes the lower hal-plane, then we consider a meromorphic function f : C − R → C − Λ(G) such that f ◦ γ = g ◦ f , for γ ∈ Γ and the same g = f*(γ)∈ G in both halfplanes. We investigate whether this situation can occur except in the trivial case that Λ(G) ⊂ R and f (z ) = J(f (J(z))), where J is the complex conjugation.

 Complex Variables: Theory and Appl. 28 (1995), 121-133. In this paper, we obtain a certain rigidity property for Kleinian groups. This asserts that if F and G (both non-elementary torsion-free Fuchsian groups) and [F,F] = [G,G],then F= G (all equalities are taken in the sense of Mbius transformations). This result is connected to Torelli's theorem (for closed Riemann surfaces) and can be a step in understanding an equivalent for more general type of Riemann surfaces.

 Revista Proyecciones 14 (1995), 75-81. In this article we provide some partial results concerning commutator subgroups of Kleinian groups and how they are determined by them.

1996

 Proc. London Math Soc. 73 (1996), 385-403. We introduce coordinates, given by fixed points, for the marked Schottky space Sg of genus g ≥ 2. With these coordinates, different to the ones given by Bers, Sato and Gerritzen, we can think of the space Sg as an open subset of CxC^{3g–4}. A partial closure of Sg, denoted by NSg and called the marked noded Schottky space of genus g, is considered. Each point in NSg corresponds to a geometrically finite free group of rank g, called a (marked) noded Schottky group of genus g, Conversely, each such group corresponds to a point in NSg. We have that each noded Schottky group of genus g uniformizes a stable Riemann surface of genus g. Moreover, we show that every stable Riemann surface is uniformized by such a group (retrosection theorem with nodes).

 C.R. Math. Rep. Acad. Sci. Canada 18 (1996), 39-42. In these notes we consider a particular family of groups of conformal automorphisms on closed Riemann surfaces of genus at least two. We see that the order of such groups are bounded by the classical $12(g-1)$.  As a consequence, we obtain the results in Zimmermann and May.We also obtain the following: (1) Let $H$ be a group of conformal automorphisms on a stable Riemann surface $R$ of genus at least two, with at least one node, with the property that: if for some $h \in H$ we have that $h^{k}$ keeps invariant a component $R_{i}$ of $R$ minus its nodes and it acts as the identity there, then $h^{k}$ is the identity. Then $H$ has order at most $12(g-1)$. (2) Let $K$ be a function group, different from a Fuchsian one, with invariant component $\Delta$. If $K$ contains a group $G$ such that $\Delta/G$ is a closed Riemann surface of genus $g \geq 2$, then the index $[K:G]$ is bounded by $12(g-1)$.

 Revista Proyecciones 15 (1996), 47-90. In this article we provide some Schottky uniformizations of closed Riemann surfaces defining dihedral groups of conformal automorphisms.

1997

 13.- (with Marcel Hagelberg)        Some generalized Coxeter groups and their orbilfolds.        Revista Matemática Iberoamericana 13 (1997), 543-566. In this note we construct examples of geometric 3-orbifolds with (orbifold) fundamental group isomorphic to a (Z− extension of a) generalized Coxeter group. Some of these orbifolds have either euclidean, spherical or hyperbolic structure. As an application, we obtain an alternative proof of a theorem of Hagelberg, Maclaughlan and Rosenberg. We also obtain a similar result for generalized Coxeter groups.

 14.- (with Gabino González-Diez: UAM Madrid, Spain)        Conformal versus topological automorphisms of compact Riemann surfaces.        Bull. of the London Math. Soc. 29 (1997), 280-284. We produce a family of algebraic curves (closed Riemann surfaces) Sadmitting two cyclic groups H_1 and H_2 of conformal automorphisms, which are topologically (but not conformally) conjugate and such that S/H_i is the Riemann sphere . The relevance of this example is that it shows that the subvarieties of moduli space consisting of points parametrizing curves which occur as cyclic coverings (of a fixed topological type) of the Riemann sphere need not be normal.

1998

 Complex Variables: Theory and Applications 36 (1998), 19-26. In these notes we generalize the following result due to R. Accola: Given a hyperelliptic Riemann surface S of genus g ≥ 2 and n a non-negative integer, there is a smooth n−sheeted covering π:R→ S, where R is a hyperelliptic Riemann surface. We show that the above result extends to the family of η−hyperelliptic Riemann surfaces as follows. Given a η−hyperelliptic Riemann surface S of genus g ≥ 2, a η−hyperelliptic involution τ : S → S and a non-negative integer n, there is a smooth n-sheeted covering π:R→ S, where R is a η*−hyperelliptic Riemann surface for which τ lifts as a η*−hyperelliptic involution, and η* = (n + 1)(η − 1) + 1, if η = 0, η* =0, if η = 0.

 Revista Matemática Complutense 11 (1998), 165-183. In these notes we construct explicit examples of degenerations on the noded Schottky space of genus g>2. The particularity of these degenerations is the invariance under the action of a dihedral group of order 2g. More precisely, we find a two-dimensional complex manifold in the Schottky space such that all groups (including the limit ones in the noded Schottky space) admit a fixed topological action of a dihedral group of order 2g  as conformal automorphisms.

 Complex Variables: Theory and Appl. 36 (1998), 45-66. We consider certain type of Kleinian groups called noded Fuchsian groups. Torsion-free noded Fuchsian groups are divided into two families called noded Schottky groups and noded πg groups, respectively. The extended region of discontinuity is defined together a cuspidal topology. We show that these groups uniformize stable Riemann surfaces and generalize some classical results on torsion-free co-compact Fuchsian groups and Schottky groups to the above ones.

 18.-  Γ-hyperelliptic Riemann surfaces.        Revista Proyecciones 17 (1998), 77-117.   We give some characterizations of γ −hyperelliptic Riemann surfaces of genus g ≥ 2, that is, pairs (S, j ) where S is a closed Riemann surface of genus g and j : S → S is a conformal involution with exactly 2g + 2 − 4γ fixed points. These characterizations are given by Schottky groups, special hyperbolic polygons and algebraic curves.

1999

 Contemporary Mathematics 240 (1999), 209-222. Given a Kleinian group $G$, we denote by $\Omega(G)$ and $P(G)$ its region of discontinuity and its set  of double-cusped parabolic fixed points, respectively. The extended region of discontinuity of $G$ is defined as $\Omega^{ext}(G)=\Omega(G) \cup P(G)$, together the cusp topology. In the present note, we consider noded function groups, that is, finitely generated kleinian groups having an invariant component of their extended region of discontinuity. We sketch an alternative proof of a theorem of Kra and Maskit which says that a noded function group with two invariant components is in fact a noded fuchsian group and, in particular, geometrically finite (extending Maskit's result that a finitely generated Kleinian group with two invariant components is in fact a quasifuchsian group). The main purpose of this note is to show that the commutator subgroup of a torsion-free, non-elementary noded function group $G$ determines it uniquely. For a general non-elementary torsion-free Kleinian group with no invariant component the above rigidity property still yet unknown.

 Revista Proyecciones 18 (1999), 165-173. In previous works we have seen that a finitely generated torsion-free  non-elementary function group is uniquely determined by its commutator subgroup.  In this note, we observe that under the presence of orientation-reversing conformal  automorphisms the above rigidity property still valid. More precisely, we see that finitely generated torsion-free reversing Fuchsian groups of the first kind, without parabolic transformations, are uniquely determined by their commutator subgroup. The arguments of the proof follows the same lines as for the orientable situation.

 Revista Proyecciones 18 (1999), 23-48. We show that a dihedral group H of conformal automorphisms of a closed Riemann surface S can be lifted for a suitable Schottky uniformization of S . In particular, this implies the existence of a suitable symplectic homology basis of S for which the symplectic representation of H has a simple form.

 Revista Proyecciones 18 (1999), 13-21. Given an abstract group of order n, we call its Schottky genus to the minimum genus g ≥ 2 on which it acts as group of conformal automorphisms of Schottky type.  In this article, we compute the Schottky genus for both cyclic and dihedral groups. In particular, we obtain that the Schottky genus of the dihedral group of order 2n is the same as for the cyclic group of order n. Since every dihedral group is of Schottky type, we have that the Schottky genus of a dihedral group of order 2n is also its minimum genus.

2000

 Complex Variables: Theory and Appl. 42 (2000), 183-192. In previous works we have seen that a finitely generated torsion-free non-elementary function group is uniquely determined by its commutator subgroup. In this note, we observe that under the presence of orientation-reversing conformal automorphisms the above rigidity property still valid. More precisely, we see that finitely generated torsion-free reversing Fuchsian groups of the first kind are uniquely determined by their commutator subgroup. The arguments of the proof follow the same lines as for the orientable situation.

 24.- (with Gustavo Labbe: U. la Serena, Chile)        Fixed point parameters for Teichmueller spacesof closed Riemann surfaces.        Revista Proyecciones 19 (2000), 65-94. We provide a set of parameters for the Teichmueller space, of genus g≥2, given by fixed points of some special set of generators for the uniformizing Fuchsian groups. Explicit computations are given in low genus.

 Revista Proyecciones 19 (2000), 157-196. Let  Γn (respectively, Γ∞) be a free group of rank n (respectively, a free group of countable infinite rank). We consider the space of algebraic representations of the group Γn (respectively, Γ∞) Hom(Γn , PGL(2, C)) (respectively, Hom(Γ∞, PGL(2, C))). Inside each of these spaces we consider a couple of open and dense subsets. These subsets contain non-discrete groups of Meobius transformations. We proceed to find complex analytic  parameters for these spaces given by fixed points.

 Revista Proyecciones 20 (2001), 93-126. We consider pairs (S, τ ), where S is a closed Riemann surface of genus five and τ : S → S is some anticonformal involution with fixed points so that K(S, τ ) = {h ∈ Aut(S) : hτ = τ h} has the maximal order 96 and S/τ is orientable, where Aut(S) denotes the full group of conforma/anticonformal automorphismsm of S. We observe that there are exactly two topological ly different choices for  τ . They give non-isomorphic groups K(S, τ ), each one acting topological ly rigid on the respective surface S . These two cases give then two (connect) real algebraic sets of real dimension one in the moduli space of genus 5. We describe these components by classical Schottky groups and with the help of these uniformizations we compute their Riemann matrices.

2001

 Notas Soc. Mat. Chile (N.S.) 1 (2001), 82-91. We extend the notion of Schottky uniformization of a closed Riemann surfaces to the class of stable Riemann surfaces. These type of uniformizations are called stable Schottky uniformizations and the uniformizing groups are called stable Schottky groups. We observe that any stable Riemann surface can be uniformized by a stable Schottky group. This result can be used to show that every anticonformal involution on a stable Riemann surface can be lifted to a suitable stable Schottky uniformization of it.

 Notas Soc. Mat. Chile (N.S.) 1 (2001), 11-21. Parameters for groups of genus two have been constructed involving fixed points and multipliers of certain geometrical generators. The presence of multipliers in the set of parameters produces complications if we want to approach the boundary of the Teichmueller space of genus two. These complications appear when some loxodromic transformations approach to parabolic ones. In this note we consider a different set of coordinates for Fuchsian and quasi-Fuchsian groups of signature (2, 0), that is, groups uniformizing closed Riemann surfaces of genus two. These new coordinates are given by six fixed points (up to normalization) of a particular set of geometrical generators.

 29.- (with A.F. Costa: UNED Madrid, Spain)        Anticonformal automorphisms and Schottky coverings.        Ann. Acad. Scie. Fenn. 26 (2001), 489-508. In this work, we consider anticonformal automorphisms of closed Riemann surfaces and Schottky groups. We study the problem of deciding when an anticonformal automorphism can be lifted for some Schottky covering (Schottky type automorphisms). This can be seen as generalization of the results due to Sibner, Heltai and Natanzon on anticonformal involutions. Also, for the conformal automorphisms, we study the relation between the condition

 Revista Proyecciones 20 (2001), 139-175. In this note we consider a class of groups of conformal automorphisms of closed Riemann surfaces containing those which can be lifted to some Schottky uniformization. These groups are those which satisfy a necessary condition for the Schottky lifting property. We find that all these groups have upper bound 12(g – 1), where g≥2is the genus of the surface. We also describe a sequence of infinite genera g1< g2 < ... for which these upper bound is attained. Also lower bounds are found, for instance, (i ) 4(g+1) for even genus and 8(g – 1) for odd genus. For cyclic groups in such a family, sharp upper bounds are given.

 31.- Γ-hyperelliptic-symmetric Schottky groups.        Complex Variables: Theory and Appl. 45 (2001), 117-141. Let S be a γ-hyperelliptic Riemann surface, with a γ-hyperelliptic involution τ. Assume that S has a symmetry  σ so that  στ=τσ. If H denotes the group generated by  τ and σ, then we show that H is of Schottky type, that is, there is a Schottky uniformization (ωGP: ω S) of S for which the group H lifts. For hyperelliptic Riemann surfaces, we describe explicitly Schottky uniformizations (hyperelliptic-symmetric Schottky groups) for which both τ and σ lift. The particularity of these uniformizations is that both τ and σ are reflected in a marking of the uniformizing groups. For g=2 we use the above groups to describe inside the Schottky space of genus two, the locus of symmetric.

 Revista Proyecciones 20 (2001), 351-365. In this note we consider hyperelliptic-M-symmetric Riemann surfaces, that is, hyperelliptic Riemann surfaces with a symmetry with maximal number of components of fixed points. These surfaces can be represented either by real algebraic curves or by real Schottky groups. To obtain one of these in terms of the other is difficult. In this note we proceed to describe explicit transcendental relations between the different sets of parameters these representations give. This can be used to obtain a computer program which permits obtain numerical approximations of the algebraic curve in terms of real Schottky group and viceversa.

2002

 Revista Matemática Complutense 15 (2002), 11-29. Let  H be a group of conformal automorphisms of a closed Riemann surface S, isomorphic to either of the alternating groups A4  or  A5 or the symmetric groups S4  or S5. We provide necessary and sufficient conditions for the existence of a Schottky uniformization of S for which  H lifts. In particular, together with the previous done works, we exhaust the list of finite groups of Möbius transformations of Schottky type.

 Revista Proyecciones 21 (2002), 21-50. Let us consider an abstract group with the fol lowing presentation G = , where k_{i} , l_{j} ∈ {2, ..., ∞}. We provide conditions in order to find a faithful, discrete and geometrical ly finite representation Θ : G→ PSL(2, C), that is, to represent G as a group of isometries of the hyperbolic three space H^{3}.

 35.- (with A. Vassiliev: U Bergern Bergen, Norway)        Harmonic moduli of family of curves on Teichmueller spaces. Geometry and analysis.        Sci. Ser. A Math. Sci. (N.S.) 8  (2002), 63-73. Study of moduli of family of curves on the universal Teichmueller curve is provided.

 Contemporary Mathematics 311 (2002), 277-299. It is a well known fact that every real Riemann surface can be uniformized by a real Schottky group. These real Schottky groups can be used in order to compute explicitly Riemann matrices of the uniformized real Riemann surfaces. We apply such a method to two family of examples. Our first family are hyperelliptic May's surfaces. As a consequence, for genera $2$ and $3$, we are able to write down algebraic curves in function of the uniformizing Schottky group. The second family is a two real dimensional family of genus $3$ non-hyperelliptic real surfaces. We also may write down for them an algebraic equation in function of the uniformizing Schottky group.

2003

 Complex Variables: Theory and Appl. 48 (2003), 43-62. In a work due to Aigon and Silhol (also Buser and Silhol) a construction of 10 genus two closed Riemann surfaces is done from a given right angled hyperbolic pentagon. In this note we construct real Schottky groups uniformizations of the corresponding constructions. In particular, we are able to write down the algebraic curves obtained in the above work in terms of the parameters of the real Schottky group. We generalize such a construction for any right angled hyperbolic polygon and also consider an example for a nonright angled pentagon.

 Revista Proyecciones 22 (2003), 117-125. We show that a non-elementary finitely generated torsion-free function group is uniquely determined by its commutator subgroup. This is well related to Torelli’s theorem for closed Riemann surfaces. For a general non-elementary torsion-free Kleinian group the above rigidity property still unknown.

2004

 39.- (with Victor Gonzalez: UTFSM and G. Labbe: U. la Serena)        Schottky Uniformizations of Genus 6 Riemann Surfaces with A5 Geometriae Dedicata 106 (1) (2004), 79-95. In this note we construct a 1-complex dimensional family of (marked) Schottky groups of genus 6 with the property that every closed Riemann surface of genus 6 admitting the group A5 as conformal group of automorphisms is uniformized by one of these Schottky groups. In the algebraic limit closure of this family we describe three noded Schottky groups uniformizing the three boundary points of the pencil described by González-Aguilera and Rodriguez. We are able to find a very particular Riemann surface of genus 6 which is a (local) extremal for a maximal set of homologically independent simple closed geodesics. We observe that it is not Wimann''s curve, the only Riemann surface of genus 6 with S5 as group of conformal automorphisms. The Schottky uniformizations permit us to compute a reducible symplectic representation of A5.

 Revista Matematica Iberoamericana 20 (2004), 737-770. Let S be a real closed Riemann surfaces together a reflection τ : S→ S , that is, an anticonformal involution with fixed points. A well known fact due to C. L. May asserts that the group K(S, τ ), consisting on all automorphisms (conformal and anticonformal) of S which commutes with τ , has order at most 24(g−1). The surface S is called maximally symmetric Riemann surface if |K(S, τ )| = 24(g−1). In this note we proceed to construct real Schottky uniformizations of all maximally symmetric Riemann surfaces of genus g ≤ 5. A method due to Burnside permits us the computation of a basis of holomorphic one forms in terms of these real Schottky groups and, in particular, to compute a Riemann period matrix for them. We also use this in genus 2 and 3 to compute an algebraic curve representing the uniformized surface S . The arguments used in this note can be programed into a computer program in order to obtain numerical approximation of Riemann period matrices and algebraic curves for the uniformized surface S in terms of the parameters defining the real Schottky groups.

 41.- (with Rubi Rodriguez: PUCCH)        Real Schottky uniformizations and Jacobians of May surfaces.        Revista Matematica Iberoamericana 20 (2004), 627-646. Given a closed Riemann surface R of genus p ≥ 2 together with an anticonfor- mal involution τ : R → R with fixed points, we consider the group K (R, τ ) consisting of the conformal and anticonformal automorphisms of R which commute with τ . It is a well known fact due to C. L. May that the order of K (R, τ ) is at most 24(p − 1) and that such an upper bound is attained for infinitely many, but not all, values of p. May also proved that for every genus p ≥ 2 there are surfaces for which the order of K (R, τ ) can be chosen to be 8p and 8(p + 1). These type of surfaces are called May surfaces. In this note we construct real Schottky uniformizations of every May surface. In par- ticular, the corresponding group K (R, τ ) lifts to such an uniformization. With the help of these real Schottky uniformizations, we obtain (extended) symplectic representations of the groups K (R, τ ). We study the families of principally polarized abelian varieties admit- ting the given group of automorphisms and compute the corresponding Riemann matrices, including those for the Jacobians of May surfaces.

 Ann. Acad. Sci. Fenn.  29 (2004), 329-344. A conformal automorphism φ: S → S of a closed Riemann surface S of genus p ≥ 2 is said to be of Schottky type if there is a Schottky uniformization of S for which φ lifts. In the case that φ is of Schottky type, we have associated a geometrically finite Kleinian group K , generated by the uniformizing Schottky group G and any of the liftings of φ . We have that K contains G as a normal subgroup and K/G is cyclic. In this note we describe, up to topological equivalence, all possible groups K obtained in this way. Equivalently, if we are given a handlebody M of genus p ≥ 2 and an orientation preserving homeomorphism of finite order φ , then we proceed to describe, up to topological equivalence, the hyperbolic structures of the orbifold M/φ having bounded by below injectivity radius.

 43.- Abelian groups of Schottky type.        Revista Proyecciones 23 (2004), 187-203. We study the problem of lifting an Abelian group H of automorphisms of a closed Riemann surface S (containing anticonformals ones) to a suitable Schottky uniformization of S (that is, when H is of Schottky type). If H+ is the index two subgroup of orientation preserving automorphisms of H and R = S/H+, then H induces an anticonformal automorphism τ : R→ R. If τ has fixed points, then we observe that H is of Schottky type. If τ has no fixed points, then we provide a sufficient condition for H to be of Schottky type. We also give partial answers for the excluded cases.

2005

 44.- Automorphisms of Schottky type.        Ann. Acad. Scie. Fenn.  30 (2005), 183-204. A group H of (conformal/anticonformal) automorphisms of a closed Riemann surface S of genus g ≥ 2 is said of Schottky type if there is a Schottky uniformization of S for which it lifts. We observe that H is of Schottky type if and only if it leaves invariant a collection of pairwise disjoint simple loops which disconnect S into genus zero surfaces. Moreover, in the case that H is a cyclic group (either generated by a conformal or an anticonformal automorphism) we provide a simple to check necessary and sufficient condition in order for it to be of Schottky type.

 45.- (with J. Figueroa: UTFSM)         Numerical Schottky uniformizations.        Geometriae Dedicata 111 (2005), 125-157. A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut+(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut+(S,τ)| is bounded above by 12(g−1). We say that (S,τ) is maximally symmetric if |Aut+(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.

 Complex Variables: Theory and Appl. 50 (2005), 401-426. A real algebraic curve of genus g is a pair (S,τ), where S is a closed Riemann surface of genus g and τ :S → S is an anticonformal involution. It was already known to Koebe that each real algebraic curve for which τ is a reflection can be uniformized by a real Schottky group, that is, a Schottky group that keeps invariant the unit circle. In the case that τ is an imaginary reflection, we produce uniformizations by either (i) real noded Klein-Schottky groups (once we have chosen some points on S as phantom nodes) or (ii) Klein-Schottky groups. We also give explicit descriptions of the real algebraic curves of genus 2 in terms of these types of uniformizing groups.

 47.- Schottky uniformizations of genus 3 and 4 reflecting ${\mathcal S}_{4}$.        Journal of the London Math. Soc. 72 (1) (2005), 185-204. Schottky uniformizations are provided of every closed Riemann surface S of genus g∈{3,4} admitting the symmetric group S4 as group of conformal automorphisms. These Schottky uniformizations reflect the group S4 and permit concrete representations of S4 to be obtained in the respective symplectic group Spg(Z). Their corresponding fixed points, in the Siegel space, give principally polarized Abelian varieties of dimension g. For g = 3, and for some cases of g = 4, they turn out to be holomorphically equivalent to the product of elliptic curves.

 48.- Schottky uniformizations of Z_{2}^{2}  actions on Riemann surfaces.        Rev. Mat. Complut. 18 (2005), no. 2, 427–453. Given a closed Riemann surface $S$ together a group of its conformal automorphisms $H \cong {\mathbb Z}_{2}^{2}$, it is known that there are Schottky uniformizations of $S$ realizing $H$. In this note we proceed to give an explicit Schottky uniformizations for each of all different topological actions of ${\mathbb Z}_{2}^{2}$ as group of conformal automorphisms on a closed Riemann surface.

 49.- (with. B. Maskit: SUNY at Stony Brook, NY, USA)         On Klein-Schottky groups.         Pacific J. of Math. (2) 220 (2005), 313-328. The retrosection theorem asserts that every closed Riemann surface of genus g ≥ 1 can be uniformized by a Schottky group of rank g. Here we define and topologically classify Klein–Schottky groups; these are the freely acting extended Kleinian groups whose orientation-preserving subgroup is a Schottky group. These groups yield uniformizations of all nonorientable closed Klein surfaces.

 50.- (with G. Rosenberger: Univ. Dortmund, Dortmund, Germany)        Torsion free commutator subgroups of generalized Coxeter groups.        Results in Mathematics 48 (2005), 50-64. In this note we consider generalized Coxeter groups and we study the problem of when their commutator subgroup is torsion free. As a consequence we describe all (i) Coxeter groups, (ii) triangle groups and (iii) index two orientation preserving subgroups of the finite co-volume hyperbolic Coxeter tetrahedra, for which the commutator subgroup is torsion free.

2006

 51.- (with B. Maskit: SUNY at Stony Brook, NY, USA)         Lowest uniformizations of compact real surfaces.         Contemporary Mathematics 397 (2006), 145-152. We look at uniformizations of compact real surfaces, where the boundary curves are required to be covered by arcs of fixed points of reflections. We investigate the lowest such uniformizations; in particular, we show that each such lowest uniformization is a combination theorem free product of cyclic groups, and that each such lowest uniformization has a Schottky group as its orientation preserving half. We also give some examples of lowest uniformizations as well as some examples showing that, for real surfaces with boundary, the uniformization corresponding to the universal covering of the interior is essentially non-unique.

 52.- (with B. Maskit: SUNY at Stony Brook, NY, USA)        On neoclassical Schottky groups.        Transactions of the AMS. 358 (2006), 4765-4792. The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are sufficiently complicated''. We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus 3 that cannot be uniformized by any neoclassical Schottky group (i.e., classical noded Schottky group); the second is an explicit example of a sufficiently complicated noded Schottky group in genus 3.

 53.- (with A. Vasiliev: U. Bergen, Bergen, Norway)        Noded Teichmueller Spaces.       Journal d'Analyse Mathematique 99 (2006), 89-107. Let G be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmüller space T(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant under G and supported in Δ. In this paper we propose a partial closure of T(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted by NT(Δ,G) and called thenoded Teichmüller space of G supported in Δ. Some concrete examples are discussed.

2007

 54.- (with I. Markina and A. Vassiliev: U Bergern Bergen, Norway)        Finite dimensional grading of the Virasoro algebra.        Georgian Mathematical Journal  14 (2007), No. 3, 419-434. The Virasoro algebra is a central extension of the Witt algebra, the complexified Lie algebra of the sense preserving diffeomorphism group of the circle Diff S^1. It appears in Quantum Field Theories as an infinite dimensional algebra generated by the coefficients of the Laurent expansion of the analytic component of the momentum-energy tensor, Virasoro generators. The background for the construction of the theory of unitary representations of Diff S^1 is found in the study of Kirillov's manifold Diff S^1/S^1. It possesses a natural Kählerian embedding into the universal Teichmüller space with the projection into the moduli space realized as an infinite-dimensional body of the coefficients of univalent quasiconformally extendable functions. The differential of this embedding leads to an analytic representation of the Virasoro algebra based on Kirillov's operators. In this paper we overview several interesting connections between the Virasoro algebra, Teichmüller theory, Löwner representation of univalent functions, and propose a finite-dimensional grading of the Virasoro algebra such that the grades form a hierarchy of finite dimensional algebras which, in their turn, are the first integrals of Liouville partially integrable systems for coefficients of univalent functions.

 55.- (with Beatriz Estrada and Ernesto Martinez: UNED Madrid, Spain)        Q-n-gonal Klein Surfaces.        Acta Mathematica Sinica 23 (2007), No. 10, 1833-1844. We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/φ of algebraic genus q. These Klein surfaces are called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the (admissible) values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given: q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.

 56.- (with Maximiliano Leyton: Univ. Grenoble, Grenoble, France)         On uniqueness of automorphisms groups of Riemann surfaces.         Revista Matematica Iberoamericana 23, No. 3 (2007), 793-810. Let γ , r, s, ≥ 1, non-negative integers. If p is a prime sufficiently large relative to the values γ , r and s, then a group H of conformal automorphisms of a closed Riemann surface S of order p^s so that S/H has signature (γ , r) is the unique such subgroup in Aut(S ). Explicit sharp lower bounds for p in the case (γ , r, s) ∈ {(1, 2, 1), (0, 4, 1)} are provided. Some consequences are also derived.

2008

 57.- On the Retrosection Theorem.       Revista Proyecciones 27, No. 1 (2008), 29-61. We survey some old and new results related to the retrosection theorem and some of its extensions to compact Klein surfaces, stable Riemann surfaces and stable Klein surfaces.

 58.- (with A. Carocca:PUCCH, Victor Gonzalez: UTFSM and Rubi Rodriguez: PUCCH)        Generalized Humbert Curves.        Israel Journal of Mathematics 164, No. 1 (2008), 165-192. In this note we consider a certain class of closed Riemann surfaces which are a natural generalization of the so called classical Humbert curves.  They are given by closed Riemann surfaces $S$ admitting $H \cong \mathbb{Z}_{2}^{k}$ as a group of conformal automorphisms so that $S/H$ is an orbifold of signature $(0,k+1;2, \ldots ,2)$. The classical ones are given by $k=4$. We mainly describe some of its generalities and provide Fuchsian, algebraic and Schottky descriptions.

 59.- (with Mauricio Godoy: U Bergern, Bergen, Norway.         Revista Proyecciones 27, No. 2 (2008), 149-161. Let Q : �l^2 → l^2 be a symmetric and positive semi-definite linear operator and f_j : R→ R (j = 1, 2, ...) be real functions so that, f_j (0) = 0 and, for every x = (x1 , x2 , ....)∈l^2, it holds that f(x) := (f_1 (x1 ), f_2 (x2 ), ...)∈l^2. Sufficient conditions for the existence of non-trivial solutions to the semilinear problem Qx = f (x) are provided. Moreover, if G is a group of orthogonal linear automorphisms of l^�2 which commute with Q, then such sufficient conditions ensure the existence of non-trivial solutions which are invariant under G. As a consequence, sufficient conditions to ensure solutions of nonlinear partial difference equations on finite degree graphs with vertex set being either finite or infinitely countable are obtained. We consider adaptations to graphs of both Matukuma type equations and Helmholtz equations and study the existence of their solutions.

 60.- Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs.       Journal of Difference Equations and Applications 14 No. 9 (2008), 953-969. Let $Q$ be a $m\times m$ real matrix and $f_{j}:{\mathbb R} \to {\mathbb R}$, $j=1,...,m$, be some given functions.  If $x$ and $f(x)$ are column vectors whose $j$-coordinates are $x_{j}$ and $f_{j}(x_{j})$, respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system $Qx=f(x)$ for  $Q$ symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by J.M. Neuberger in \cite{Neuberger}. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph ${\mathcal G}$ is just any linear operator $D:C^{0}({\mathcal G}) \to C^{0}({\mathcal G})$, where $C^{0}({\mathcal G})$ is the real vector space of real maps defined on the vertex set $V$ of the graph. Given a derivation $D$ and a function $F:V \times {\mathbb R} \to {\mathbb R}$, one has associated  a partial difference equation $D \mu = F(v,\mu)$, and one searchs for solutions $\mu \in C^{0}({\mathcal G})$. Sufficient conditions in order to have non-trivial solutions of  partial difference equations on any  finite, connected simple graph for $D$ symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair $({\mathcal G},d)$, where ${\mathcal G}$ is a connected finite degree simple graph and $d$ is a positive function on the set of edges of the graph. The metric $d$ permits to consider some classical derivations, such as the Laplacian operator $\triangle_{2}$. In \cite{Neuberger} was  considered the nonlinear elliptic partial difference equations $\triangle_{2} u=F(u)$, for  the metric $d=1$.

2009

 61.- (with Gabino Gonzalez: UAM, Madrid, Spain and Maximiliano Leyton: Univ. Grenoble, Grenoble, France)         Generalized Fermat Curves.        Journal of Algebra 321 (2009), 1643-1660. A closed Riemann surface $S$ is a generalized Fermat curve of type $(k,n)$ if it admits a group of automorphisms $H \cong Z_{k}^{n}$  such that the quotient ${\mathcal O}=S/H$ is an orbifold with signature $(0,n+1;k,...,k)$, that is, the Riemann sphere with $(n+1)$ conical points, all of same order $k$. The group $H$ is called a generalized Fermat group of type $(k,n)$ and the pair $(S,H)$ is called a generalized Fermat pair of type $(k,n)$. We study some of the properties of generalized Fermat curves and, in particular, we provide simple algebraic curve realization of a generalized Fermat pair $(S,H)$ in a lower dimensional projective space than the usual canonical curve of $S$ so that the normalizer of $H$ in $Aut(S)$ is still linear. We (partially) study the problem of the uniqueness of a generalized Fermat group on a fixed Riemann surface. It is noted that the moduli space of generalized Fermat curves of type $(p,n)$, where $p$ is a prime, is isomorphic to the moduli space of orbifolds of signature $(0,n+1;p,...,p)$. Some applications are: (i) an example of a pencil consisting of only non-hyperelliptic Riemann surfaces of genus $g_{k}=1+k^{3}-2k^{2}$, for every integer $k \geq 3$, admitting exactly three singular fibers, (ii) an injective holomorphic map $\psi:{\mathbb C}-\{0,1\} \to {\mathcal M}_{g}$, where ${\mathcal M}_{g}$ is the moduli space of genus $g \geq 2$ (for infinitely many values of $g$), and (iii) a description of all  complex surfaces isogenous to a product with invariants $p_{g}=q=0$ and covering group equal to ${\mathbb Z}_{5}^{2}$ or ${\mathbb Z}_{2}^{4}$.

 62.- A theoretical algorithm to get a Schottky uniformization from a Fuchsian one.        pp. 193-204.        Analysis and Mathematica Physics        Series: Trends in Mathematics        Eds.: Gustafsson, Bjorn; Vassiliev Alexander        2009, Approx. 525 p., Hardcover        ISBN: 978-3-7643-9905-4        Birkhauser. Riemann surfaces appear in many diﬀerent areas of mathematics  and physics, as in algebraic geometry, the theory of moduli spaces, topological ﬁeld theories, cosmology, quantum chaos and integrable systems. A closed Riemann surface may be described in many diﬀerent forms; for instance, as algebraic curves and by means of diﬀerent topological classes of  uniformizations. The highest uniformization corresponds to Fuchsian groups and the lowest ones to Schottky groups. In this note we discuss a theoretical algorithm which relates a Schottky group from a given Fuchsian group both uniformizing the same closed Riemann surface.

 63.- Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals.          Archiv der Mathematik 93 (2009), 219-222. The known examples of explicit equations for Riemann surfaces, whose field of moduli is different from their field of definition, are all hyperelliptic. In this paper we construct a family of equations for non-hyperelliptic Riemann surfaces, each of them is isomorphic to its conjugate Riemann surface, but none of them admit an anticonformal automorphism of order 2; that is, each of them has its field of moduli, but not a field of definition, contained in R. These appear to be the first explicit such examples in the non-hyperelliptic case.

 64.- (with Bernard Maskit: SUNY at Stony Brook, NY, USA)        A Note on the Lifting of Automorphisms.        In Geometry of Riemann Surfaces.        Lecture Notes of the London Mathematics Society 368, 2009.       Edited by Fred Gehring, Gabino Gonzalez and Christos Kourouniotis.       ISBN-13: 9780521733076 The goal of this paper is to study uniformizations of surfaces and orbifolds (either Riemann or Klein). There is of course a well developed theory of regular coverings based on the correspondence with subgroups of the fundamental group. This theory can be applied to branched regular coverings by removing the discrete set of branch points and their preimages. There is however no known extension of the theory of regular coverings to include folded coverings; this is the case where the covering group contains elements with real co-dimension one sets of ﬁxed points. In this paper we take a ﬁrst step towards laying a foundation for the study of such coverings.

2010

 65.- Double Schottky Covers.        Revista Matematica Complutense No.1 23 (2010), 37-48. Let R → S be an unbranched covering of the degree two between closed Riemann surfaces and let a generate the group of the deck transformations. If S is hyperelliptic, then its hyperelliptic involution lifts to two holomorphic involutions and each of the two corresponding quotient orbifolds has underlying structure of a hyperelliptic Riemann surface with the hyperelliptic involution induced by a. In this paper we present a Schottky description of this picture and provide some applications to handlebodies.

 66.- (with Bernard Maskit: SUNY at Stony Brook, NY, USA)        Fixed points of imaginary reflections on hyperbolic handlebodies.        Mathematical Proceedings of the Cambridge Philosophical Society 148 (2010), 135-158. A Klein-Schottky group is an extended Kleinian group, containing no reflections and whose orientation-preserving half is a Schottky group. A dihedral-Klein-Schottky group is an extended Kleinian group generated by two different Klein-Schottky groups, both with the same orientation-preserving half.  We provide a structural description of the dihedral-Klein-Schottky groups.  Let $M$ be a handlebody of genus $g$, with a Schottky structure. An imaginary reflection $\tau$ of $M$  is  an orientation-reversing homeomorphism of $M$, of order two, whose restriction to its interior is an hyperbolic isometry having at most isolated fixed points. It is known that the number of fixed points of $\tau$  is at most $g+1$; $\tau$ is called a maximal imaginary reflection if it has $g+1$ fixed points. As a consequence of the structural description of the dihedral-Klein-Schottky groups, we are able to provide  upper bounds for the cardinality of the set of fixed points of two or three different imaginary reflections acting on a handlebody with a Schottky structure. In particular, we show that maximal imaginary reflections are unique.

 67.- (with Alexander Mednykh: Novosibirsk Institute of Math, Russia)       Geometric orbifolds with torsion free derived subgroup.       Siberian Math. Journal. 51 No.1 (2010), 38-47. A geometric orbifold of dimension $d$ is of the form ${\mathcal O}=X/K$, were $(X,G)$ is a geometry of dimension $d$ and $K  68.-A short note on M-symmetric hyperelliptic Riemann Surfaces. Cubo A Mathematical Journal 12 No.1 (2010), 175-179. We provide an argument, based on Schottky groups, of a result due to B. Maskit which states a necessary and sufficient condition for the double oriented cover of a planar compact Klein surface of algebraic genus at least two to be a hyperelliptic Riemann surface.  69.- Maximal Schottky extension groups. Geometriae Dedicata. 146 No1 (2010), 141-158. A Schottky extension group is a Kleinian group$K$containing a Schottky group$G$of rank$g \geq 2$as a normal subgroup. It is well known that the index of$G$in$K$is at most$12(g-1)$; if the index is$12(g-1)$, then we say that$K$is a maximal Schottky extension group. A structural description of the maximal Schottky extension groups using$2$-dimensional arguments, internal to Riemann surfaces and classical Kleinian groups in spirit, is provided. As a consequence, we re-obtain Zimmermann's result which states that a maximal Schottky extension group is isomorphic to one of the following groups $$D_{2}*_{{\mathbb Z}_{2}} D_{3}, \; D_{3}*_{{\mathbb Z}_{3}} {\mathcal A}_{4}, \; D_{4}*_{{\mathbb Z}_{4}} {\mathfrak S}_{4}, \; D_{5}*_{{\mathbb Z}_{5}} {\mathcal A}_{5},$$ where$D_{r}$is the dihedral group of order$2r$,${\mathcal A}_{r}$is the alternating group in$r$letters and${\mathfrak S}_{4}$is the symmetric group in$4$letters. The methods used by Zimmermann are from combinatorial group theory (finite extensions of free groups) and also dimension three, so our arguments are different.  70.- Lowest uniformizations of closed Riemann orbifolds. Revista Matematica Iberoamericana 26 No.2 (2010), 639-649. A Kleinian group containing a Schottky group as a finite index subgroup is called a Schottky extension group. If$\Omega$is the region of discontinuity of a Schottky extension group$K$, then the quotient$\Omega/K$is a closed Riemann orbifold; called a Schottky orbifold. Closed Riemann surfaces are examples of Schottky orbifolds as a consequence of the Retrosection Theorem. Necessary and sufficient conditions for a Riemann orbifold to be a Schottky orbifold are due to M. Reni and B. Zimmermann in terms of the signature of the orbifold. It is well known that the lowest uniformizations of a closed Riemann surface are exactly those for which the Deck group is a Schottky group. In this paper we extend such a result to the class of Schottky orbifolds, that is, we prove that the lowest uniformizations of a Schottky orbifold are exactly those for which the Deck group is a Schottky extension group.  71.- Maximal virtual Schottky groups: Explicit constructions. Revista Colombiana de Matematicas 44 (2010), 41-57. A Schottky group of rank$g$is a purely loxodromic Kleinian group isomorphic to the free group of rank$g$. A virtual Schottky group is a Kleinian group containing a Schottky group as a finite index subgroup. A virtual Schottky group$K$containing as finite index a Schottky group$G$of rank$g \in \{0,1\}$is an elementary Kleinian group. Moreover, for each$g \in \{0,1\}$and for every integer$n \geq 2$, it is possible to find$K$and$G$as above for which the index of$G$in$K$is$n$. It is known that if$\Gamma$is a Schottky subgroup of rank$g \geq 2$of finite index in$K$, then the index is at most$12(g-1)$. If$K$contains a Schottky subgroup of rank$g \geq 2$and index$12(g-1)$, then$K$is called a maximal virtual Schottky group. We provide explicit examples of maximal virtual Schottky groups and corresponding explicit Schottky normal subgroups of rank$g \geq 2$of lowest rank and index$12(g-1)$. Every maximal Schottky extension Schottky group is quasiconformally conjugate to one of these explicit examples. Schottky space of rank$g$is a finite dimensional complex manifold that parametrizes quasiconformal deformations of Schottky groups of rank$g$. If$g \geq 2$, Schottky space of rank$g$has dimension is$3(g-1)$. Each virtual Schottky group, containing a Schottky group of rank$g$as a finite index subgroup, produces a sublocus in Schottky space, called a Schottky strata. The maximal virtual Schottky groups produce the maximal Schottky strata. We provide some facts about these stratum and, in particular, the case of maximal ones.  72.- (with Mauricio Godoy: University of Bergen) Navier-Stokes equations on weighted graphs. Complex Analysis and Operator Theory 4 (2010), 525-540. Navier-Stokes equations arise in the study of incompressible fluid mechanics, star movement inside a galaxy, dynamics of airplane wings, etc. In the case of Newtonian incompressible fluids, we propose an adaptation of such equations to finite connected weighted graphs such that it produces an ordinary differential equation with solutions contained in a linear subspace, this subspace corresponding to the Newtonian conservation law. We discuss the particular case when the graph is the complete graph$K_{m}$, with constant weight, and provide a necessary and sufficient condition for it to have solutions.  73.- (with Raquel Diaz: UCM, Ignacio Garijo: UNED and Grzegorz Gromadzki: U. Gdansk) Structure of Whittaker groups and application to conformal involutions on handlebodies. Topology and its Applications 157 (2010), 2347-2361. The geometrically finite complete hyperbolic Riemannian metrics in the interior of a handlebody of genus$g$, having injectivity radius bounded away from zero, are exactly those produced by Schottky groups of rank$g$; these are called Schottky structures. A Whittaker group of rank$g$is by definition a Kleinian group$K$containing, as an index two subgroup, a Schottky group$\Gamma$of rank$g$. In this case,$K$corresponds exactly to a conformal involution on the handlebody with Schottky structure given by$\Gamma$. In this paper we provide a structural description of Whittaker groups and, as a consequence of this, we provide some facts concerning conformal involutions on handlebodies. For instance, we provide a formula to count the type and the number of connected components of the set of fixed points of a conformal involution of a handlebody with a Schottky structure in terms of a group of automorphisms containing the conformal involution. 2011  74.- (with Mika Seppala) Numerical Schotky Uniformizations: Myrberg's Opening Process. In: Lecture Notes in Mathematics 2013 (2011), 195-209. We provide a proof for the convergence of an algorithm due to P.J. Myberg, which allows one to numerically approximate a Schottky uniformization for a given hyperelliptic Riemann surface. If the branch points of the hyperelliptic Riemann surface are real, the algorithm approximates a classical Schottky uniformization.  75.- The fiber product of Riemann surfaces: A Kleinian group point of view. Analysis and Mathematical Physics 1 (2011), 37-45. Let$P_{1}:S_{1} \to S$and$P_{2}:S_{2} \to S$be non-constant holomorpic maps between closed Riemann surfaces. Associated to the previous data is the fiber product$S_{1} \times_{S} S_{2}=\{(x,y) \in S_{1} \times S_{2}: P_{1}(x)=P_{2}(y)\}$. This is a compact space which, in general, fails to be a Riemann surface at some (possible empty) finite set of points$F \subset S_{1} \times_{S} S_{2}$. One has that$S_{1} \times_{S} S_{2}-F$is a finite collection of analytically finite Riemann surfaces. By filling out all the punctures of these analytically finite Riemann surfaces, we obtain a finite collection of closed Riemann surfaces; whose disjoint union is the normal fiber product$\widetilde{S_{1} \times_{S} S_{2}}$. In this paper we prove that the connected components of$\widetilde{S_{1} \times_{S} S_{2}}$of lowest genus are conformally equivalents if they have genus different from one (isogenous if the genus is one). A description of these lowest genus components are provided in terms of certain class of Kleinian groups; B-groups.  76.- On the inverse uniformization problem: real Schottky uniformizations. Revista Matematica Complutense 24 (2011), 391-420. Classical uniformization theorem, together with Torelli's theorem, assert that each closed Riemann surface may be described in terms of Fuchsian uniformizations (highests uniformizations), Schottky uniformizations (lowest uniformizations), algebraic curves, Riemann period matrices, etc. The inverse uniformization problem may be stated as to provide any or some of the other descriptions once one of them is given. In general (part of) this inverse problem has been (numerically) partially solved for hyperelliptic Riemann surfaces. In this paper we provide a family of real Riemann surfaces, in general non-hyperelliptic ones, which are described in terms of Fuchsian and Schottky uniformizations. The explicit relation between these two uniformizations is given. The Schottky uniformization is used to compute a suitable Riemann period matrix of the uniformized surface so that its coefficients are given in terms of the corresponding Schottky group. Two explicit examples, one of genus$2$and the other of genus$3$(non-hyperelliptic), are provided and (numerically) algebraic curve representations are given for them.  77.- Conjugacy classes of Automorphisms$p$-Groups. Bulletin of the Korean Math. Society. 48 No.4 (2011), 847-851. In this paper we provide examples of pairs of conformally non-equivalent, but topologically equivalent,$p$-groups$H_{1}, H_{2}<{\rm Aut}(S)$, where$S$is a closed Riemann surface of genus$g \geq 2$, so that$S/H_{j}$has genus zero and all its cone points are of order equal to$p$.  78.- (with Raquel Díaz: UCM and Ignacio Garijo: UNED (España)) Uniformization of conformal involutions on stable Riemann surfaces. Israel Journal of Math. 186 (2011), 297-331. Let$S$be a closed Riemann surface of genus$g$. It is well known that there are Schottky groups producing uniformizations of$S$(Retrosection Theorem). Moreover, if$\tau\colon S \to S$is a conformal involution, it is also known that there is a Kleinian group$K$containing, as an index two subgroup, a Schottky group$G$that uniformizes$S$and so that$K/G$induces the cyclic group$\langle \tau \rangle$. Let us now assume$S$is a stable Riemann surface and$\tau\colon S \to S$is a conformal involution. Again, it is known that$S$can be uniformized by a suitable noded Schottky group, but is not known whether or not there is a Kleinian group$K$, containing a noded Schottky group$G$of index two, so that$G$uniformizes$S$and$K/G$induces$\langle \tau \rangle$. In this paper we discuss this existence problem and provide some partial answers: (1) a complete positive answer for genus$g \leq 2$and for the case that$S/\langle \tau \rangle$is of genus zero; (2) the existence of a Kleinian group$K$uniformizing the quotient stable Riemann orbifold$S/\langle\tau\rangle$. Applications to handlebodies with orientation-preserving involutions are also provided.  79.- (with Yolanda Fuertes: UAM (España)) On unbranched non-normal degree four coverings of hyperelliptic Riemann surfaces. Quarterly Journal of Math. 62 (2011), 593–605; doi:10.1093/qmath/haq017 The number of unramified normal coverings of a closed Riemann surface$C$with group of covering transformations isomorphic to$\mathbb{Z}_{2}^{2}$or$\mathbb{Z}_2$is well known. If$C$is hyperelliptic, then Horiouchi has given the explicit algebraic\ equations of the subset of those covers which turn out to be hyperelliptic themselves and, recently, G. Gonz\'alez-Diez and the first author have obtained the remaining ones for both cases. In this paper, we obtain the algebraic equations for all unbranched four degree coverings of a hyperelliptic Riemann surface with monodromy group isomorphic to the dihedral group$D_4$and such that the hyperelliptic involution lifts to the covering. As an immediate application we obtain examples of curves whose field of moduli is$\mathbb{Q}$but they can not be defined over$\mathbb{Q}$. The examples in this paper can be defined over a real quadratic extension of$\mathbb{Q}$.  80- Lifting di-analytic involutions of compact Klein surfaces to extended-Schottky uniformizations. Fundamenta Mathematicae 214 (2011), 161-180 Let$S$be a compact Klein surface together with a di-analytic involution$\kappa:S \to S$. The lowest uniformizations of$S$are those whose deck group is an extended-Schottky group, that is, an extended Kleinian group whose orientation preserving half is a Schottky group. If$S$is a bordered compact Klein surface, then it is well known that$\kappa$can be lifted with respect to a suitable extended-Schottky uniformization of$S$. In this paper, we complete the above lifting property by proving that if$S$is a closed Klein surface, then$\kappa$can also be lifted to a suitable extended-Schottky uniformization. 2012  81- A simple remark on fields of definition Proyecciones Journal of Mathematics 31 No. 1 (2012), 25-28. Let$K

 82.- On Conjugacy of p-gonal automorphisms.       Bulletin of the Korean Math. Society 49 No. 2 (2012), 411-415. In 1995 it was proved by Gonz\'alez-Diez that the cyclic group generated by a $p$-gonal automorphism of a closed Riemann surface of genus at least two is unique up to conjugation in the full group of conformal automorphisms. Later, in 2008, Gromadzki provided a different and shorter proof of the same fact using the Castelnuovo-Severi theorem. In this paper we provide another proof which is shorter and is just a simple use of Sylow's theorem together with the Castelnuovo-Severi theorem. This method permits also to obtain the following generalization: the cyclic group generated by a conformal automorphism of order $p$ of a handlebody with a Kleinian structure  and quotient the three-ball is unique up to conjugation in the full group of conformal automorphisms.

 83.- (with Mauricio Godoy)         Parabolic and Elliptic Partial Difference Equations: Towards a Discrete Solution of Navier-Stokes Equations (Capitulo de Libro).       Navier-Stokes Equations: Properties, Description and Applications. Nova Publishers.       Series: Mathematics Research Developments Physics Research and Technology (2012)        ISBN: 978-1-61324-590-3 In the present chapter we survey some results about partial difference equations (PdEs) in weighted graphs obtained previously by the authors, in the case of finite graphs and countable graphs of finite degree. We discuss the existence of solution of elliptic PdEs via an associated semilinear matrix equation and under growth conditions of the forcing term. As applications of these techniques, we study the discrete analogues of some classical elliptic partial differential equations such as Matukuma, Helmholtz and Lane-Emden-Fowler. Finally we discuss our results concerning the discrete Navier-Stokes equation and give some explicit solutions for concrete weighted graphs and discuss possible graphs that simplify the applications in the planar case.

 84.- Erratum to: Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals        Archive der Mathematik 98 (2012), 449–45. Published online May 15, 2012        DOI 10.1007/s00013-012-0378-y Erratum to: Arch. Math. 93 (2009), 219–224 DOI 10.1007/s00013-009-0025-4. In this note we correct a computation of the original paper and, moreover, we state the main result in a more general form.

 85.- Numerical Schottky uniformizations of certain cyclic L-gonal curves        Computational Algebraic and Analytic Geometry.        Contemporary Mathematics 572 (2012)        Edited by: Mika Seppälä, Florida State University, Tallahassee, FL and Emil Volcheck        ISBNs: 978-0-8218-6869-0 (p); 978-0-8218-8989-3 (e)        DOI: http://dx.doi.org/10.1090/conm/572 We generalize Myrberg's algorithm to provide numerical Schottky uniformizations of algebraic curves of the form $$y^{L}=\prod_{j=1}^{r} (x-a_{j})^{L/n_{j}} (x-b_{j})^{L-L/n_{j}},$$ where $n_{j} \geq 2$ are integers, $L=\mbox{lcm}(n_{1},...,n_{r})$, where lcm stands for lowest common multiple" and $a_{1}$,..., $a_{r}$, $b_{1}$,..., $b_{r}$ are pairwise different points in the complex plane ${\mathbb C}$. This will be a consequence of a numerical algorithm that permits to approximate certain type of uniformizations, called Whittaker uniformizations, of Riemann orbifolds with signatures of the form $(0;n_{1},n_{1},n_{2},n_{2},...,n_{r},n_{r})$.

 86- A remark on (p,q)-gonal quasiplatonic Riemann surfaces       Geometriae Dedicata 160 (2012), 309-312.       (http://www.springer.com/alert/urltracking.do?id=L4f5879M92f8a4Sa821618)       doi: 10.1007/s10711-011-9683-z We provide some remarks to a recent paper of Gromadzki, Weaver and Wootton about quasiplatonic $(p,n)$-gonal surfaces, where, among the others,  they prove that for every prime $p$ and $n>1$ there are just finitely many quasiplatonic strongly $(p,n)$-gonal surfaces. They remarked  that this does not hold for $n=0,1$ and $p=2$.  We provide  examples to see that the above property fails also for such $n$ for  every prime $p$. The authors also conjectured that the strong hypothesis is essential which is false since for given  genus $g \geq 2$ there are  only finitely many  quasiplatonic surfaces up to conformal equivalence.

 87.- (with Sebastian Reyes)      Fields of moduli of classical Humbert curves.      Quarterly Journal of Math. 63 (2012), 919–930; doi:10.1093/qmath/har017 The computation of the field of moduli of a closed Riemann surface seems to be a very difficult problem and even more difficult is to determine if the field of moduli is a field of definition. In this paper we consider the family of closed Riemann surfaces $S$ of genus five admitting a group $H$ of conformal automorphisms isomorphic to ${\mathbb Z}_{2}^{4}$. It turns out that $S$ is non-hyperelliptic and that $S/H$ is an orbifold of signature $(0;2,2,2,2,2)$. We compute the field of moduli of these surfaces and we prove that they are fields of definition. This result is in contrast with the case of the closed Riemann surfaces $R$ admitting a group $K$ of conformal automorphisms isomorphic to ${\mathbb Z}_{2}^{5}$ with $R/K$ of signature $(0;2,2,2,2,2,2)$ as there are cases for which the above property fails.

 88- Homology closed Riemann surfaces     Quarterly Journal of Math. 63 (2012), 931-952; doi: 10.1093/qmath/har026 A closed Riemann surface of genus at least two can be described by many different objects, for instance, by algebraic curves and  by torsion free co-compact Fuchsian groups. If a torsion free co-compact Fuchsian group is provided, then in general it is a difficult task to obtain an algebraic curve describing the surface uniformized by the given Fuchsian group. We consider those closed Riemann surfaces appearing as a maximal Abelian cover of an orbifold; called homology closed Riemann surfaces. These surfaces are uniformized by the (torsion free) derived subgroup of certain co-compact Fuchsian groups of genus zero. We describe a general method to obtain algebraic curves for homology closed Riemann surfaces. We make this explicit for the case of (i) hyperelliptic homology closed Riemann surfaces and (ii) homology closed Riemann surfaces being the highest Abelian covers of orbifolds with triangular signature. The structure of the cover groups are also provided. As a simple application, we notice that if $S$ is a closed Riemann surface and $A<{\rm Aut}(S)$ is an Abelian group so that $S/A$ has triangular signature, then  $S$ (and a Galois cover with $A$ as its deck group) can be defined over ${\mathbb Q}$. This says, in other words, that Abelian regular dessins d'enfants are definable over ${\mathbb Q}$.  We also prove that if two orbifolds with triangular signatures have conformally equivalent homology closed Riemann surfaces, then they are necessarily conformally equivalent as orbifolds.

2013

 89.- The bipartite graphs of Abelian dessins d'enfants      Ars Mathematica Contemporanea 6 No.2 (2013), 301-304 Let $S$ be a closed Riemann surface and let $\beta:S \to \widehat{\mathbb C}$ be a regular branched holomorphic covering, with an abelian group as deck group, whose branch values are contained in the set $\{\infty,0,1\}$. Three dessins d'enfants are provided by $\beta^{-1}([0,1])$, $\beta^{-1}([1,\infty])$ and $\beta^{-1}([0,\infty])$. In this paper we provide a description of the bipartite graphs associated to these dessins d'enfants using simple arguments.

 90.- Lowest uniformizaton of compact Klein surfaces      Revista Matematica Iberoamericana 29 No.1 (2013), 53-85. A Schottky group is a purely loxodromic Kleinian group, with non-empty region of discontinuity, isomorphic to a free group of finite rank.  An extended Schottky group is an extended Kleinian group whose orientation-preserving half is a Schottky group. The collection of uniformizations of either a closed Riemann surface or a compact Klein surface is partially ordered. In the case of closed Riemann surfaces, the lowest uniformizations are provided by Schottky groups. We provide simple arguments to see that the lowest uniformizations of compact Klein surfaces are exactly those produced by extended Schottky groups.

 91- (with Y. Fuertes, G. Gonzalez-Diez and M. Leyton)      Automorphisms group of generalized Fermat curves of type (k,3)      Journal of Pure and Applied Algebra 217 (2013), 1791-1806. The determination of the full group of automorphisms of a closed Riemann surface is in general a very complicated task. For hyperelliptic curves, the uniqueness of the hyperelliptic involution permits one to compute these groups in a very simple manner. Similarly, as classical Fermat curves of degree $k$ admit a unique subgroup of automorphisms isomorphic to ${\mathbb Z}_{k}^{2}$, the determination of the group of automorphisms is not difficult. In this paper we consider a family of non-hyperelliptic Riemann surfaces, obtained as the fiber product of two classical Fermat curves of the same degree $k$, which exhibit behaviors of both elliptic and hyperelliptic curves. These curves, called  generalized Fermat curves of type $(k,3)$, are the highest regular abelian branched covers of orbifolds of genus zero with four cone points, all of the same order  $k$. More precisely, a generalized Fermat curve of type $(k,3)$ is a closed Riemann surface $S$ admitting a group $H<{\rm Aut}(S)$, called a generalized Fermat group of type $(k,3)$, so that $H \cong {\mathbb Z}_{k}^{3}$ and $S/H$ is an orbifold with signature $(0,4;k,k,k,k)$.  In this paper we prove the uniqueness of generalized Fermat groups of type $(k,3)$. In particular, this allows the explicit computation of the full group of automorphisms of $S$.

 92.- (with A. Carocca and R. Rodriguez)        Orbifolds with signarure $(0;k,k^{n-1},k^{n},k^{n})$.      Pacific Journal of Math. 263 No.1 (2013), 53-85. Two interesting problems that arise in the theory of closed Riemann surfaces are the following: (i) the computation of algebraic curves representing the surface, and (ii) to decide if the field of moduli is a field of definition. In this paper we consider pairs $(S,H)$, where $S$ is a closed Riemann surface and $H$ is a subgroup of $\Aut(S)$, the group of automorphisms of $S$, so that $S/H$ is an orbifold with signature $(0;k,k^{n-1},k^{n},k^{n})\;$ where $\; k, \: n \geq 2$ are integers. In the case that $S$ is the highest Abelian branched cover of $S/H$ we provide explicit algebraic curves representing $S$. In the case that $k$ is an odd prime, we also describe algebraic curves for some intermediate Abelian covers. For $k = p\geq 3$  a prime and $H$  a $p$-group, we prove that $H$ is a $p$-Sylow subgroup of $\Aut(S)$,  and if $p \geq 7$ we prove that $H$ is normal in $\Aut(S)$. Also, when  $n \neq 3$ we prove that the field of moduli in such cases  is a field of definition. If, moreover, $S$ is the highest Abelian branched cover of $S/H$, then we compute explicitly the field of moduli.

 93.- (with Grzegorz Gromadzki)        Schottky uniformizations of symmetries        Glasgow Mathematical Journal 55 (2013), 591-613. A real algebraic curve of genus $g$ is a pair $(S,\langle \tau \rangle)$, where $S$ is a closed Riemann surface of genus $g$ and $\tau:S \to S$ is a symmetry, that is, an anti-conformal involution. A Schottky uniformization of $(S,\langle \tau \rangle)$ is a tuple $(\Omega,\Gamma,P:\Omega \to S)$, where $\Gamma$ is a Schottky group with region of discontinuity $\Omega$ and $P:\Omega \to S$ is a regular holomorphic cover map with $\Gamma$ as its deck group, so that there exists an extended M\"obius transformation $\widehat{\tau}$ keeping $\Omega$ invariant with $P \circ \widehat{\tau}=\tau \circ P$. The extended Kleinian group $K=\langle \Gamma, \widehat{\tau}\rangle$ is called an extended Schottky groups of rank $g$. The interest on Schottky uniformizations rely on the fact that they provide the lowest uniformizations of closed Riemann surfaces. In this paper we obtain a structural picture of extended Schottky groups in terms of  Klein-Maskit's combination theorems and some basic extended Schottky groups. We also provide some insight of the structural picture in terms of the group of automorphisms of $S$ which are reflected by the Schottky uniformization. As a consequence of our structural description of extended Schottky groups we get alternative proofs to results due to Kalliongis and McCullough on orientation reversing involutions on handlebodies.

 94.- Almost abelian regular dessins d'enfants.      Fundamenta Mathematicae 222 (2013), 269-278. A regular dessin d'enfant, in this paper, will be a pair $(S, \beta)$, where $S$ is a closed Riemann surface and $\beta:S \to \widehat{\mathbb C}$ is a  regular branched cover whose branch values are contained in the set $\{\infty,0,1\}$. Let ${\rm Aut}(S,\beta)$ be the group of automorphisms of $(S,\beta)$, that is, the deck group of $\beta$.  If ${\rm Aut}(S,\beta)$ is Abelian, then it is known that $(S,\beta)$ can be defined over the field of rational numbers ${\mathbb Q}$.  In this paper we prove that, if $A$ is an Abelian group and ${\rm Aut}(S,\beta) \cong A \rtimes {\mathbb Z}_{2}$, then $(S,\beta)$ is also definable over ${\mathbb Q}$. Moreover, if $A \cong {\mathbb Z}_{n}$, then we provide explicitly these dessins over ${\mathbb Q}$.

 95.- (with M. E. Valdes and S. Reyes-Carocca)      Field of moduli and generalized Fermat curves.      Revista Colombiana de Matematicas (2) 47 (2013), 205-221. A generalized Fermat curve of type $(p,n)$ is a closed Riemann surface $S$ admitting a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with $S/H$ being the Riemann sphere with exactly $n+1$ cone points, each one of order $p$. If $(p-1)(n-1) \geq 3$, then $S$ is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by ${\rm Aut}_{H}(S)$ the normalizer of $H$ in ${\rm Aut}(S)$. If $p$ is a prime, and either (i) $n=4$ or (ii) $n$ is even and ${\rm Aut}_{H}(S)/H$ is not a non-trivial cyclic group or (iii) $n$ is odd and ${\rm Aut}_{H}(S)/H$ is not a cyclic group, then we prove that $S$ can be defined over its field of moduli. Moreover, if $n \in \{3,4\}$, then we also compute the field of moduli of $S$.

2014

 96.- Computing the field of moduli of the KFT family.      Proyecciones Journal of Mathematics (1) 33 (2014), 61-75. The computation of the field of moduli of a closed Riemann surface is in general a difficult task. Akso it is difficult  to see if it is a field of definition. In this article we consider the collection of Riemann surfafdes of genus 3 admitting the symmetric group in 4 letters as a group of automorphisms. We describe, in each case, the field of moduli and a model over such a field.

 97.- A simple remark on the field of moduli of rational maps.      Quarterly Journal of Math. 65 (2014), 627-635. A complex rational map $R \in {\mathbb C}[z]$ has associated its field of moduli ${\mathcal M}_{R}$, an invariant under conjugation by M\"obius transformations, which is contained in every field of definition of $R$. In general ${\mathcal M}_{R}$ is not a field of definition of $R$ as it is shown by explicit examples due to Silverman. In these examples, the rational maps are definable over a degree two extension of the field of moduli. In this paper we observe that such a property always holds, that is, every rational map is definable over an extension of degree at most two of its field of moduli. The main ingredient in the proof is Weil's descent theorem, applied to the Riemann sphere, and the fact that the group of automorphisms of degree at least two rational maps are well known.

 98.- (with M. Izquierdo)      On the connectivity of the branch locus of the Schottky space.      Annales Academiae Scientiarum Fennicae 39 (2014), 635-654. Let $M$ be a handlebody of genus $g \geq 2$. The space ${\mathcal T}(M)$, that parametrizes marked Kleinian structures on $M$ up to isomorphisms, can be identified with the space ${\mathcal MS}_{g}$ of marked Schottky groups of rank $g$, so it carries a complex manifold of finite dimension $3(g-1)$. The space ${\mathcal M}(M)$, that parametrizes Kleinian structures on $M$ up to isomorphisms, can be identified with ${\mathcal S}_{g}$, the Schottky space of rank $g$, and it carries the structure of a complex orbifold. In these identifications, the projection map $\pi:{\mathcal T}(M) \to {\mathcal M}(M)$ corresponds to the map from ${\mathcal MS}_{g}$ onto ${\mathcal S}_{g}$ that forget about the marking. In this paper we observe that the singular (branch) locus ${\mathcal B}(M)$ of ${\mathcal M}(M)$, that is, the branch locus of $\pi$,  has (i) exactly two connected components for $g=2$, (ii) at most two connected components for $g \geq 4$ even, and (iii) for $g \geq 3$ odd, that ${\mathcal M}(M)$ is connected.

 99.- (with G. Gromadzki)      Conjugacy classes of symmetries of compact Kleinian 3-manifolds.      Contemporary Mathematics 629 (2014), 181-188. Let $M$ be a compact Kleinian $3$-manifold with non-empty andonnected boundary $S$. Then  $S$ carries the  structure of a closed Riemann surface and by  symmetries of $S$ or $M$ we understand their antiholomorphic involutions. In this paper we provide upper bounds for the number of conjugacy classes of symmetries  of $M$ in terms of the genus of $S$. Furthermore, we show that our bounds are sharp for hyperbolic handlebodies for infinitely many genera, by explicit constructions of finite normal extensions of certain Schottky groups, using  at decisive stage of construction, quasi-conformal deformation theory of Riemann surfaces and Teichm\"uller theory of Fuchsian groups. In particular, we obtain that  a Kleinian $3$-manifolds of even genus $g$ has at most four non-conjugate symmetries and that this bound is achieved for arbitrary even $g$. Motivating by the behavior  of Riemann surfaces we propose the problem of the validity of our  bounds, obtained  for hyperbolic handlebodies, in a purely topological setting.

 100.- (with J. Cirre)      Normal coverings of hyperelliptic real Riemann surfaces.      Contemporary Mathematics 629 (2014),  59-76. A real Riemann surface is a pair $(R,\tau)$, where $R$ is a compact Riemann surface and $\tau:R \to R$ is an anticonformal involution, called a real structure of $R.$ If $R$ is hyperelliptic then we say that $(R,\tau)$ is a hyperelliptic real Riemann surface. In this paper we describe in terms of algebraic equations all normal (possibly branched) coverings $\pi:(R,\tau)\to(S,\eta)$ between hyperelliptic real Riemann surfaces. This extends results due to Bujalance-Cirre-Gamboa, where either the number of ovals fixed by the real structure $\tau$ is maximal, or the degree of the covering is two. In this paper we consider any real structure $\tau$ and any degree of the covering.

2015

 101.-  Edmonds maps on the Fricke-Macbeath curve.       Ars Mathematica Contemporanea 8 (2015), 275-289       http://amc-journal.eu/index.php/amc/article/view/496 In 1985, L. D. James and  G. A. Jones proved that the complete graph $K_{n}$ defines a clean dessin d'enfant (the bipartite graph is given by taking as the black vertices the vertices of $K_{n}$ and the white vertices as middle points of edges) if and only if $n=p^{e}$, where $p$ is a prime.  Later, in 2010, G. A. Jones, M. Streit and J. Wolfart computed the minimal field of definition of them. The minimal genus $g>1$ of these types of clean dessins d'enfant is $g=7$, obtained for $p=2$ and $e=3$. In that case, there are exactly two such clean dessins d'enfant (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus $7$) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over ${\mathbb Q}$, but both Edmonds maps cannot be defined over ${\mathbb Q}$; in fact they have as minimal field of definition the quadratic field ${\mathbb Q}(\sqrt{-7})$. It seems that no explicit models for the Edmonds maps over ${\mathbb Q}(\sqrt{-7})$ are written in the literature. In this paper we start with an explicit model $X$ for the Fricke-Macbeath curve provided by Macbeath, which is defined over ${\mathbb Q}(e^{2 \pi i/7})$, and we construct an explicit birational isomorphism $L:X \to Z$, where $Z$ is defined over ${\mathbb Q}(\sqrt{-7})$, so that both Edmonds maps are also defined over that field.

 102.- (with P. Johnson)      Field of moduli of generalized Fermat curves of type (k,3) with an application to non-hyperelliptic dessins  d'enfants.      Journal of Symbolic Computation 77 (2015) 60-72. http://dx.doi.org/10.1016/j.jsc.2014.09.042 A generalized Fermat curve of type $(k,3)$, where $k \geq 2$, is a closed Riemann surface admitting a group $H \cong {\mathbb Z}_{k}^{3}$ as a group of conformal automorphisms so that the quotient orbifold $S/H$ is the Riemann sphere and it has exactly $4$ cone points, each one of order $k$. Every genus one Riemann surface is a generalized Fermat curve of type $(2,3)$ and, if $k \geq 3$, then a generalized Fermat curve of type $(k,3)$ is non-hyperelliptic. For each generalized Fermat curve, we compute its field of moduli and note that it is a field of definition. Moreover, for $k=e^{2 i \pi/p}$, where $p \geq 5$ is a prime integer, we produce explicit algebraic models over the corresponding field of moduli. As a byproduct, we observe that the absolute Galois group ${\rm Gal}(\overline{\mathbb Q}/{\mathbb Q})$ acts faithfully at the level of non-hyperelliptic dessins d'enfants. This last fact was already known for dessins of genus $0$, $1$ and for hyperelliptic ones, but it seems that the non-hyperelliptic situation is not explicitly given in the existent literature.

 103.- (with R. Rodriguez)     A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree.     Archiv der Mathematik 105 (2015), 333-341. Recently, Barraza-Rojas have described the action of the full automorphisms group on the Fermat curve of degree $p$, for $p$ a prime integer, and obtained the group algebra decomposition of the corresponding Jacobian variety. In this short note we observe that the factors in such a decomposition are given by the Jacobian varieties of certain $p$-gonal curves.

2016

 104.- (with Saul Quispe)     Fields of moduli of some special curves.     Journal of Pure and Applied Algebra.  220 (2016), 55-60. DOI information: 10.1016/j.jsc.2014.09.042 In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic $p$-gonal curves by A. Kontergeorgis.

 105.- (with M. Carvacho and S. Quispe)       Jacobian variety of generalized Fermat curves.       Quarterly Journal of Math. 67 (2016), 261-284. DOI information: 10.1093/qmath/haw009 The isogenous decomposition of the Jacobian variety of classical Fermat curve of prime degree $p \geq 5$ has been obtained by Aoki using techniques of number theory, by Barraza and Rojas in terms of decompositions of the algebra of groups and by Hidalgo and Rodr\'{\i}guez using Kani-Rosen results. In the last, it was seen that all factors in the isogenous decomposition are Jacobian varieties of certain cyclic $p$-gonal curves. The highest Abelian branched covers of an orbifold of genus zero with exactly $n+1$ branch points, each one of order $p$, are provided by the so called generalized Fermat curves of type $(p,n)$; these being a suitable fiber product of $n-1$ Fermat curves of degree $p$.  In this paper, we provide a decomposition, up to isogeny, of the Jacobian variety of a generalized Fermat curve $S$ of type $(p,n)$ as a product of Jacobian varieties of certain cyclic $p$-gonal curves whose explicit equations are provided in terms of the equations for $S$. As a consequence of this decomposition, we are able to provide explicit positive-dimensional families of closed Riemann surfaces whose Jacobian variety is isogenous to the product of elliptic curves.

 106.- (with M. Carvacho, G. Gromadzki and CB. Baginski)          On periodic self-homeomorphisms of closed orientable surfaces determined by their orders.          Collectanea Mathematica 67 (2016), 415-429. doi:10.1007/s13348-015-0151-1 The fundamentals for the topological classification of periodic orientation preserving self-homeomorphisms of a closed orientable topological surface $X=X_g$ of genus $g \geq 2$ have been established, by Nielsen, in the thirties of the last century. Here we consider two concepts related to this classification; rigidity and weak rigidity. A cyclic action $G$ of order $N$ on $X$ is said to be {topologically rigid} if any other cyclic action of order $N$ on $X$ is topologically conjugate to it. If this assertion holds for arbitrary other action having, in addition, the same orbit genus and the same structure of singular orbits, then $G$ is said to be {weakly topologically rigid}. Here we give a precise description of rigid and weakly rigid quasi-platonic actions.

 107.- (with A. F. Costa)      Automorphisms of non-cyclic p-gonal Riemann surfaces.      Moscow Mathematical Journal 16 (2016), 659-674. In this paper we prove that the order of a holomorphic automorphism of a non-cyclic $p$-gonal compact Riemann surface $S$ of genus $g>(p-1)^{2}$ is bounded above by $2(g+p-1)$. We also show that this maximal order is attained for infinitely many genera. This generalises the similar result for the particular case $p=3$ recently obtained by Costa-Izquierdo. Moreover,  we also observe that the full group of holomorphic automorphisms of $S$ is either the trivial group or is a finite cyclic group or a dihedral group or one of the Platonic groups ${\mathcal A}_{4}$, ${\mathcal A}_{5}$ and $\Sigma_{4}$. Examples in each case are also provided. In the case that $S$ admits a holomorphic automorphism of order $2(g+p-1)$, then its full group of automorphisms is the cyclic group generated by it and every $p$-gonal map of $S$ is necessarily simply branched.  Finally, we note that each pair $(S,\pi)$, where $S$ is a non-cyclic $p$-gonal Riemann surface and $\pi$ is a $p$-gonal map, can be defined over its field of moduli. Also, if the group of automorphisms of $S$ is different from a non-trivial cyclic group and $g>(p-1)^{2}$, then $S$ can be also be defined over its field of moduli. In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic $p$-gonal curves by A. Kontergeorgis.

2017

 108.- (with L. Jimenez, S. Quispe and S. reyes-Carocca)       Quasiplatonic curves with symmetry group $\mathbb{Z}_2^2 \rtimes \mathbb{Z}_m$ are definable over $\mathbb{Q}$.      Bull. of the London Math. Soc. 49 (2017), 165-183. It is well known that every closed Riemann surface $S$ of genus $g \geq 2$, admitting a group $G$ of conformal automorphisms so that $S/G$ has triangular signature, can be defined over a finite extension of ${\mathbb Q}$. It is interesting to know, in terms of the algebraic structure of $G$, if $S$ can in fact be defined over ${\mathbb Q}$. This is the situation if $G$ is either abelian or isomorphic to $A \rtimes {\mathbb Z}_{2}$, where $A$ is an abelian group. On the other hand, as shown by Streit and Wolfart, if $G \cong {\mathbb Z}_{p} \rtimes {\mathbb Z}_{q}$ where $p,q>3$ are prime integers, then $S$ is not necessarily definable over ${\mathbb Q}$. In this paper, we observe that if $G\cong{\mathbb Z}_{2}^{2} \rtimes {\mathbb Z}_{m}$ with $m \geq 3$, then $S$ can be defined over ${\mathbb Q}$. Moreover, we describe explicit models for $S$, the corresponding groups of automorphisms and an isogenous decomposition of their Jacobian varieties as product of Jacobians of hyperelliptic Riemann surfaces.

 109.- (with S. Quispe)       Regular dessins d'enfants with field of moduli $\mathbb{Q}(\sqrt[p]{2})$.      ARS Mathematica Contemporanea 13 No. 2 (2017), 323-330. Herradon has recently provided an example of a regular dessin d'enfant whose field of moduli is the non-abelian extension ${\mathbb Q}(\sqrt[3]{2})$ answering in this way a question due to Conder, Jones, Streit and Wolfart. In this paper we observe that Herradon's example belongs naturally to an infinite series of such kind of examples; for each prime integer $p \geq 3$ we construct a regular dessin d'enfant whose field of moduli is the non-abelian extension ${\mathbb Q}(\sqrt[p]{2})$; for $p=3$ it coincides with Herradon's example.

ACCEPTED ARTICLES

 1.- (with M. Artebani, M. Carvacho, and S. Quispe)     A Tower of Riemann Surfaces which cannot be defined over their Field of Moduli.     Glasgow Math. J. Explicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and  non-hyperelliptic curves3. In this paper we provide necessary conditions for a curve to be definable over its field of moduli. These conditions generalize the results known for the hyperelliptic case by B. Huggins and for normal cyclic $p$-gonal curves by A. Kontergeorgis.

 2.- (with A. Kontogeorgis, M. Leyton and P. Paramantzouglou)     Automorphisms of generalized Fermat curves.     Journal of Pure and Applied Agebra Let $K$ be an algebraically closed field of characteristic $p \geq 0$. A generalized Fermat curve of type $(k,n)$, where $k,n \geq 2$ are integers (for $p \neq 0$ we also assume that $k$ is relatively prime to $p$), is a non-singular irreducible projective algebraic curve $F_{k,n}$ defined over $K$ admitting a group of automorphisms $H \cong {\mathbb Z}_{k}^{n}$ so that $F_{k,n}/H$ is the projective line with exactly $(n+1)$ cone points, each one of order $k$. Such a group $H$ is called a generalized Fermat group of type $(k,n)$. If $(n-1)(k-1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type $(k,n)$ has a unique generalized Fermat group of type $(k,n)$ if $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$).  Generalized Fermat curves of type $(k,n)$ can be described as a suitable fiber product of $(n-1)$ classical Fermat curves of degree $k$. We prove that, for $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$), each automorphism of such a fiber product curve can be extended to an automorphism of the ambient projective space. In the case that $p>0$  and $k-1$ is a power of $p$, we use tools from the theory  of complete projective intersections in order to prove that, for $k$ and $n+1$ relatively prime,  every automorphism of the fiber product curve can also be extended to an automorphism of  the ambient projective space. In this article we also prove that the set of fixed points of the non-trivial elements  of the generalized Fermat group coincide with  the hyper-osculating points of the fiber product model  under the assumption that the characteristic $p$ is either zero or $p>k^{n-1}$.Let $K$ be an algebraically closed field of characteristic $p \geq 0$. A generalized Fermat curve of type $(k,n)$, where $k,n \geq 2$ are integers (for $p \neq 0$ we also assume that $k$ is relatively prime to $p$), is a non-singular irreducible projective algebraic curve $F_{k,n}$ defined over $K$ admitting a group of automorphisms $H \cong {\mathbb Z}_{k}^{n}$ so that $F_{k,n}/H$ is the projective line with exactly $(n+1)$ cone points, each one of order $k$. Such a group $H$ is called a generalized Fermat group of type $(k,n)$. If $(n-1)(k-1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type $(k,n)$ has a unique generalized Fermat group of type $(k,n)$ if $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$).

 3.- (with T. Shaska)     On the field of moduli of superelliptic curves.     Contemporary Math. Beshaj and Thompson have proved that a superelliptic curve $\X$ can always be defined over a quadratic extension of its field of moduli. If $\X$ has no extra automorphisms, then equations over the minimal field of definition can be determined. In this case, using ideas of Clebsh, it can be decided if $\X$ can be defined over its field of moduli. If $\X$ has extra automorphisms, then to determine if $\X$ can be defined over its field of moduli is more difficult. In this case, Beshaj and Thompson provided equations over the minimal field of definition using the dihedral (or Shaska) invariants. In this paper we observe that if the reduced group is different from the trivial or cyclic group, then $\X$ can be defined over its field of moduli; in the cyclic situation we provide a sufficient condition for this to happen. We also determine those of genus at most $10$ which might not be definable over their field of moduli.

 4.- (with S. Quispe)     A note on the connectedness of the branch locus of rational points.     Glasgow Math. J. Milnor proved that the moduli space ${\rm M}_{d}$ of rational maps of degree $d \geq 2$ has a complex orbifold structure of dimension $2(d-1)$. Let us denote by ${\mathcal S}_{d}$ the singular locus of ${\rm M}_{d}$ and by ${\mathcal B}_{d}$ the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify ${\rm M}_2$ with ${\mathbb C}^2$ and, within that identification, that ${\mathcal B}_{2}$ is a cubic curve; so ${\mathcal B}_{2}$ is connected and ${\mathcal S}_{2}=\emptyset$. If $d \geq 3$, then it is well known that  ${\mathcal S}_{d}={\mathcal B}_{d}$. In this paper we use simple arguments to prove the connectivity of ${\mathcal S}_{d}$.