PUBLICATIONS
1989
1.On Γhyperelliptic
Schottky groups. Notas Soc. Mat. Chile 8 (1) (1989), 2736. 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
2. Homology
Covering of Riemann Surfaces. 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 torsionfree, finitely generated Fuchsian group of the first kind determines it uniquely. 
1994
3. On
the Schottky
groups
with automorphisms. Ann. Acad. Fenn. 19 (1994), 259289. 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 nontrivial 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), 247258. 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 MaskitKlein 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. 
5. Schottky
Uniformation with Abelian Groups of Conformal
Automorphism. 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
6. Kleinian
groups with an invariant Jordan curve:
Jgroups. The Pacific Journal of Math. 169 (1995), 291310. 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: TUBerlin, Berlin, Germany) On simultaneous homomorphisms onto Kleinian groups. Complex Variables: Theory and Applications 27 (1995), 309315. Let Γ be a Fuchsian group acting on the upper halfplane H and let G be a Kleinian group with 0, 1, ∞ in its limit set Λ(G). If L denotes the lower halplane, 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. 
8. Kleinian
groups with common commutator
subgroup. Complex Variables: Theory and Appl. 28 (1995), 121133. In this paper, we obtain a certain rigidity property for Kleinian groups. This asserts that if F and G (both nonelementary torsionfree 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. 
9. A
Commutator Rigidity for Kleinian
Groups. Revista Proyecciones 14 (1995), 7581. In this article
we
provide some partial results concerning commutator subgroups of
Kleinian groups
and how they are determined by them. 
1996
10. The
noded Schottky Space. Proc. London Math Soc. 73 (1996), 385403. 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).

11. On the 12(g1)
Bound. C.R. Math. Rep. Acad. Sci. Canada 18 (1996), 3942. 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 $12(g1)$. 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 $12(g1)$. (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 $12(g1)$. 
12. Closed
Riemann Surfaces with Dihedral Groups of Conformal
Automorphisms. Revista Proyecciones 15 (1996), 4790. 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), 543566. In this note we construct examples of geometric 3orbifolds 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 GonzálezDiez: UAM
Madrid, Spain) Conformal versus topological automorphisms of compact Riemann surfaces. Bull. of the London Math. Soc. 29 (1997), 280284. 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
15.
On a Theorem of Accola. Complex Variables: Theory and Applications 36 (1998), 1926. 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 nonnegative 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 nonnegative integer n, there is a smooth nsheeted 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. 
16. An
example
of degeneration on the noded Schottky space. Revista Matemática Complutense 11 (1998), 165183. 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 twodimensional 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. 
17.
Noded Fuchsian groups I. Complex Variables: Theory and Appl. 36 (1998), 4566. We consider certain type of Kleinian groups called noded Fuchsian groups. Torsionfree 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 torsionfree cocompact Fuchsian groups and Schottky groups to the above ones. 
18.
Γhyperelliptic
Riemann surfaces. Revista Proyecciones 17 (1998), 77117. 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
19. Noded function
groups. Contemporary Mathematics 240 (1999), 209222. Given a Kleinian group $G$, we denote by $\Omega(G)$ and $P(G)$ its region of discontinuity and its set of doublecusped 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 torsionfree, nonelementary noded function group $G$ determines it uniquely. For a general nonelementary torsionfree Kleinian group with no invariant component the above rigidity property still yet unknown. 
20.
A note on the homology covering of closed Klein
surfaces. Revista Proyecciones 18 (1999), 165173. In previous works we have seen that a finitely generated torsionfree nonelementary function group is uniquely determined by its commutator subgroup. In this note, we observe that under the presence of orientationreversing conformal automorphisms the above rigidity property still valid. More precisely, we see that finitely generated torsionfree 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. 
21.
Dihedral groups are of Schottky type. Revista Proyecciones 18 (1999), 2348. 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. 
22.
Cyclic groups of automorphiss of Schottky type. Revista Proyecciones 18 (1999), 1321. 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
23.
A note on the Homology Covering of Analytically finite Klein
surfaces. Complex Variables: Theory and Appl. 42 (2000), 183192. In previous works
we
have seen that a finitely generated torsionfree nonelementary
function group is uniquely determined by its commutator subgroup.
In this note, we observe that under the presence of
orientationreversing conformal automorphisms the above rigidity
property still valid. More precisely, we see that finitely
generated torsionfree 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), 6594. 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.

25.
Fixed point parameters for Moebius Groups. Revista Proyecciones 19 (2000), 157196. 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 nondiscrete groups of Meobius transformations. We proceed to find complex analytic parameters for these spaces given by fixed points. 
26.
Schottky uniformizations and Riemann matrices of maximally
symmetric Riemann surfaces of genus 5. Revista Proyecciones 20 (2001), 93126. 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 nonisomorphic 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
27. Schottky
uniformization of stable symmetric Riemann
surfaces. Notas Soc. Mat. Chile (N.S.) 1 (2001), 8291. 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. 
28. Fixed point
parameters for Fuchsian groups of signature (2,0). Notas Soc. Mat. Chile (N.S.) 1 (2001), 1121. 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 quasiFuchsian 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), 489508. 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 
30.
Bounds for Conformal automorphisms of Riemann surfaces with
condition (A). Revista Proyecciones 20 (2001), 139175. 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. Γhyperellipticsymmetric
Schottky groups. Complex Variables: Theory and Appl. 45 (2001), 117141. Let S be a
γhyperelliptic Riemann surface, with a γhyperelliptic
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 (hyperellipticsymmetric 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.

32.
Numerical Uniformation of Hyperelliptic  M  Symmetric Riemann
Surfaces. Revista Proyecciones 20 (2001), 351365. In this note we
consider hyperellipticMsymmetric 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
33. A4, A5,
S4
and S5 of Schottky type. Revista Matemática Complutense 15 (2002), 1129. 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. 
34.
Some Special Kleinian Groups and their Orbifolds. Revista Proyecciones 21 (2002), 2150. Let us consider an abstract group with the fol lowing presentation G = <x_{1} , ..., x_{n} ; x_{i}^{k_{i}}= (x_{j+1} x_{j )^{1})^{l _{j}} = 1>, 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), 6373. Study of moduli of family of curves on the universal Teichmueller curve is provided. 
36. Real Surfaces, Riemann
Matrices and Algebraic Curves. Contemporary Mathematics 311 (2002), 277299. 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 $3$ nonhyperelliptic real surfaces. We also may write down for them an algebraic equation in function of the uniformizing Schottky group. 
2003
37.
Hyperbolic polygons and real Schottky groups. Complex Variables: Theory and Appl. 48 (2003), 4362. 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.

38.
A commutator rigidity for function groups and Torelli’s
theorem. Revista Proyecciones 22 (2003), 117125. We show that a nonelementary finitely generated torsionfree function group is uniquely determined by its commutator subgroup. This is well related to Torelli’s theorem for closed Riemann surfaces. For a general nonelementary torsionfree 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 as Group of Automorphisms. Geometriae Dedicata 106 (1) (2004), 7995. In this note we construct a 1complex 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 GonzálezAguilera 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. 
40.
Maximal real Schottky groups. Revista Matematica Iberoamericana 20 (2004), 737770. 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), 627646. 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. 
42. Cyclic extensions of Schottky
uniformizations. Ann. Acad. Sci. Fenn. 29 (2004), 329344. 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), 187203. 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), 183204. 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), 125157. 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.

46.
Real Schottky Parametrizations. Complex Variables: Theory and Appl. 50 (2005), 401426. 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 KleinSchottky groups (once we have chosen some points on S as phantom nodes) or (ii) KleinSchottky 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), 185204. 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 KleinSchottky groups. Pacific J. of Math. (2) 220 (2005), 313328. 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 orientationpreserving 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), 5064. 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 covolume 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), 145152. 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
nonunique.

52. (with B.
Maskit: SUNY at Stony Brook, NY,
USA) On neoclassical Schottky groups. Transactions of the AMS. 358 (2006), 47654792. The goal of this paper is to describe a theoretical construction of an infinite collection of nonclassical Schottky groups. We first show that there are infinitely many nonclassical 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 nonclassical. 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), 89107. 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, 419434. 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 momentumenergy 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 infinitedimensional 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 finitedimensional 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) Qngonal Klein Surfaces. Acta Mathematica Sinica 23 (2007), No. 10, 18331844. 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 qngonal surfaces and they are nsheeted 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 qngonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of qngonal surfaces with algebraic genus p. Several examples are given: qpentagonal surfaces and qngonal 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), 793810. Let γ , r, s, ≥ 1, nonnegative 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), 2961. 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), 165192. 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) Existence of solutions of semilinear systems in l^{2}. Revista Proyecciones 27, No. 2 (2008), 149161. Let Q : �l^2
→ l^2 be a symmetric and positive semidefinite 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 nontrivial 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 nontrivial 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), 953969. 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 semidefinite. 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 nontrivial solutions of partial difference equations on any finite, connected simple graph for $D$ symmetric and positive semidefinite 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), 16431660. 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 nonhyperelliptic 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. 193204. Analysis and Mathematica Physics Series: Trends in Mathematics Eds.: Gustafsson, Bjorn; Vassiliev Alexander 2009, Approx. 525 p., Hardcover ISBN: 9783764399054 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. Nonhyperelliptic
Riemann surfaces with real field of moduli but not definable over
the reals. Archiv der Mathematik 93 (2009), 219222. 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 nonhyperelliptic 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 nonhyperelliptic 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. ISBN13: 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 codimension 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), 3748. 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), 135158. A KleinSchottky group is an extended Kleinian group, containing no reflections and whose orientationpreserving half is a Schottky group. A dihedralKleinSchottky group is an extended Kleinian group generated by two different KleinSchottky groups, both with the same orientationpreserving half. We provide a structural description of the dihedralKleinSchottky groups. Let $M$ be a handlebody of genus $g$, with a Schottky structure. An imaginary reflection $\tau$ of $M$ is an orientationreversing 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 dihedralKleinSchottky 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), 3847. A geometric orbifold of dimension $d$ is of the form ${\mathcal O}=X/K$, were $(X,G)$ is a geometry of dimension $d$ and $K<G$ is a cocompact discrete subgroup. In this case, $\pi_{1}^{orb}({\mathcal O})=K$ is called the orbifold fundamental group of ${\mathcal O}$. In general, the derived subgroup $K'$ of $K$ may have elements acting with fixed points, that is, it may happen that the homology cover $M_{\mathcal O}=X/K'$ of ${\mathcal O}$ may not be a geometric manifold; it may have geometric singular points. We are concerned with the problem of deciding when $K'$ acts freely on $X$, that is, when the homology cover $M_{\mathcal O}$ is a geometric manifold. In the case $d=2$ a complete answer is due to C. Maclachlan. In the case $d=3$, under the assumption that the underlying topological space of the orbifold ${\mathcal O}$ is $S^{3}$, we provide a necessary and sufficient condition for the derived subgroup $K'$ to act freely. 
68.A
short note on Msymmetric hyperelliptic Riemann
Surfaces. Cubo A Mathematical Journal 12 No.1 (2010), 175179. 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), 141158. 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 $12(g1)$; if the index is $12(g1)$, 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 reobtain 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), 639649. 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), 4157. 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 $12(g1)$. If $K$ contains a Schottky subgroup of rank $g \geq 2$ and index $12(g1)$, 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 $12(g1)$. 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 $3(g1)$. 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) NavierStokes equations on weighted graphs. Complex Analysis and Operator Theory 4 (2010), 525540. NavierStokes 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), 23472361. 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), 195209. 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), 3745. Let $P_{1}:S_{1} \to S$ and $P_{2}:S_{2} \to S$ be nonconstant 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; Bgroups. 
76. On
the
inverse uniformization problem: real Schottky
uniformizations. Revista Matematica Complutense 24 (2011), 391420. 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 nonhyperelliptic 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 $3$ (nonhyperelliptic), 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), 847851. In this paper we provide examples of pairs of conformally nonequivalent, 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), 297331. 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 orientationpreserving involutions are also provided. 
79. (with
Yolanda Fuertes: UAM (España)) On unbranched nonnormal 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\'alezDiez 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 dianalytic
involutions
of compact Klein surfaces to extendedSchottky
uniformizations. Fundamenta Mathematicae 214 (2011), 161180 Let $S$ be a compact Klein surface together with a dianalytic involution $\kappa:S \to S$. The lowest uniformizations of $S$ are those whose deck group is an extendedSchottky 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 extendedSchottky 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 extendedSchottky uniformization. 
2012
81 A simple remark on fields of
definition Proyecciones Journal of Mathematics 31 No. 1 (2012), 2528. Let $K<L$ be an extension of fields, in characteristic zero, with $L$ algebraically closed and let $\overline{K}<L$ be the algebraic closure of $K$ in $L$. Let $X$ and $Y$ be irreducible projective algebraic varieties, $X$ defined over $\overline{K}$ and $Y$ defined over $L$, and let $\pi:X \to Y$ be a nonconstant morphism, defined over $L$. If we assume that $\overline{K} \neq L$, then one may wonder if $Y$ is definable over $\overline{K}$. In the case that $K={\mathbb Q}$, $L={\mathbb C}$ and that $X$ and $Y$ are smooth curves, a positive answer was obtained by Gonz\'alezDiez. In this short note we provide simple conditions to have a positive answer to the above question. We also state a conjecture for a class of varieties of general type. 
82. On Conjugacy of pgonal
automorphisms. Bulletin of the Korean Math. Society 49 No. 2 (2012), 411415. In 1995 it was proved by Gonz\'alezDiez 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 CastelnuovoSeveri theorem. In this paper we provide another proof which is shorter and is just a simple use of Sylow's theorem together with the CastelnuovoSeveri 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 threeball 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 NavierStokes Equations (Capitulo de Libro). NavierStokes Equations: Properties, Description and Applications. Nova Publishers. Series: Mathematics Research Developments Physics Research and Technology (2012) ISBN: 9781613245903 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 LaneEmdenFowler. Finally we discuss our results concerning the discrete NavierStokes 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:
Nonhyperelliptic 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/s000130120378y Erratum to: Arch. Math. 93 (2009), 219–224 DOI 10.1007/s0001300900254. 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 Lgonal curves Computational Algebraic and Analytic Geometry. Contemporary Mathematics 572 (2012) Edited by: Mika Seppälä, Florida State University, Tallahassee, FL and Emil Volcheck ISBNs: 9780821868690 (p); 9780821889893 (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 $$y^{L}=\prod_{j=1}^{r} (xa_{j})^{L/n_{j}} (xb_{j})^{LL/n_{j}},$$ 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), 309312. (http://www.springer.com/alert/urltracking.do?id=L4f5879M92f8a4Sa821618) doi: 10.1007/s107110119683z 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 nonhyperelliptic 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), 931952; 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 cocompact Fuchsian groups. If a torsion free cocompact 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 cocompact 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), 301304 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), 5385. A Schottky group is a purely loxodromic Kleinian group, with nonempty region of discontinuity, isomorphic to a free group of finite rank. An extended Schottky group is an extended Kleinian group whose orientationpreserving 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.
GonzalezDiez and M. Leyton) Automorphisms group of generalized Fermat curves of type (k,3) Journal of Pure and Applied Algebra 217 (2013), 17911806. 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 nonhyperelliptic 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 $(0;k,k^{n1},k^{n},k^{n})$. Pacific Journal of Math. 263 No.1 (2013), 5385. 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^{n1},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), 591613. 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 anticonformal 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 KleinMaskit'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), 269278. 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. ReyesCarocca) Field of moduli and generalized Fermat curves. Revista Colombiana de Matematicas (2) 47 (2013), 205221. 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 $(p1)(n1) \geq 3$, then $S$ is known to be nonhyperelliptic 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 nontrivial 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), 6175. 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), 627635. 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), 635654. 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 $3(g1)$. 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 3manifolds. Contemporary Mathematics 629 (2014), 181188. Let $M$ be a compact Kleinian $3$manifold with nonempty 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, quasiconformal 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 nonconjugate 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), 5976. 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 BujalanceCirreGamboa, 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
FrickeMacbeath
curve. Ars Mathematica Contemporanea 8 (2015), 275289 http://amcjournal.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 FrickeMacbeath curve (the only Hurwitz curve of genus $7$) and both forming a chiral pair. The uniqueness of the FrickeMacbeath 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 FrickeMacbeath 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 nonhyperelliptic dessins d'enfants. Journal of Symbolic Computation 77 (2015) 6072. 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 nonhyperelliptic. 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 nonhyperelliptic dessins d'enfants. This last fact was already known for dessins of genus $0$, $1$ and for hyperelliptic ones, but it seems that the nonhyperelliptic 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), 333341. Recently, BarrazaRojas 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), 5560. 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), 261284. 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 KaniRosen 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 $n1$ 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 positivedimensional 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 selfhomeomorphisms of closed orientable surfaces determined by their orders. Collectanea Mathematica 67 (2016), 415429. doi:10.1007/s1334801501511 The fundamentals for the topological classification of periodic orientation preserving selfhomeomorphisms 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 quasiplatonic actions. 
107. (with
A.
F. Costa) Automorphisms of noncyclic pgonal Riemann surfaces. Moscow Mathematical Journal 16 (2016), 659674. In this paper we prove that the order of a holomorphic automorphism of a noncyclic $p$gonal compact Riemann surface $S$ of genus $g>(p1)^{2}$ is bounded above by $2(g+p1)$. 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 CostaIzquierdo. 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 $2(g+p1)$, 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 noncyclic $p$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 nontrivial cyclic group and $g>(p1)^{2}$, 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. reyesCarocca) 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), 165183. 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), 323330. Herradon has recently provided an example of a regular dessin d'enfant whose field of moduli is the nonabelian 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 nonabelian extension ${\mathbb Q}(\sqrt[p]{2})$; for $p=3$ it coincides with Herradon's example. 
110. (with A.
Kontogeorgis, M. Leyton and P. Paramantzouglou) Automorphisms of generalized Fermat curves. Journal of Pure and Applied Agebra 221 (2017), 23122337 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 nonsingular 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 $(n1)(k1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it is known to be nonhyperelliptic. 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 $(k1)(n1)>2$ (for $p>0$ we also assume that $k1$ is not a power of $p$). Generalized Fermat curves of type $(k,n)$ can be described as a suitable fiber product of $(n1)$ classical Fermat curves of degree $k$. We prove that, for $(k1)(n1)>2$ (for $p>0$ we also assume that $k1$ 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 $k1$ 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 nontrivial elements of the generalized Fermat group coincide with the hyperosculating points of the fiber product model under the assumption that the characteristic $p$ is either zero or $p>k^{n1}$.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 nonsingular 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 $(n1)(k1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it is known to be nonhyperelliptic. 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 $(k1)(n1)>2$ (for $p>0$ we also assume that $k1$ is not a power of $p$). 
111. (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. 59 No. 2 (2017), 379393. Explicit examples of both hyperelliptic and nonhyperelliptic 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 nonhyperelliptic 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. 
112. (with C. Baginski
and G.
Gromadzki). On purely nonfree finite actions of abelian groups on compact surfaces. Archiv der Mathematik 109 (2017), 311321 A finite group of conformal automorphisms of a closed orientable Riemann surface is said to act on it {\it purely nonfreely} if each of its elements has a fixed point; we also called it a {\it gpnf}action. In this paper we observe that {\it gpnf}actions exist for an arbitrary finite group and we discuss on the minimum genus problem for such actions; we solve it for cyclic and abelian noncyclic groups. In the first case we prove that this minimal {\it gpnf}action genus coincides with Harvey's minimal genus. 
2018
113. (with S. Quispe) A note on the connectedness of the branch locus of rational points. Glasgow Math. J. 60 No. 1 (2018), 199207. DOI: https://doi.org/10.1017/S0017089516000665 Milnor proved that the moduli space ${\rm M}_{d}$ of rational maps of degree $d \geq 2$ has a complex orbifold structure of dimension $2(d1)$. 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 nontrivial 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 ${\mathcal S}_{d}$. 
114. (with A. F. Costa) On the connectedness of the set of Riemann surfaces with real moduli. Archiv der Mathematik 110 (2018), 305310. The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g1)$ which carries a natural real structure i.e. it admits an antiholomorphic involution $\sigma$. The involution $\sigma$ maps each point corresponding to a Riemann surface $S$ to its complex conjugate $\overline{S}$. The fixed point set of $\sigma$ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside $\mathrm{Fix}(\sigma)$ is the locus ${\mathcal{M}}_{g}(\mathbb{R})$, the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Sepp\"{a}l\"{a} and R. Silhol. The complement $\mathrm{Fix}(\sigma){\mathcal{M}}_{g}(\mathbb{R})$ consists of the so called pseudoreal Riemann surfaces, which is known to be nonconnected. In this short note we provide a simple argument to observe that $\mathrm{Fix}(\sigma)$ is connected. 
115. About
the FrickeMacbeath curve. European Journal of Math. 4 (2018), 313325. DOI 10.1007/s4087901701660 A closed Riemann surface of genus $g \geq 2$ is called a Hurwitz curve if its group of conformal automorphisms has order $84(g1)$. In 1895, A. Wiman noticed that there is no Hurwitz curve of genus $g=2,4,5,6$ and, up to isomorphisms, there is a unique Hurwitz curve of genus $g=3$; this being Klein's plane quartic curve. Later, in 1965, A. Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus $g=7$; this known as the FrickeMacbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, W. Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a $4$dimensional family of genus seven closed Riemann surfaces $S_{\mu}$ admitting a group $G_{\mu} \cong {\mathbb Z}_{2}^{3}$ of conformal automorphisms so that $S_{\mu}/G_{\mu}$ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath's description. We also observe that the jacobian variety of each $S_{\mu}$ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular FrickeMacbeath curve, we obtain the well known fact that its jacobian variety is isogenous to $E^{7}$ for a suitable elliptic curve $E$. 
116. (with E. Badr and
S.
Quispe) Riemann surfaces defined over the reals. Archive der Mathematik 110 No. 4 (2018), 351362 ISSN 0003889X https://doi.org/10.1007/s0001301711469 The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit families of real Riemann surfaces which cannot be defined over the field of moduli. 
117. (with T. Shaska) On the field of moduli of superelliptic curves. Contemporary Mathematics 703 (2018), 4762. ISBN: 9781470428563 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. 
118. (with L. Beshaj,
S.
Kruk, A. Malmendier, S. Quispe and T. Shaska) Rational points in the moduli space of genus two. Contemporary Mathematics 703 (2018), 83115. ISBN: 9781470428563 A database of over 2 million isomorphism classes of genus 2 curves defined over ${\mathbb Q}$ ordered based on their minimal absolute height, moduli height. In addition for each isomorphism class, a minimal equation over the minimal field of definition is provided, the automorphism group, the conductor, and all the twists. Some interesting statistics about the distribution of rational points in the moduli space $M_2$ are also presented. Such database organized in a dictionary in Sagemath is made public including a search utility. 
119. pgroups
acting on Riemann surfaces. Journal of Pure and Applied Algebra. 222 (2018), 41734188. ISSN: 00224049 Let $p$ be a prime integer and let $r \geq 3$ be an integer so that $p \geq 5r7$ and $p \neq 5r6$. We show that a closed Riemann surface $S$ of genus $g \geq 2$ has at most one $p$group $H$ of conformal automorphisms so that $S/H$ has genus zero and exactly $r$ cone points. This, in particular, asserts that, for $r=3$ and $p \geq 11$, the minimal field of definition of $S$ coincides with that of $(S,H)$. Another application of this fact, for the case that $S$ is pseudoreal, is that ${\rm Aut}(S)/H$ must be either trivial or a cyclic group and that $r$ is necessarily even. This generalizes a result due to BujalanceCosta for the case of pseudoreal cyclic $p$gonal Riemann surfaces. 
ACCEPTED
ARTICLES
On the
minimal
field of definition of rational maps: rational maps of odd
signature. Annales Academiae Scientiarum Fennicae. ISSN 1239629X The field of moduli of a rational map is an invariant under conjugation by M\"obius transformations. J. H. Silverman proved that a rational map, either of even degree or equivalent to a polynomial, is definable over its field of moduli and he also provided examples of rational maps of odd degree for which such a property fails. We introduce the notion for a rational map to have odd signature and prove that this condition ensures for the field of moduli to be a field of definition. Rational maps being either of even degree or equivalent to polynomials are examples of odd signature ones. 
Totally degenerate extended
Kleinian groups. Revista CUBO ISSN: 07167776 The theoretical existence of totally degenerate Kleinian groups is originally due to Bers and Maskit. In fact, Maskit proved that for any cocompact nontriangle Fuchsian group acting on the hyperbolic plane ${\mathbb H}^{2}$ there is a totally degenerate Kleinian group algebraically isomorphic to it. In this paper, by making a subtle modification to Maskit's construction, we show that for any nonEuclidean crystallographic group $F$, such that ${\mathbb H}^{2}/F$ is not homeomorphic to a pant, there exists an extended Kleinian group $G$ which is algebraically isomorphic to $F$ and whose orientationpreserving half is a totally degenerate Kleinian group. Moreover, such an isomorphism is provided by conjugation by an orientationpreserving homeomorphism $\phi:{\mathbb H}^{2} \to \Omega$, where $\Omega$ is the region of discontinuity of $G$. In particular, this also provides another proof to Miyachi's existence of totally degenerate finitely generated Kleinian groups whose limit set contains arcs of Euclidean circles. 
Automorphisms
of dessins d'enfants. Archiv der Mathematik. (2018), ISSN: 1239629X In this paper we observe that given any finite group $G$, together a fixed topological action of it of some genus $g \geq 2$, there is a dessin d'enfant having it as its full groiup of automorphisms. In particular, the symmetric strong genus of $G$ equals to the minimal genus (at least two) action on a dessin d'enfant. 
(wih
Sebastian Sarmiento) Real Structures on Marked Schottky Space. Journal of the London Math. Soc. ISSN:14697750 Marked Schottky space $\MS$ is an explicit model of the quasiconformal deformation space of a Schottky group of rank $g$, this being isomorphic to the punctured unit disc for $g=1$ and, for $g \geq 2$, a connected complex manifold of dimension $3(g1)$. This space is an intermediate nonGalois cover of moduli space of genus $g$ curves. In this paper we provide a description of the real structures of $\MS$, up to holomorphic automorphisms, together with their real parts. 