Persona: Costa González, Antonio Félix
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Costa González
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Antonio Félix
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Publicación The NSSC (Nec Subgroups Signature Calculation) Package(2017) Cortázar, Ismael; Costa González, Antonio FélixPublicación On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary(2015-01-01) Izquierdo, Milagros; Costa González, Antonio Félix; Porto Ferreira da Silva, Ana MaríaAbstract. In this work we prove that the hyperelliptic branch locus of ori- entable Klein surfaces of genus g with one boundary component is connected and in the case of non-orientable Klein surfaces it has g+1 2 components, if g is odd, and g+2 2 components for even g. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.Publicación On Riemann surfaces of genus g with 4g automorphisms(2016-04-01) Izquierdo Barrios, Milagros; Bujalance García, Emilio; Costa González, Antonio FélixWe determine, for all genus g 2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g 6= 3; 6; 12; 15 or 30, this sur- faces form a real Riemann surface Fg in the moduli space Mg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the fam- ily. Furthermore we determine the topological types of the real forms of real Riemann surfaces in Fg. The set of real Riemann surfaces in Fg consists of three intervals its closure in the Deligne-Mumford com- pacti cation of Mg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in Fg.Publicación Computing the signatures of subgroups of non-Euclidean crystallographic groups(2017) Cortázar, Ismael; Costa González, Antonio FélixA (planar and cocompact) non-Euclidean crystallographic (NEC) group is a subgroup of the group of (conformal and anti-conformal) isometries of the hyperbolic plane H2 such that H2= is compact. NEC groups are classifed alge- braically by a symbol called signature. In this symbol there is a sign + or and, in the case of sign +, some cycles of integers called period-cycles have an essential direction. In 1990 A.H.M. Hoare gives an algorithm to obtain the signature of a nite index subgroup of an NEC group. The process of Hoare fails in some cases in the task of computing the direction of period-cycles. In this work we complete the algorithm of Hoare, this allows us to construct a program for computing the signature of subgroups of NEC groups in all cases.Publicación Periodicity and free periodicity of alternating knots(ELSEVIER, 2023-05-16) Costa González, Antonio Félix; Quach Honglerb, Cam VamIn a previous paper [6], we obtained, as a consequence of Flyping Theorem due to Menasco and Thislethwaite, that the q-periodicity (q>2) of an alternating knot can be visualized in an alternating projection as a rotation of the projection sphere. See also [2]. In this paper, we show that the free q-periodicity (q>2) of an alternating knot can be represented on some alternating projection as a composition of a rotation of order qwith some flypes all occurring on the same twisted band diagram of its essential Conway decomposition. Therefore, for an alternating knot to be freely periodic, its essential decomposition must satisfy certain conditions. We show that any free or non-free q-action is some way visible (virtually visible) and give some sufficient criteria to determine from virtually visible projections the existence of a q-action. Finally, we show how the Murasugi decomposition into atoms as initiated in [12]and [13]enables us to determine the visibility type (q, r)of the freely q-periodic alternating knots ((q, r)-lens knots [3]); in fact, we only need to focus on a certain atom of their Murasugi decomposition to deduce their visibility type.Publicación Periodic projections of alternating knots(2021-08-15) Costa González, Antonio Félix; Quach Honglerb, Cam VanThis paper is devoted to the proof of existence of q-periodic alternating projections of prime alternating q-periodic knots. The main tool is the Menasco-Thistlethwaite’s Flyping Theorem. Let Kbe an oriented prime alternating knot that is q-periodic with q≥3, i.e. that admits a rotation of order qas a symmetry. Then Khas an alternating projection Π(K)such that the rotational symmetry of Kis visualized as a rotation of the projection sphere leaving Π(K)invariant. As an application, we obtain that the crossing number of a q-periodic alternating knot with q≥3is a multiple of q. Furthermore we give an elementary proof that the knot 12a634is not 3-periodic; our proof does not depend on computer calculations as in [11].Publicación Note on topologically singular points in themoduli space of Riemann surfaces of genus 2(Springer, 2019-06-20) Costa González, Antonio Félix; Porto Ferreira da Silva, Ana MaríaLet Mg be the moduli space of Riemann surfaces of genus g. Rauch (Bull Am Math Soc 68:390–394, 1962) focused his attention on and determined the so-called topological singular points ofMg: these are the points ofMg whose neighbourhoods are not homeomorphic to a ball. In a previous paper, the authors produced a topological proof for Rauch’s result for genera > 2; however, the methods used there do not apply to the genus 2 case. The only known proof for the remaining and important case, i.e., the case of singular points inM2, is to be found in an article by Igusa (Ann Math 72(3):612–649, 1960) and it lays on methods from algebraic geometry. Here, we present a topological proof for this case too.Publicación On the topological type of anticonformal square roots of automorphisms of even order of Riemann surfaces(Springer, 2024-06-04) Costa González, Antonio FélixLet S be a (compact) Riemann surface of genus greather that one. Two automorphism of S are topologically equivalent if they are conjugated by a homeomorphism. The topological classi cation of automorphisms is a classical problem and its study was initiated by J. Nielsen who in the thirties classi ed conformal ones. The case of anticonformal automorphisms is more involved and was solved by K. Yocoyama in the 80s-90s. In order to decide whether two anti-conformal automorphisms are equivalent, it is usually necessary to take into account many invariants, some of which are di cult to compute. In this work we present some situations where the topological equivalence is mainly due to the genus of some quotient surfaces and the algebraic structure of the automorphism group. An anticonformal square root of a conformal automorphism f is an anticonformal automorphism g such that g2 = f . Let g1 and g2 be anticonformal square roots of the same conformal automorphism of order m, where m is an even integer. If genus of S= hg1; g2i is even and genus of S= hgii is 6= 2 we prove that hg1i and hg2i are topologically equivalent. If genus of S= hg1; g2i is odd and hg1; g2i is abelian we obtain that hg1i and hg2i are topologically equivalent. We give examples to justify the condition genus of S= hgii 6= 2 and hg1; g2i abelian in each case.Publicación One dimensional equisymmetric strata in moduli space with genus 1 quotient surfaces(Springer, 2024) S. Allen Broughton; Costa González, Antonio Félix; Izquierdo, Milagros; https://orcid.org/0000-0002-9557-9566The complex orbifold structure of the moduli space of Riemann surfaces of genus g (g ≥2) produces a stratification into complex subvarieties named equisymmetric strata. Eachequisymmetric stratum is formed by the surfaces where the group ofautomorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata of dimension one are of two structurally different types. Type 1 equisymmetric strata correspond to Riemann surfaces where the group of automorphisms produces a quotient surface of genus zero, while those of Type 2 appear when such a quotient is a surface of genus one. Type 1 equisymmetric strata have been extensively studied by the authors of the present work in a previous recent paper, we now focus on Type 2 strata. We first establish the existence of such strata and their frequency of occurrence in moduli spaces. As a main result we obtain a complete description of Type 2 strata as coverings of the sphere branched over three point (Belyi curves) and where certain isolated points (punctures) have to be eliminated. Finally, we study in detail the doubly infinite family of Type 2 strata whose automorphism groupshave order the product of two primes.