arXiv subjects

Antonio F. Costa

Publications and source records attributed to Antonio F. Costa.

18 recordsLinked to original sources

arXiv2025

Fenchel's conjecture on NEC groups

A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group $F$ has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable surface, or in algebraic terms, that $F$ has a normal subgroup of finite index that contains no element of finite order other than the identity. In this paper we initiate and make progress on an extension of Fenchel's conjecture by considering the following question: Does every planar non-Euclidean crystallographic group $Γ$ containing transformations that reverse orientation have a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered non-orientable surface? We answer this question in the affirmative in the case where the orbit space of $Γ$ is a nonorientable surface, and also in the case where this orbit space is a bordered orientable surface of positive genus. In the case where the genus of the quotient is $0$, we have an affirmative answer in many subcases, but the question is still open for others.

math.GT↗

arXiv2025

Modular companions in planar one-dimensional equisymmetric strata

Consider, in the moduli space of Riemann surfaces of a fixed genus, the subset of surfaces with non-trivial automorphisms. Of special interest are the numerous subsets of surfaces admitting an action of a given finite group, $G$, acting with a specific signature. In a previous study we declared two Riemann surfaces to be \emph{modular companions} if they have topologically equivalent $G$ actions, and that their $G$ quotients are conformally equivalent orbifolds. In this article we present a geometrically-inspired measure to decide whether two modular companions are conformally equivalent (or how different), respecting the $G$ action. Along the way, we construct a moduli space for surfaces with the specified $G$ action and associated equivariant tilings on these surfaces. We specifically apply the ideas to planar, finite group actions whose quotient orbifold is a sphere with four cone points.

math.GT↗

arXiv2024

On the topological type of anticonformal square roots of automorphisms of even order of Riemann surfaces

Let $S$ be a (compact)\ Riemann surface of genus greater than one. Two automorphism of $S$ are topologically equivalent if they are conjugated by a homeomorphism. The topological classification of automorphisms is a classical problem and its study was initiated by J. Nielsen who in the thirties classified 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 difficult 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 $g^{2}=f$ . Let $g_{1}$ and $g_{2}$ be anticonformal square roots of the same conformal automorphism of order $m$, where $m$ is an even integer. If genus of $S/\left\langle g_{1},g_{2} \right\rangle $ is even and genus of $S/\left\langle g_{i}\right\rangle $ is $\neq2$ we prove that $\left\langle g_{1}\right\rangle $ and $\left\langle g_{2}\right\rangle $ are topologically equivalent. If genus of $S/\left\langle g_{1},g_{2}\right\rangle $ is odd and $\left\langle g_{1},g_{2}\right\rangle $ is abelian we obtain that $\left\langle g_{1}\right\rangle $ and $\left\langle g_{2}\right\rangle $ are topologically equivalent. We give examples to justify the condition genus of $S/\left\langle g_{i}\right\rangle $ $\neq2$ and $\left\langle g_{1},g_{2}\right\rangle $ abelian in each case.

math.GT↗

arXiv2022

Periodicity and Free Periodicity of Alternating Knots

A knot $K$ in $S^3$ is $q$-periodic if it admits a symmetry that is conjugate to a rotation of order $q$ of $S^3$. If $K$ admits a symmetry which is a homeomorphism without fixed point of period $q$ of $S^3$, then $K$ is called freely $q$-periodic. In a previous paper, we obtained, as a consequence of Flyping Theorem due to Menasco and Thislethwaite, that the $q$-periodicity with $q>2$ can be visualized in an alternating projection as a rotation of the projection sphere. In this paper, we show that the free $q$-action of an alternating knot can be represented on some alternating projection as a composition of a rotation of order $q$ with 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 somehow visible (virtually visible) and give some sufficient criteria to detect the existence of $q$-actions from virtually visible projections. Finally, we show how the Murasugi decomposition into atoms enables us to determine the visibility type $(q,r)$ of the freely $q$-periodic alternating knots ($(q,r)$-lens knots); in fact, we only need to focus on a certain atom of their Murasugi decomposition to deduce their visibility type.

math.GT↗

arXiv2021

Periodic projections of alternating knots

This paper is devoted to prove the existence of $q$-periodic alternating projections of prime alternating $q$-periodic knots. The main tool is the Menasco-Thistlethwaite's Flyping theorem. Let $K$ be an oriented prime alternating knot that is $q$-periodic with $q\geq 3$, i.e. $K$ admits a symmetry that is a rotation of order $q$. Then $K$ has an alternating $q$-periodic projection. As applications, we obtain the crossing number of a $q$ -periodic alternating knot with $q\geq 3$ is a multiple of $q$ and we give an elementary proof that the knot $12_{a634} $ is not 3-periodic; this proof does not depend on computer computations as in "Periodic knots and Heegaard Floer correction terms" by Stanilav Jabuka and Swatee Naik (arXiv:1307.5116 [math.GT]).

math.GT↗

arXiv2020

Note on the classification of the orientation reversing homeomorphisms of finite order of surfaces

The aim of this note is to stablish the topological classification of finite period orientation reversing autohomeomorphims of a closed oriented surface when the period is 2q, with q even. The classification of periodic orientation reversing autohomeomorphims of a closed oriented surface has been made by Kazuo Yokoyama in Complete classification of periodic maps on compact surfaces, Tokyo J. Math. 15 (1992), no. 2, 247--279, and by the author in Classification of the orientation reversing homeomorphisms of finite order of surfaces, Topology and its applications 62 (1995) 145--162 (reference [1] of this paper) following different approaches. In [1] there are some errors for the case considered in this note: orientation reversing homeomorphims of period multiple of 4. The errors have been pointed out by Weibiao Wang of the School of Mathematical Sciences of Peking University and Chao Wang of the School of Mathematical Sciences of East China Normal University in Shanghai. The approach to this problem in [1] is useful and in Section 3 we obtain the classification for the case when the orientation reversing homemorphism has period multiple of 4 following the ideas of [1]. The results in [1] has been used in references [2] and [3]. The last Section include the corrections to these articles following Section 3.

math.GT↗

arXiv2018

Topological classification of generic real meromorphic functions

In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines the topological type of the generic real meromorphic function and that the set of all parks is in $1-1$-correspondence with the set of all connected components in the space of generic real meromorphic functions. For any of the above components, we introduce and calculate the corresponding Hurwitz number. Finally as a consequence of our results we determine the topological types of real meromorphic functions from monodromy of orbifold coverings.

math.CV↗

arXiv2018

Topologically singular points in the moduli space of Riemann surfaces

In 1962 E. H. Rauch established the existence of points in the moduli space of Riemann surfaces not having a neighbourhood homeomorphic to a ball. These points are called here topologically singular. We give a different proof of the results of Rauch and also determine the topologically singular and non-singular points in the branch locus of some equisymmetric families of Riemann surfaces.

math.AG↗

arXiv2017

On the connectedness of the set of Riemann surfaces with real moduli

The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural real structure i.e. it admits an anti-holomorphic involution $σ$. The involution $σ$ maps each point corresponding to a Riemann surface $S$ to its complex conjugate $\overline{S}$. The fixed point set of $σ$ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside $\mathrm{Fix}(σ)$ 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. Seppälä and R. Silhol. The complement $\mathrm{Fix}(σ)-{\mathcal{M}}_{g}(\mathbb{R})$ consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that $\mathrm{Fix}(σ)$ is connected.

math.CV↗

arXiv2015

Automorphism Groups of Cyclic p-gonal Pseudo-real Riemann Surfaces

In this article we prove that the full automorphism group of a cyclic $p$-gonal pseudo-real Riemann surface of genus $g$ is either a semidirect product $C_{n}\ltimes C_{p}$ or a cyclic group, where $p$ is a prime $>2$ and $g>(p-1)^{2}$. We obtain necessary and sufficient conditions for the existence of a cyclic $p$-gonal pseudo-real Riemann surface with full\ automorphism group isomorphic to a given finite group. Finally we describe some families of cyclic $p$-gonal pseudo-real Riemann surfaces where the order of the full automorphism group is maximal and show that such families determine some real 2-manifolds embbeded in the branch locus of moduli space.

math.AG↗

arXiv2015

The double of the doubles of Klein surfaces

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface $X$ is a Klein surface $Y$ such that there is a degree two morphism (of Klein surfaces) $Y\rightarrow X$. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if $X$ is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: "the double of doubles" denoted by $DX$. We describe how to use the double of doubles in the study of both moduli spaces and automorphisms of Klein surfaces. Furthermore, we show that the morphism from $DX$ to $X$ is not given by the action of an isometry group on classical surfaces.

math.GT↗

arXiv2013

Connecting $p$-gonal loci in the compactification of moduli space

Consider the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g\geq 2$ and its Deligne-Munford compactification $\bar{\mathcal{M}_{g}}$. We are interested in the branch locus ${\mathcal{B}_{g}}$ for $g>2$, i.e., the subset of $\mathcal{M}_{g}$ consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in $\mathcal{M}_{g}$ but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for $g\geq 5$ the set of (cyclic) trigonal surfaces is connected in $\bar{\mathcal{M}_{g}}$. To do so we exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of 3-gonal Riemann surfaces. For $p>3$ the connectivity of the $p$-gonal loci becomes more involved. We show that for $p\geq 11$ prime and genus $g=p-1$ there are one-dimensional strata of cyclic $p$-gonal surfaces that are completely isolated in the completion $\bar{\mathcal{B}_{g}}$ of the branch locus in $\bar{\mathcal{M}_{g}}$.

math.GT↗

arXiv2010

Prime order automorphisms of Klein surfaces representable by rotations of the euclidean space

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S,f) to be conformally equivalent to (S',f') where S' is a bordered orientable Klein surface embedded in the Euclidean space and f' is the restriction to S' of a prime order rotation. We represent two famous automorphisms using rotations of R^4 and S^4 : the order seven automorphisms of the Klein quartic and the Wiman surface.

math.GT↗

arXiv2010

Klein Foams

Klein foams are analogues of Riemann and Klein surfaces with one-dimensional singularities. We prove that the field of dianalytic functions on a Klein foam $Ω$ coincides with the field of dianalytic functions on a Klein surface $K_Ω$. We construct the moduli space of Klein foams and we prove that the set of classes of topologically equivalent Klein foams form an analytic space homeomorphic to $\mathbb{R}^{n}/\mathsf{Mod}$, where $\mathsf{Mod}$ is a discrete group.

math.AG↗

arXiv2006

Poincaré's theorem for the modular group of real Riemann surfaces

Let $Mod_{g}$ be the modular group of surfaces of genus $g$. Each element $[h]\in Mod_{g}$ induces in the integer homology of a surface of genus $g$ a symplectic automorphism $H([h])$ and Poincaré shown that $H:Mod_{g}\to Sp(2g,\mathbb{Z})$ is an epimorphism. The theory of real algebraic curves justify the definition of real Riemann surface as a Riemann surface $S$ with an anticonformal involution $σ$. Let $(S,σ)$ be a real Riemann surface, the subgroup $Mod_{g}^σ$ of $Mod_{g}$ that consists of the elements $[h]\in Mod_{g}$ that have a representant $h$ such that $h\circσ=σ\circ h$, plays the rôle of the modular group in the theory of real Riemann surfaces. In this work we describe the image by $H$ of $Mod_{g}^σ$. Such image depends on the topological type of the involution $σ$.

math.AG↗

arXiv2005

A geometric characterization of orientation reversing involutions

We give a geometric characterization of compact Riemann surfaces admitting orientation reversing involutions with fixed points. Such surfaces are generally called real surfaces and can be represented by real algebraic curves with non-empty real part. We show that there is a family of disjoint simple closed geodesics that intersect all geodesics of a partition at least twice in uniquely right angles if and only if the involution exists. This implies that a surface is real if and only if there is a pants decomposition of the surface with all Fenchel-Nielsen twist parameters equal to 0 or 1/2.

math.DG↗

arXiv2004

Orientation-reversing free actions on handlebodies

We examine free orientation-reversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free actions in terms of algebraic invariants that involve Nielsen equivalence. This is applied to describe the sets of free actions in various cases, including a complete classification for many (and conjecturally all) cases above the minimum genus. For abelian groups, the free actions are classified for all genera.

math.GT↗

arXiv2000

Topological classification of Z_{p}^{m} actions on surfaces

Let $\widetilde{S}$ be a closed (compact without boundary) oriented surface with genus $g$, and $G$ be a group isomorphic to $% \mathbf{Z}_{p}^{m}$, where $p$ is a prime integer. An action of $G$ on $S$ is a pair $(\widetilde{S},f)$, where $f$ is a representation of $G$ in the group of orientation preserving autohomeomorphisms of $\widetilde{S}$. Two actions $(\widetilde{S},f)$ and $(\widetilde{S^{\prime}},f^{\prime})$ are called strongly (resp. weakly) equivalent if there is a homeomorphism$,$ $% \widetildeψ:\widetilde{S}\to \widetilde{S}^{\prime},$ sending the orientation of $\widetilde{S}$ to the orientation of $\widetilde{S}% ^{\prime},$ such that $f^{\prime}(h)=\widetildeψ\circ f(h)\circ \widetildeψ^{-1},$ (resp. there is an automorphism $α\in Aut(G)$ such that $f^{\prime}\circ α(h)=\widetildeψ\circ f(h)\circ \widetildeψ^{-1}$) for all $h\in G.$ We give the full description of strong and weak equivalence classes. The main idea of our work is the fact that a fixed point free action of $\mathbf{Z}_{p}^{m}$ on a surface provides a bilinear antisymmetric form on $\mathbf{Z}_{p}^{m}.$ For instance, we prove that the weakly equivalence classes of actions of $G$ on surfaces with orbit space of genus $g$ are in one to one correspondence with the set of pairs which consist in a positive integer number $k$, $k\leq m-n,$ $k=(m-n)% \func{mod}2,$ $g\geq {1/2}(m-n+k),$ and an orbit of the action of $% Aut(G)$ on the set of unordered $r$-tuples $[C_{1},...,C_{r}]$ of non-trivial elements generating a subgroup isomorphic to $\mathbf{Z}_{p}^{n}$ and such that $\sum_{1}^{r}C_{i}=0$. We use this result in describing the moduli space of complex algebraic curves admitting a group of automorphisms isomorphic to $\mathbf{Z}_{p}^{m}.$

math.GT↗