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Rubén A. Hidalgo

Publications and source records attributed to Rubén A. Hidalgo.

At least 19 recordsLinked to original sources
arXiv2026

Purely non-free actions on Riemann surfaces

A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If $G$ is an infinite countable group, and $X$ is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on $X$ admitting a gpnf-action by $G$. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus $g \geq 2$. If $G$ is a finite group, then its purely non-free genus $σ^{pnf}_{hyp}(G)$ is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by $G$. This parameter is already known for cyclic, abelian non-cyclic groups, ${\mathcal A}_{4}$, ${\mathcal A}_{5}$ and ${\mathfrak S}_{4}$. For general $G$, we provide the value of $σ^{pnf}_{hyp}(G)$, correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups $G^{0}_{s}(n)=\langle x,y: x^{n}=y^{2}=1, yxy=x^{s}\rangle,$ where $n \geq 3$ and $2 \leq s \leq n-1$ is such that $s^{2} \equiv 1 \mod(n)$.

math.GT↗
arXiv2026

Infinite-type Schottky groups and group actions on infinite-type surfaces

We introduce a certain class of purely loxodromic free Kleinian groups, called infinite-type Schottky groups, which are defined by a suitable collection of simple loops on the Riemann sphere, in a similar way as in the case of Schottky groups of finite rank. An infinite-type Schottky group $Γ$ admits a $Γ$-invariant connected component $Ω_Γ$ of its region of discontinuity $Ω(Γ)$, such that every other connected component of $Ω(Γ) \setminus Ω_Γ$ is a topological disc with trivial $Γ$-stabilizer, and $Ω_Γ/Γ$ is an infinite-type Riemann surface without planar ends. Let $F$ be a torsion-free purely hyperbolic Fuchsian group of the first kind such that $Σ_{F}={\mathbb H}^{2}/F$ is an infinite-type Riemann surface with no planar ends. Then there exists an infinite-type Schottky group $Γ$ such that $Σ_{F}$ is isomorphic to $Ω_Γ/F$ (retrosection theorem). If $G < {\rm Aut}(Σ_{F})$ acts freely and $Σ_{F}/G$ is of finite-type, then we observe that (i) the existence of some infinite Schottky $Γ$ such that $Ω_Γ/Γ$ and $Σ_{F}$ are conformally equivalent and for which $G$ lifts to a group of automorphisms of $Ω_Γ$, is equivalent to (ii) the existence of a $G$-invariant collection ${\mathcal F}$ of pairwise disjoint essential simple loops on $Σ_{F}$ such that each connected component of $Σ_{F} \setminus {\mathcal F}$ is a finite planar surface. This generalizes the situation for the case of closed Riemann surfaces and Schottky groups of finite rank.

math.GT↗
arXiv2026

${\mathbb Z}_{k}^{m}$-actions of signature $(0;k,\stackrel{n+1}{\ldots},k)$

An action of a finite group $G$ is a pair $(S,\hat{G})$, where $S$ is a compact Riemann surface of genus $g \geqslant 2$ and $\hat{G} \leqslant {\rm Aut}(S)$ is isomorphic to $G$. To each action $(S,\hat{G})$ there is associated a signature $(γ;k_{1},\ldots,k_{r})$ that codifies the orbifold structure of $S/\hat{G}$. Two actions of $G$, say $(S_{1},G_{1})$ and $(S_{2},G_{2})$, are topologically equivalent if there is an orientation-preserving homeomorphism $φ:S_{1} \to S_{2}$ such that $φG_{1} φ^{-1}=G_{2}$. Topologically equivalent actions necessarily must have the same signature. The problem of determining the number of different topological actions of $G$ for a given signature is in general a difficult task. In this article, we describe, up to topological equivalence, those actions when $G$ is an abelian group and quotient genus $γ=0$. We are particularly interested in the case $G={\mathbb Z}_{k}^{m}$ and the quotient signature of the action to be of the form $(0;k,\stackrel{n+1}{\ldots},k)$.

math.AG↗
arXiv2025

Towards a proof of the Classical Schottky Uniformization Conjecture

By Koebe's retrosection theorem, every closed Riemann surface of genus $g \geq 2$ is uniformized by a Schottky group. Marden observed that there are Schottky groups that are not classical ones, that is, they cannot be defined by a suitable collection of circles. This opened the question of whether every closed Riemann surface can be uniformized by a classical Schottky group. In this paper, we observe that every Belyi curve can be uniformized by a classical Schottky group. Since Belyi curves form a dense locus in the moduli space ${\mathcal M}_{g}$ and the locus ${\mathcal M}_{g}^{cs} \subset {\mathcal M}_{g}$ of those Riemann surfaces uniformized by classical Schottky groups is a non-empty open set, this ensures that ${\mathcal M}_{g}^{cs}$ is open and dense in ${\mathcal M}_{g}$.

math.CV↗
arXiv2024

Homology Covers and Automorphisms: Examples

Let $S$ be a Riemann surface with a non-abelian fundamental group and for each integer $k \geq 2$ or $k=\infty$, let $\widetilde{S}_{k}$ be its $k$-homology cover. The surface $\widetilde{S}_{k}$ admits a group of conformal automorphisms $M_{k} \cong {\rm H}_{1}(S;{\mathbb Z}_{k})$, where ${\mathbb Z}_{\infty}:={\mathbb Z}$, such that $S=\widetilde{S}_{k}/M_{k}$. If $L \leq {\rm Aut}(S)$, then there is a short exact sequence $1 \to M_{k} \to \widetilde{L}_{k} \to L \to 1$, where $\widetilde{L}_{k}$ is a subgroup of conformal automorphisms of $\widetilde{S}_{k}$. In general, the above exact sequence does not need to be split. This paper investigates situations when the splitting is or is not obtained.

math.GT↗
arXiv2024

The structure of MDC-Schottky extension groups

Let $M^{0}$ be a complete hyperbolic $3$-manifold whose conformal boundary is a closed Riemann surface $S$ of genus $g \geq 2$. If $M=M^{0} \cup S$, then let ${\rm Aut}(S;M)$ be the group of conformal automorphisms of $S$ which extend to hyperbolic isometries of $M^{0}$. If the natural homomorphism at fundamental groups, induced by the natural inclusion of $S$ into $M$, is not injective, then it is known that $|{\rm Aut}(S;M)| \leq 12(g-1)$. If $M$ is a handlebody, then it is also known that the upper bound is attained. In this paper, we consider the case when $M$ is homeomorphic to the connected sum of $g \geq 2$ copies of $D^{*} \times S^{1}$, where $D^{*}$ denotes the punctured closed unit disc and $S^{1}$ the unit circle. In this case, we obtain that: (i) if $g=2$, then $|{\rm Aut}(S;M)| \leq 12$ and the equality is attained, this happening for ${\rm Aut}(S;M)$ isomorphic to the dihedral group of order $12$, and (ii) if $g \geq 3$, then $|{\rm Aut}(S;M)|<12(g-1)$, in particular, the above upper bound is not attained.

math.GT↗
arXiv2024

Uniqueness of Generalized Fermat Groups in positive characteristic

Let $X\subset {\mathbb P}_{K}^{m}$ be a smooth irreducible projective algebraic variety of dimension $d$, defined over an algebraically closed field $K$ of characteristic $p>0$. We say that $X$ is a generalized Fermat variety of type $(d;k,n)$, where $n \geq d+1$ and $k \geq 2$ is relatively prime to $p$, if there is a Galois branched covering $π\colon X\to {\mathbb P}_{K}^{d}$, with deck group ${\mathbb Z}_k^n\cong H<\rm{Aut}(X)$, whose branch divisor consists of $n+1$ hyperplanes in general position (each one of branch order $k$). In this case, the group $H$ is called a generalized Fermat group of type $(d;k,n)$. We prove that, if $k-1$ is not a power of $p$ and either (i) $p=2$ or (ii) $p>2$ and $(d;k,n) \notin \{(2;2,5), (2;4,3)\}$, then a generalized Fermat variety of type $(d;k,n)$ has a unique generalized Fermat group of that type.

math.AG↗
arXiv2024

On non-normal subvarieties of the moduli space of Riemann surfaces

In this article, we consider certain irreducible subvarieties of the moduli space of compact Riemann surfaces determined by the specification of actions of finite groups. We address the general problem of determining which among them are non-normal subvarieties of the moduli space.

math.AG↗
arXiv2024

Extending finite free actions of surfaces

We prove the existence of finite groups of orientation-preserving homeomorphisms of some closed orientable surface $S$ that act freely and which extends as a group of homeomorphisms of some compact orientable $3$-manifold with boundary $S$, but which cannot extend to a handlebody.

math.GT↗
arXiv2023

Description of Origamis by Schottky groups

Let $(S,η)$ be an origami pair, that is, $S$ is a closed Riemann surface of genus $g \geq1$ and $η:S \to E$ is a holomorphic branched covering, with at most one branch value, where $E$ is a genus one Riemann surface. As the lowest uniformizations of $S$ are provided by Schottky groups, we are interested in describing origami pairs in terms of virtual Schottky groups. In other words, we are interested in those Kleinian groups $K$ which contain, as a finite index subgroup, a Schottky group $Γ$ such that $S=Ω/Γ$ and such that $η$ is induced by the inclusion $Γ\leq K$. We say that $K$ is an origami-Schottky group. We provide a geometrical structural picture, in terms of the Klein-Maskit combination theorems, of these origami-Schottky groups.

math.GT↗
arXiv2023

The minima of the geodesic length functions of uniform filling curves

There is a natural link between (multi-)curves that fill up a closed oriented surface and dessins d'enfants. We use this approach to exhibit explicitly the minima of the geodesic length function of a kind of curves (uniform filling curves) which include those that admit a homotopy equivalent representative such that all self-intersection points as well as all faces of their complement have the same multiplicity. We show that these minima are attained at the Grothendieck-Belyi surfaces determined by a natural dessin d'enfant associated to these filling curves. In particular they are all Riemann surfaces defined over number fields.

math.GT↗
arXiv2023

Quasi-abelian group as automorphism group of Riemann surfaces

Conformal/anticonformal actions of the quasi-abelian group $QA_{n}$ of order $2^n$, for $n\geq 4$, on closed Riemann surfaces, pseudo-real Riemann surfaces and closed Klein surfaces are considered. We obtain several consequences, such as the solution of the minimum genus problem for the $QA_n$-actions, and for each of these actions, we study the topological rigidity action problem. In the case of pseudo-real surfaces, attention was typically restricted to group actions that admit anticonformal elements. In this paper we consider two cases: either $QA_n$ has anticonformal elements or only contains conformal elements.

math.AG↗
arXiv2022

Generalized quasi-dihedral group as automorphism group of Riemann surfaces

In this paper, we discuss certain types of conformal/anticonformal actions of the generalized quasi-dihedral group $G_{n}$ of order $8n$, for $n\geq 2$, on closed Riemann surfaces, pseudo-real Riemann surfaces and compact Klein surfaces, and in each of these actions we study the uniqueness (up to homeomorphisms) action problem.

math.AG↗
arXiv2022

Smooth quotients of generalized Fermat curves

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient orbifold $S/H$ of genus zero with exactly $n+1$ cone points, each one of order $p$; in this case $H$ is called a generalized Fermat group of type $(p,n)$. In this case, it is known that $S$ is non-hyperelliptic and that $H$ is its unique generalized Fermat group of type $(p,n)$. Also, explicit equations for them, as a fiber product of classical Fermat curves of degree $p$, are known. For $p$ a prime integer, we describe those subgroups $K$ of $H$ acting freely on $S$, together with algebraic equations for $S/K$, and determine those $K$ such that $S/K$ is hyperelliptic.

math.AG↗
arXiv2021

Dessins d'enfants and some holomorphic structures on the Loch Ness Monster

The classical theory of dessin d'enfants, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (the surface of infinite genus with exactly one end) admits infinitely many regular dessins d'enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the Loch Ness monster, which come from homology covers of compact Riemann surfaces, infinite hyperelliptic and infinite superelliptic curves.

math.GT↗
arXiv2020

Homology group automorphisms of Riemann surfaces

If $Γ$ is a finitely generated Fuchsian group such that its derived subgroup $Γ'$ is co-compact and torsion free, then $S={\mathbb H}^{2}/Γ'$ is a closed Riemann surface of genus $g \geq 2$ admitting the abelian group $A=Γ/Γ'$ as a group of conformal automorphisms. We say that $A$ is a homology group of $S$. A natural question is if $S$ admits unique homology groups or not, in other words, is there are different Fuchsian groups $Γ_{1}$ and $Γ_{2}$ with $Γ_{1}'=Γ'_{2}$? It is known that if $Γ_{1}$ and $Γ_{2}$ are both of the same signature $(0;k,\ldots,k)$, for some $k \geq 2$, then the equality $Γ_{1}'=Γ_{2}'$ ensures that $Γ_{1}=Γ_{2}$. Generalizing this, we observe that if $Γ_{j}$ has signature $(0;k_{j},\ldots,k_{j})$ and $Γ_{1}'=Γ'_{2}$, then $Γ_{1}=Γ_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${\rm Aut}(S)$ of each homology group $A$ is also obtained.

math.GT↗
arXiv2020

Weil's Galois Descent Theorem from a computational point of view

Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over ${\mathcal K}$, that is, for the existence of an algebraic variety $Y$ defined over ${\mathcal K}$ together with a birational isomorphism $R:X \to Y$ defined over ${\mathcal L}$. Weil's proof does not provide a method to construct the birational isomorphism $R.$ The aim of this paper is to give an explicit construction of $R$.

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