arXiv subjects

Camilo Ramírez Maluendas

Publications and source records attributed to Camilo Ramírez Maluendas.

12 recordsLinked to original sources

arXiv2026

Stratification and realization of singularity data on the Loch Ness monster

The classical theory of compact translation surfaces is organized through the stratification of moduli spaces according to the orders of the zeros of the associated Abelian differentials. In the setting of tame translation surfaces of infinite topological type, no analogous notion of strata has been systematically developed. In this article, we introduce a stratification theory for tame translation surfaces based on their singularity data. To each non-compact tame translation surface, we associate a sequence recording the cardinalities of finite-angle cone singularities of each possible cone angle together with the cardinality of the set of infinite-angle singularities. We call this sequence the singularity data of the surface. We study the realization problem for singularity data on the Loch Ness monster. Our first main result shows that there is no tame translation structure on the Loch Ness monster having only a finite number of finite cone angle singularities. We then prove that this is the only obstruction: every singularity data not excluded by the previous result is realized by a tame translation structure on the Loch Ness monster. As a consequence, we obtain a complete characterization of the strata of tame translation structures on this surface in terms of their singularity data. These results provide the first realization theorem for strata of non-compact tame translation surfaces and reveal a strong interaction between singularity data and the ends space.

math.GT↗

arXiv2025

On the Ends of groups and the Veech groups of infinite-genus surfaces

In this paper, we study the PSV construction, which provides a step by step method for obtaining tame translation surfaces with a suitable Veech group. In addition, we modify slightly this construction, and for each finitely generated subgroup $G<{\rm GL}_{+}(2,\mathbb{R})$ without contracting elements, we produce a tame translation surface $S$ with infinite genus such that its Veech group is $G$. Furthermore, the ends space of $S$ can be written as $\mathcal{B}\sqcup \mathcal{U}$, where $\mathcal{B}$ is homeomorphic to the ends space of the group $G$, and $\mathcal{U}$ is a countable, discrete, dense, and open subset of the ends space of $S$.

math.DG↗

arXiv2023

A brief tour through the history of complex numbers

In this paper, we chronologically recount several situations that have contributed to the development and formalization of the objects known as imaginary or complex numbers. We will begin by introducing the earliest documented knowing for calculating the square root of a negative quantity, attributed to the Greek mathematician Heron of Alexandria. From there, we will progress through history to explore the formal concept of complex numbers given by William Rowan Hamilton.

math.HO↗

arXiv2023

On Veech groups of infinite superelliptic curves

We study infinite superelliptic curves as translation surfaces and explore their Veech groups. These objects are branched covering of the complex plane with branching over infinitely many points. We provide a criterion for isomorphism between a special family of infinite superelliptic curves. We show geometric descriptions of saddle connections and holonomy vectors on these infinite superelliptic curves. We prove that the Veech group of an infinite superelliptic curve are all the matrices arising from the differential of the affine mappings $\mathbb{C}$ to itself, permuting the branched points. We obtain necessary and sufficient conditions to guarantee that the Veech group of an infinite superelliptic curve is uncountable. We establish a trichotomy on the holonomy vector set and from it, we give a precise characterization of some countable groups that can appear as Veech group of an infinite superelliptic curve. We also construct and study several examples of interesting infinite superelliptic curves illustrating our results.

math.DG↗

arXiv2023

On the Cauchy Integral Theorem and Polish spaces

We prove that if a function $f$ is continuous in an open subset $U\subset\mathbb{C}$ and analytic in $U\setminus X$, where $X\subset U$ is a Polish space having characteristic system $(i,n)$, such that $i\in\{0,1\}$ and $n\in\mathbb{N}$, then the complex integral line of $f$ along the boundary of any triangle in $U$ vanishes.

math.CV↗

arXiv2022

On the fiber product over infinite-genus Riemann surfaces

Considering non-constant holomorphic maps $β_{i}:S_{i}\to S_{0}$, $i\in\{1,2\}$, between non-compact Riemann surfaces for which it is associated its fiber product $S_{1}\times_{(β_{1},β_{2})}S_{2}$. With this setting, in this paper we relate the ends space of such fiber product to the ends space of its normal fiber product. Moreover, we provide conditions on the maps $β_{1}$ and $β_{2}$ to guarantee connectednes on the fiber product. From these conditions, we link the ends space of fiber product with the topological type of the Riemann surfaces $S_{1}$ and $S_{2}$. We also study the fiber product over infinite hyperelliptic curves and discuss its connectedness and ends space.

math.AG↗

arXiv2021

Parabolicity of zero-twist tight flute surfaces and uniformization of the Loch Ness monster

We study the zero-twist flute surface and we associate to each one of them a sequence of positive real numbers $\mathbf{x}=(x_{n})_{n\in\mathbb{N}_{0}}$, with a torsion-free Fuchsian group $Γ_{\mathbf{x}}$ such that the convex core of $\mathbb{H}^2/Γ_{\mathbf{x}}$ is isometric to a zero-twist tight flute surface $S_{\mathbf{x}}$. Moreover, we prove that the Fuchsian group $Γ_{\mathbf{x}}$ is of the first kind if and only if the series $\sum x_{n}$ diverges. As consequence of the recent work of Basmajian, Hakobian and {Š}ari{ć}, we obtain that the zero-twist flute surface $S_{\mathbf{x}}$ is of parabolic type if and only $\sum x_{n}$ diverges. In addition, we present an uncountable family of hyperbolic surfaces homeomorphic to the Loch Ness Monster. More precisely, we associate to each sequence $\mathbf{y}=(y_{n})_{n\in\mathbb{Z}}$, where $y_{n}=(a_{n},b_{n},c_{n},d_{n},e_{n})\in \mathbb{R}^5$ and $a_n\leq b_n \leq c_n \leq d_n \leq e_n \leq a_{n+1}$, a Fuchsian group $G_{\mathbf{y}}$ such that $\mathbb{H}^2/G_{\mathbf{y}}$ is homeomorphic to the Loch Ness Monster.

math.DG↗

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↗

arXiv2021

Symmetry Type Graphs on 4-Orbit maps

It is well known that there exist twenty two symmetry type graphs associated to 4-orbit maps. For this ones we give the feasible values taken by the degree of the vertices and the number appropriate of edges in the boundary of each face of the map, by introducing the concepts of vertex type graph, face type graph and characteristic system.

math.CO↗

arXiv2019

Geometric Schotkky groups and non compact hyperbolic surface with infinite genus

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface $S_{m,s}$ with $s$ ends and exactly $m$ of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\leq s$, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group $Γ_{m,s}$ such that the quotient space $\mathbb{H}/ Γ_{m, s}$ is homeomorphic to $S_{m,s}$ and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

math.DG↗

arXiv2018

On Infinitely generated Fuchsian groups of some infinite genus surfaces

In this paper, for a non compact and orientable surface $S$ been either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we construct explicitly an infinitely generated Fuchsian group $Γ<PSL(2,\mathbb{R})$, such that the quotient $\mathbb{H}/Γ$ is a hyperbolic surface homeomorphic to $S$.

math.DG↗

arXiv2017

On the Infinite Loch Ness monster

In this paper we present in a topological way the construction of the orientable surface with only one end and infinite genus, called \emph{The Infinite Loch Ness Monster}. In fact, we introduce a flat and hyperbolic construction of this surface. We discuss how the name of this surface has evolved and how it has been historically understood.

math.GT↗