Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2007.01781

Description of Origamis by Schottky groups

Rubén A. Hidalgo
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rubén A. Hidalgo. 2023-10-23. Description of Origamis by Schottky groups. https://arxiv.org/abs/2007.01781

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Word Length Formulae, Normal Forms, Conjugation and Root-finding Algorithms in Surface Groups

In this paper, we mainly study the following symmetric presentation of the surface group $$π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle.$$ For every nontrivial element $x\in π_1(Σ_g)$ and $k\geq 2$, we obtain a uniform representative of the normal forms $\mathfrak{nf}(x^k)$ of $x^k$ under the length-lexicographical order: $$\mathfrak{nf}(x^k) = \overline{LW^{k-2}R}.$$ Building on this result, we establish a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Furthermore, we extend these results to obtain a coarser analogue for every minimal geometric presentation. We then define normal forms of conjugacy classes in $π_1(Σ_g)$ and provide a criterion for determining the conjugacy of group elements. As a consequence, we provide efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications to the computation of several growth rates.

math.GT↗
arXiv2026

Barbell twists are natural

For any oriented smooth $4$--manifold $X$ diffeomorphic to $(S^2\times D^2)^{\natural n}$ ($n\geq0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times\wedge^2H_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fixing mapping class group of $X$. For $n=2$, the Budney--Gabai barbell twist $φ\in\mathrm{Mod}(\mathcal{N},\partial\mathcal{N})$ is identified with a generator of the factor subgroup $\wedge^2H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}$. Up to boundary-fixing diffeotopy, the barbell spines of $\mathcal{N}$ are completely classified by the bases of $H_2(\mathcal{N};\mathbb{Z})\cong\mathbb{Z}^2$, forming a homogeneous set modeled on the group $\mathrm{GL}(H_2(\mathcal{N};\mathbb{Z}))\cong\mathrm{GL}(2,\mathbb{Z})$. Any barbell spine of $\mathcal{N}$ gives rise to an implanted barbell twist equal to $φ$ or $φ^{-1}$ in $\mathrm{Mod}(\mathcal{N},\partial \mathcal{N})$, according to the sign of the homological basis orientation.

math.GT↗
arXiv2026

Plane separating continua inscribe rectangles

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

math.GT↗