Extremal disc packings in compact hyperbolic surfaces
type: Item
instance of: scholarly article
title: Extremal disc packings in compact hyperbolic surfaces
zbMATH Open document ID: 1401.30047
author: Ernesto Girondo
published in: Revista Matemática Complutense
publication date: 2018-06-01T00:00:00Z
review text: The author introduces extremal \(k\)-packings of compact Riemann surfaces as a natural generalization of extremal discs. A compact Riemann surface of genus \(g\geq 2\) is called a \(k\)-extremal surface if it contains an extremal \(k\)-packing, namely, a collection of \(k\) metric discs of the largest radius \(R=R(g,k)\) such that the discs are pairwise disjoint and each of them is homeomorphic to a topological disc. The radius \(R\) is described by \(g\) and \(k\) explicitly. For \(k\)-extremal surfaces of genus \(g\), the author obtains some interesting properties: the relation between \(k\) and \(g\) which \(k\)-extremal surfaces of genus \(g\) fulfill, a characterization of a surface group uniformizing a \(k\)-extremal surface as a subgroup of Fuchsian triangle groups of a certain type, determination of pairs \(\{k_1,k_2\}\) such that a Riemann surface of genus \(g\) is \(k_1\)-extremal and \(k_2\)-extremal simultaneously, and the relation between \(k\) and \(g\) such that extremal \(k\)-packings in \(k\)-extremal surfaces are unique or not. The author also shows that quasiplatonic surfaces do not contain extremal discs, that is, they are not (1-)extremal surfaces while there are infinitely many \(k\)'s for a surface to be quasiplatonic and \(k\)-extremal. Moreover, the actions of the automorphism groups on the extremal \(k\)-packings are explained by giving examples of compact Riemann surfaces.
reviewed by: Gou Nakamura
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 52C26
zbMATH DE Number: 6879859
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: extremal \(k\)-packings
zbMATH Keywords: uniform Belyi functions
zbMATH Keywords: triangle groups
MaRDI profile type: Publication
full work available at URL: s13163-017-0252-3
OpenAlex ID: W2771934456
cites work: Extremal disks and modular surfaces
cites work: Equality in Laszlo Fejes Toth's triangle bound for hyperbolic surfaces
cites work: Q3310101
cites work: Packing of spheres in spaces of constant curvature
cites work: Torsion free subgroups of Fuchsian groups and tessellations of surfaces
cites work: On extremal discs inside compact hyperbolic surfaces
cites work: Genus of two extremal surfaces: Extremal discs, isometries and Weierstrass points
cites work: On extremal Riemann surfaces and their uniformizing fuchsian groups
cites work: Q3110201
cites work: Compact non-orientable hyperbolic surfaces with an extremal metric disc
cites work: Shimura curves with many uniform dessins
cites work: Dessins d'Enfants on Riemann Surfaces
cites work: Q4027286
cites work: Extremal disks and extremal surfaces of genus three
cites work: Compact non-orientable surfaces of genus 4 with extremal metric discs
cites work: Compact non-orientable surfaces of genus 5 with extremal metric discs
cites work: Compact Klein surfaces of genus 5 with a unique extremal disc
cites work: Arithmetic triangle groups
cites work: Q4141415
cites work: Q4364760
DOI: 10.1007/S13163-017-0252-3
description: scientific article; zbMATH DE number 6879859