On the orbifold structure of the moduli space of Riemann surfaces of genera four and five
type: Item
instance of: scholarly article
title: On the orbifold structure of the moduli space of Riemann surfaces of genera four and five
zbMATH Open document ID: 1297.14031
author: Gabriel Bartolini
author: Antonio F. Costas
author: Milagros Izquierdo
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2014-09-09T00:00:00Z
full work available at URL: http://urn.kb.se/resolve
review text: For a given genus \(g \geq 4\), the moduli space \(\mathcal M_g\) of compact Riemann surfaces of genus \(g\) is the quotient of the Teichmüller space by the action of the mapping class group. It has the structure of a complex orbifold, and its set of singular points is called the branch locus \(\mathcal B_g\). When the actions of the automorphism groups of two surfaces of the same genus are topologically equivalent, the surfaces are said to be equisymmetric. Then the moduli space \(\mathcal M_g\) has an equisymmetric stratification in which each stratum consists of the points corresponding to equisymmetric surfaces. So the branch locus is formed by the strata corresponding to surfaces whose automorphism group is not trivial. The goal of the paper under review is to determine the orbifold structure of the moduli spaces \(\mathcal{M}_4\) and \(\mathcal{M}_5\), by means of the topological classification of the actions of the automorphism groups of Riemann surfaces of both genera \(4\) and \(5\). These actions are determined using the uniformization of Riemann surfaces by Fuchsian groups. The strata of \(\mathcal{M}_4\) and \(\mathcal{M}_5\) correspond to the actions of the full automorphism groups of Riemann surfaces of the respective genus. So the authors obtain firstly all triples \((G,\Delta,\theta)\) satisfying that \(\theta : \Delta \rightarrow G\) is an epimorphism from the Fuchsian group \(\Delta\) onto the finite group \(G\) such that \(\ker(\theta)\) is a surface group of genus \(4\) (resp. \(5\)). Then they determine when these actions are maximal, and so the group \(G\) is the full group of automorphisms of a surface, what gives the list of equisymmetric strata of \(\mathcal{B}_4\) and \(\mathcal{B}_5\). The last step is to determine the inclusion relations of the strata for each of the two branch loci \(\mathcal{B}_4\) and \(\mathcal{B}_5\). The key point to obtain it is the fact that if the action \((G, \theta)\) is an extension of the action \((G', \theta…
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 57N60
Mathematics Subject Classification ID: 57N80
zbMATH DE Number: 6341672
zbMATH Keywords: moduli spaces
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: equisymmetric stratification
zbMATH Keywords: orbifold structure
reviewed by: José Javier Etayo
describes a project that uses: GAP
MaRDI profile type: Publication
OpenAlex ID: W2092781827
cites work: On the connectedness of the branch locus of the moduli space of Riemann surfaces
cites work: On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus
cites work: Classifying the actions of finite groups on orientable surfaces of genus 4
cites work: Q4502840
cites work: The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups
cites work: Classifying finite group actions on surfaces of low genus
cites work: Finite abelian subgroups of the mapping class group
cites work: On extendability of group actions on compact Riemann surfaces
cites work: On symmetries of compact Riemann surfaces with cyclic groups of automorphisms
cites work: Groups of automorphisms of cyclic trigonal Riemann surfaces
cites work: Q4400425
cites work: ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF THE MODULI SPACE OF RIEMANN SURFACES OF GENUS 4
cites work: Q3567906
cites work: Maximal order of automorphisms of trigonal Riemann surfaces
cites work: On Riemann surfaces with non-unique cyclic trigonal morphism
cites work: Applications of a theorem of singerman about Fuchsian groups
cites work: On some Galois invariants of branched coverings
cites work: On the real nerve of the moduli space of complex algebraic curves of even genus
cites work: On Branch Loci in Teichmuller Space
cites work: EQUISYMMETRIC STRATA OF THE MODULI SPACE OF CYCLIC TRIGONAL RIEMANN SURFACES OF GENUS 4
cites work: Classification of automorphism groups, up to topological equivalence, of compact Riemann surfaces of genus 4
cites work: Automorphism groups of compact Riemann surfaces of genus five
cites work: Isolated points in the branch locus of the moduli space of compact Riemann surfaces
cites work: Q3994371
cites work: Subgroups of Fuchsian Groups and Finite Permutation Groups
cites work: Finitely Maximal Fuchsian Groups
cites work: Abelian actions on 2-manifolds
cites work: Q3582260
cites work: Hyperelliptic surfaces and their moduli
DOI: 10.1007/S13398-013-0140-8
description: scientific article; zbMATH DE number 6341672