Groups of automorphisms of bordered orientable Klein surfaces of topological genus 2
type: Item
instance of: scholarly article
title: Groups of automorphisms of bordered orientable Klein surfaces of topological genus 2
zbMATH Open document ID: 1383.57025
published in: Osaka Journal of Mathematics
publication date: 2017-12-22T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1508486582
review text: Let \(X\) be a Klein surface, that is a compact surface endowed with a dianalytic structure. This type of surface is determined topologically by three data: the topological genus \(g\), the number of boundary components \(k\) and its orientability. With these data, the algebraic genus \(p\) can be calculated. If \(p \geq 2\), the automorphism group of the surface is finite and its order is bounded above by \(12(p-1)\). Regarding group actions, a major problem is to determine which groups act on a surface of a given topological type. Some data is known for this topic: bounds for the maximal order, groups of automorphisms of Riemann surfaces of low topological genus, groups of automorphisms of bordered Klein surfaces of low algebraic genus, groups of automorphisms of orientable Klein surfaces of topological genus 0 and 1. In this paper, the authors obtain the groups of automorphisms of bordered orientable Klein surfaces of topological genus 2, expanding the existing abelian list for this type of surfaces. Section 2 is on the groups of automorphisms of compact orientable Klein surfaces of topological genus 2, using and expanding previous results. Also in this section, the notation is established for all the rest of the paper. Section 3 contains the main result, in which the authors establish the admissible values of \(k\) for a given group \(G\) that is the group of automorphisms of an orientable Klein surface of topological genus 2 with \(k \geq 1\) boundary components. The proof is highly complex. The authors prove the theorem for a sample of groups that illustrate in a very clear way the procedure. The remaining groups follow by a proof similar to the one given in the paper. The proof is detailed and well explained. In general, the paper is well written and the results are very interesting and novel.
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F50
zbMATH DE Number: 6821140
zbMATH Keywords: group of automorphisms
zbMATH Keywords: Klein surfaces
zbMATH Keywords: topological genus 2
author: Emilio Bujalance
author: José Javier Etayo
author: Ernesto Martínez
reviewed by: Adrián Bacelo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6821140