Maximal group actions on compact oriented surfaces
type: Item
instance of: scholarly article
title: Maximal group actions on compact oriented surfaces
zbMATH Open document ID: 1368.14042
author: Valerie Peterson
author: Jacob Russell
author: Aaron Wootton
published in: Journal of Algebra
publication date: 2016-12-22T00:00:00Z
review text: Let \(p\) be a prime integer and let \(C_{p}\) be the cyclic group of order \(p\). It is well known the existence of closed Riemann surfaces of genus at least two on which \(C_{p}\) acts as a group of conformal automorphisms. Assume \(S\) is such a closed Riemann surface of genus \(\sigma \geq 2\) and that \(p \geq 5\). The quotient orbifold \(S/C_{p}\) has genus \(h\) and exactly \(r \neq 1\) cone points, each one of order \(p\); its signature is then defined by \((h;p^{r})\). In the paper under review, it is observed that: (i) if \(h \leq (p-5)/2\) and \(r>p+7\), then \(\text{Aut}(S)=C_{p}\), and (ii) if \(h \geq (p-3)/2\) and \(r \geq 3\), then there is a (possible different) closed Riemann surface \(R\) of same genus \(\sigma\) admitting also a conformal action of \(C_{p}\) with quotient orbifold \(R/C_{p}\) of the same signature \((h;p^{r})\) and \(C_{p} \neq \text{Aut}(R)\). The arguments are given by studying the possible generating vectors for \(C_{p}\) and the use of the Riemann-Hurwitz formula. At the end of the paper, two explicit examples are provided. The first one corresponds to the two possible actions of \(C_{7}\) in genus three, in both cases the group of conformal automorphisms being bigger, and the second one is a \(C_{7}\) action of genus nine which turns out to be the full group of conformal automorphisms.
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 37E30
zbMATH DE Number: 6668095
zbMATH Keywords: automorphisms
zbMATH Keywords: compact Riemann surfaces
zbMATH Keywords: mapping class group
reviewed by: Rubén A. Hidalgo
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2016.10.004
OpenAlex ID: W2535983554
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DOI: 10.1016/J.JALGEBRA.2016.10.004
description: scientific article; zbMATH DE number 6668095