Riemann surfaces with real forms which have maximal cyclic symmetry
type: Item
instance of: scholarly article
title: Riemann surfaces with real forms which have maximal cyclic symmetry
zbMATH Open document ID: 1068.30031
author: Francisco Javier Cirre
author: Emilio Bujalance
author: Peter Turbek
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: It was proved by \textit{C. L. May} [Cyclic automorphism groups of compact bordered Klein surfaces, Houston J. Math. 3, 395--405 (1977; Zbl 0379.14012)] and \textit{E. Bujalance} [Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary, Pac. J. Math. 109, 279--289 (1983; Zbl 0545.30033)] that the maximum order of an automorphism of a compact Klein surface of algebraic genus \(g\geq 2\) is \(\mu(g)= 2g+1+(-1)^g\). The authors of the paper under review prove that, unlike the classical case, there exist non-isomorphic Klein surfaces for which this upper bound is attained. More precisely, they describe explicitely the moduli space of compact Riemann surfaces \(X\) having a symmetry \(\tau\) such that the compact Klein surface \(Y= X/\tau\) admits an automorphism of maximum order \(\mu(g)\) as a real one dimensional semianalytic space. Another striking results proves that, with the notations above, the full automorphism group of \(Y\) is not cyclic; even more, with a finite number of exceptions, it is dihedral and coincides with the full automorphism group of its double cover. This work, whose proofs and structure are limpid, provides also algebraic equations of all involved objects, i.e, the genus \(g\) Riemann surface \(X\), the symmetry \(\tau\), the cyclic automorphism of maximum order u\(\mu(g)\) of \(Y= X/\tau\), and a set of generators of the full automorphism group of \(Y\). To finish, let us point out that the article contains also some interesting contributions to the knowledge of the funcion \(v:\{g\in \mathbb{Z},g\geq 2\}\to\mathbb{N}\), where \(v(g)\) is the order of the largest group of automorphisms of a compact Klein surface of algebraic genus \(g\).
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14P25
zbMATH DE Number: 2136591
zbMATH Keywords: Riemann surface
zbMATH Keywords: Klein surface
zbMATH Keywords: Real form
zbMATH Keywords: Automorphism group
reviewed by: Jose Manuel Gamboa
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.09.010
OpenAlex ID: W2052675770
cites work: On the Number of Automorphisms of a Closed Riemann Surface
cites work: Klein surfaces and real algebraic function fields
cites work: Symmetries of Accola-Maclachlan and Kulkarni surfaces
cites work: Normal N.E.C. signatures
cites work: Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary
cites work: Q4340482
cites work: Symmetry types of hyperelliptic Riemann surfaces
cites work: On compact Riemann surfaces with dihedral groups of automorphisms
cites work: On non-orientable Riemann surfaces with \(2p\) or \(2p+2\) automorphisms.
cites work: On Klein surfaces with \(2p\) or \(2p+2\) automorphisms
cites work: Complex automorphism groups of real algebraic curves of genus 2
cites work: Automorphism groups of real algebraic curves which are double covers of the real projective plane
cites work: BOUNDS FOR THE NUMBER OF AUTOMORPHISMS OF A COMPACT NON-ORIENTABLE SURFACE
cites work: CYCLIC GROUPS OF AUTOMORPHISMS OF A COMPACT RIEMANN SURFACE
cites work: The Classification of Non-Euclidean Plane Crystallographic Groups
cites work: A Bound for the Number of Automorphisms of a Compact Riemann Surface
cites work: Q4159474
cites work: A Bound for the Number of Automorphisms of a Compact Klein Surface with Boundary
cites work: Finitely Maximal Fuchsian Groups
cites work: On non-Euclidean crystallographic groups
DOI: 10.1016/J.JALGEBRA.2004.09.010
Recommended article: Q4340482
Recommended article: Minimal genus of Klein surfaces admitting an automorphism of a given order
Recommended article: Complex doubles of bordered Klein surfaces with maximal symmetry
Recommended article: Nilpotent Automorphism Groups of Bordered Klein Surfaces
Recommended article: On soluble groups of automorphisms of nonorientable Klein surfaces
Recommended article: On automorphism groups of Riemann double covers of Klein surfaces
Recommended article: On non-orientable Riemann surfaces with \(2p\) or \(2p+2\) automorphisms.
Recommended article: Automorphisms of Klein surfaces with fixed points
Recommended article: On the problem of finding the full automorphism group of a compact Klein surface
Recommended article: On the order of automorphism group of a compact bordered Riemann surface of genus four
description: scientific article; zbMATH DE number 2136591