On the automorphism group of the canonical double covering of bordered Klein surfaces with large automorphism group
type: Item
instance of: scholarly article
title: On the automorphism group of the canonical double covering of bordered Klein surfaces with large automorphism group
zbMATH Open document ID: 0889.30032
DOI: 10.1007/BF02362635
author: A. M. Porto Fereira da Silva
author: Antonio F. Costas
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-06-08T00:00:00Z
review text: Let \(X\) be a Klein surface with nonempty boundary and double covering \(X^+\). The aim of this article is to get information about the group \(\Aut(X^+)\) of the conformal and anticonformal automorphisms of \(X^+\) from the group \(\Aut(X)\) of automorphisms of \(X\). Consequently, it is proved here that if \(\Aut X>6(g- 1)\), where \(g\) is the genus of \(X^+\), then \(\Aut(X^+)= \Aut(X)\times C_2\) except for a finite number of cases. This result improves a previous one by \textit{C. L. May} [Glasg. Math. J. 33, No. 1, 61-71 (1991; Zbl 0718.30032)] and \textit{E. Bujalance}, \textit{A. F. Costa}, \textit{G. Gromadzki} and \textit{D. Singerman} [Glasg. Math. J. 36, No. 3, 313-330 (1994; Zbl 0810.30035)]. The Porto-Costa new result is the best possible in this line because, in the paper under review, the existence of infinitely many Klein surfaces \(X\) with different topological type satisfying \(\#\Aut(X)= 6(g- 1)\) and \(\Aut(X^+)\neq \Aut(X)\times C_2\), is established. This fact is a nice consequence of the theory of irreflexible maps on surfaces. In particular, for \(g=2\), families of surfaces with \(\#\Aut(X)= 6\) and \(\Aut(X^+)\neq \Aut(X)\times C_2\) were constructed recently by \textit{J. Cirre} in his Ph.D. thesis in 1997. Finally, the authors conclude that the existence of an automorphism of \(X\) with fixed points and prime order \(p\) implies, provided the order \(p\) is large enough, that \(\Aut(X^+)\neq \Aut(X)\times C_2\) with a unique exception for each value of \(g\).
Mathematics Subject Classification ID: 30F50
zbMATH DE Number: 1017601
zbMATH Keywords: conformal automorphisms
zbMATH Keywords: Klein surface
zbMATH Keywords: anticonformal automorphisms
reviewed by: Jose Manuel Gamboa
MaRDI profile type: Publication
cites work: Q3310101
cites work: Automorphism groups of complex doubles of Klein surfaces
cites work: Automorphism groups of compact bordered Klein surfaces. A combinatorial approach
cites work: Complex doubles of bordered Klein surfaces with maximal symmetry
cites work: Q3912902
cites work: Automorphisms of compact non-orientable Riemann surfaces
Wikidata QID: Q125849870
full work available at URL: https://doi.org/10.1007/bf02362635
OpenAlex ID: W2070013610
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description: scientific article; zbMATH DE number 1017601