Hyperelliptic compact non-orientable Klein surfaces without boundary
type: Item
instance of: scholarly article
title: Hyperelliptic compact non-orientable Klein surfaces without boundary
zbMATH Open document ID: 0688.14024
author: J. A. Bujalance
published in: Kodai Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let X be a compact non-orientable Klein surface (KS). X may be represented as \(D/\Gamma\), D being the hyperbolic plane and \(\Gamma\) a non-Euclidean crystallographic group (NEC group), i.e. a discrete subgroup of the group G of all isometries of the hyperbolic plane, including orientation-reversing ones. X is q-hyperelliptic if there exists an involution \(\Phi\) of X such that \(X/<\Phi >\) has algebraic genus q. In the theorem 2.2 a characterisation of those compact non- orientable KS's without boundary is given, which are q-hyperelliptic. If such a q-hyperelliptic KS has the algebraic genus \(p>4q+1,\) then: (1) the automorphism \(\Phi\) (the automorphism of q-hyperellipticity) is central and unique (see proposition 2.3); (2) if additionally \(X/<\Phi >\) has boundary, then \(|\text{Aut}(X)| \leq 12(p-1)\). The equality is possible only for \(p=2\) and in this case Aut\((X)\) is the dihedron group \(D_ 6\) with 12 elements (s. theorem 3.1).
reviewed by: Andrei Duma
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 4127384
zbMATH Keywords: q-hyperelliptic Klein surface
zbMATH Keywords: compact non-orientable Klein surface
zbMATH Keywords: non- Euclidean crystallographic group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138038984
OpenAlex ID: W2122721200
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DOI: 10.2996/KMJ/1138038984
description: scientific article; zbMATH DE number 4127384