On \(pq\)-hyperelliptic Klein surfaces
type: Item
instance of: scholarly article
title: On \(pq\)-hyperelliptic Klein surfaces
zbMATH Open document ID: 1155.30021
author: G. Gromadzki
author: Ewa Tyszkowska
published in: Manuscripta Mathematica
publication date: 2008-09-25T00:00:00Z
review text: A compact Klein surface \(X\) is a compact surface, orientable or non-orientable, with \( k\geq 0\) boundary components. If \(X\) is orientable and \(k=0\), then \(X\) is a Riemann surface. The algebraic genus \(d\geq2\) of \(X\) is the genus of the Riemann surface \(X^+\) that is its complex double. The algebraic genus \( d=\eta g+k-1\), where \(g\) is the topological genus of \(X\) and \(\eta=2\) or \( \eta=1\) according to \(X\) be orientable or not. Klein surfaces are uniformized by means of non-Euclidean crystallographic groups (NEC groups, in short). Given a Klein surface \(X\) of algebraic genus \(d\geq2\), \(X\) can be represented as \(\mathcal{H}/\Gamma\) for some NEC group \(\Gamma\), where \(\mathcal{ H}\) denotes the hyperbolic plane. A Klein surface is said to be \(p\)-hyperelliptic if there exists an involution \(\rho\) such that the quotient \(X/\rho\) has algebraic genus \(p\). In the paper under review \(pq\)-hyperelliptic Klein surfaces, surfaces which are simultaneously \(p\)- and \(q \)-hyperelliptic, are studied. It is proved that given integers \(0\leq p\leq q\), \(q\neq 0\) then there exists a \(pq\)-hyperelliptic Klein surface of algebraic genus \(d\) if and only if \(2q-1\leq d\leq 2p+2q+1\). This result generalizes for Klein surfaces a similar result for Riemann surfaces from the second author in [\textit{E. Tyszkowska}, Colloq. Math. 103, No. 1, 115--120 (2005; Zbl 1080.30037)]. The authors study the product of \(p\)- and \(q\)- involutions obtaining some results about conditions such that these involutions commute. The last section of the paper is devoted to the study of the topological types of Klein surfaces with commuting \(p\)- and \(q\)- involutions. It is proved that with few exceptions a \(pq\)-hyperelliptic Klein surface can have arbitrary number \(k\) of boundary components \(0\leq k\leq d+1\). The authors start from a Klein surface \(X\) of genus \(d\) with maximal number of components. Let \(\Lambda \) be an NEC group of signat…
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 5347174
zbMATH Keywords: NEC groups
zbMATH Keywords: p-hyperellipticity
zbMATH Keywords: involutions
reviewed by: Ernesto Martínez
MaRDI profile type: Publication
full work available at URL: s00229-006-0072-0
OpenAlex ID: W2005963423
cites work: Topics in the theory of Riemann surfaces
cites work: Foundations of the theory of Klein surfaces
cites work: Topological types of p-hyperelliptic real algebraic curves
cites work: Automorphism groups of compact bordered Klein surfaces. A combinatorial approach
cites work: Q3931656
cites work: The Classification of Non-Euclidean Plane Crystallographic Groups
cites work: Q4771443
cites work: On pq-hyperelliptic Riemann surfaces
cites work: On non-Euclidean crystallographic groups
DOI: 10.1007/S00229-006-0072-0
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description: scientific article; zbMATH DE number 5347174