Geometrical characterization of \(p\)-hyperelliptic planar Klein surfaces.
type: Item
instance of: scholarly article
title: Geometrical characterization of \(p\)-hyperelliptic planar Klein surfaces.
zbMATH Open document ID: 1043.20022
DOI: 10.1007/BF03321021
author: B. Estrada
published in: Computational Methods and Function Theory
publication date: 2003-10-22T00:00:00Z
review text: Let us consider \(q\)-hyperelliptic Klein surfaces \(X=H^2/\Gamma\), where \(\Gamma\) is a surface NEC-group with signature \(\sigma(\Gamma)=(0,+,[-1],\{(-1)^k\})\), called planar surfaces. These surfaces are characterized by the existence of an order two automorphism \(\phi\), called \(q\)-hyperelliptic involution, such that the quotient \(X/\langle\phi\rangle\) has algebraic genus \(q\). The main purpose of this work is the geometrical study of \(q\)-hyperelliptic involutions. For this aim, first the author constructs fundamental regions of the NEC-group \(\Gamma\) that are right-angled hyperbolic polygons. After that the author obtains canonical regions from Wilkie regions. By considering the existence of a symmetric canonical right-angled region so the \(q\)-hyperelliptic character of planar surfaces is characterized. Besides these the author considers the Teichmüller spaces of \(q\)-hyperelliptic planar Klein surfaces. Then the dimension of these spaces is calculated and it is described geometrically by means of congruence classes of polygons.
reviewed by: Turgut Başkan
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 1995506
zbMATH Keywords: non-Euclidean crystallographic groups
zbMATH Keywords: Klein surfaces
zbMATH Keywords: fundamental regions
MaRDI profile type: Publication
cites work: Foundations of the theory of Klein surfaces
cites work: A characterization of q-hyperelliptic compact planar Klein surfaces
cites work: Geometry and spectra of compact Riemann surfaces
cites work: Q4027274
cites work: On \(q\)-hyperelliptic \(k\)-bordered tori
cites work: The Classification of Non-Euclidean Plane Crystallographic Groups
cites work: SMOOTH COVERINGS OF HYPERELLIPTIC SURFACES
cites work: Convex fundamental regions for N.E.C. groups
cites work: Large automorphism groups of compact Klein surfaces with boundary, I
cites work: Q4771443
cites work: On non-Euclidean crystallographic groups
cites work: Surfaces and planar discontinuous groups. Revised and expanded transl. from the German by J. Stillwell
full work available at URL: https://doi.org/10.1007/bf03321021
OpenAlex ID: W2045330218
description: scientific article; zbMATH DE number 1995506