Geometric moduli for Klein surfaces
type: Item
instance of: scholarly article
title: Geometric moduli for Klein surfaces
zbMATH Open document ID: 0723.30040
DOI: 10.1216/RMJ-1989-19-3-939
author: Tuomas Sorvali
author: Mika Seppälä
published in: Rocky Mountain Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: This paper is concerned with the parametrisation of Teichmüller spaces for Klein surfaces. More precisely, let \(\Sigma\) be the interior of a compact topological surface \(\Sigma^*\) which is either non-orientable or has non-empty boundary or both; one then obtains an (open) Klein surface by endowing \(\Sigma\) with an analytic structure for which none of the boundary curves (if any) of \(\Sigma^*\) degenerates to a point, or equivalently with a complete hyperbolic metric d for which the boundary curves have infinite length. The problem is to find parameters which determine d up to an isometry homotopic to the identity mapping of \(\Sigma\). The authors show that, if \(\Sigma\) has genus \(g\geq 2\), then there exists 3g-3 closed curves on \(\Sigma\) such that the associated geodesic length functions do indeed determine d up to the required equivalence; in fact, they prove this when \(\Sigma\) is non-orientable with non-empty boundary, the other cases being already known. The result is interesting because (i) the Teichmüller space T(\(\Sigma\)) (which can be identified with a component of the Teichmüller space of real algebraic curves of genus g) has dimension exactly 3g-3, (ii) in the compact oriented case, more than 3g-3 curves are required to determine d.
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 4192100
zbMATH Keywords: Klein surfaces
zbMATH Keywords: geodesic length
zbMATH Keywords: Teichmüller space
reviewed by: P. E. Newstead
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4192100