Length spectra as moduli for hyperbolic surfaces
type: Item
instance of: scholarly article
title: Length spectra as moduli for hyperbolic surfaces
zbMATH Open document ID: 0595.30052
DOI: 10.1215/S0012-7094-85-05249-4
author: Andrew Haas
published in: Duke Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let S be a hyperbolic Riemann surface with a metric of constant negative curvature -1. It is known that in general even for compact S the geodesic length spectrum does not determine S up to isometry. By contrast, the main result of this paper is that if S is a torus with one puncture or one infinite area end (an h-torus) then the length spectrum does completely determine S. When S has genus zero and three punctures or holes, a similar result follows easily from the facts that boundary geodesics have minimal length and that the lengths of the boundary geodesics determine the surface. The key result in the case that S is an h-torus, of interest in its own right, is that a simple smooth closed geodesic on S minimises length in its homology class. Two geodesics on S are a generating pair for the fundamental group if and only if they are simple and have a single intersection. From results of Fricke and Keen, the lengths of a generating pair and of the boundary geodesic determine S. It is shown that the geodesic \(\alpha\) of minimal length on S is simple, and that together with the next shortest simple geodesic \(\beta\) it forms a generating pair. Using the result above about minimising length in the homology class, geodesics shorter than \(\beta\) are characterized as belonging to Alt(\(\alpha\),\(\beta)\), the subgroup of \(\pi_ 1(S)\) generated by \(\alpha,\beta^{-1}\alpha \beta\). (By abuse of notation \(\alpha\) denotes both a geodesic and its representative in \(\pi_ 1.)\) Finally, lengths of geodesics corresponding to elements of Alt(\(\alpha\),\(\beta)\) are shown to be given by a two variable polynomial in the lengths of \(\alpha\) and \(\alpha \beta \alpha^{-1}\beta^{-1}\).
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 3957573
zbMATH Keywords: hyperbolic Riemann surface
zbMATH Keywords: length spectrum
zbMATH Keywords: boundary geodesics
MaRDI profile type: Publication
cites work: The real analytic theory of Teichmüller space
cites work: Q3958531
cites work: Q4061209
cites work: Q4135772
cites work: Q4102056
cites work: Correction to: Silberg' trace formua as applied to a compact riemann surface by H. P. Mckean, comm. pure appl. math. 25, pp. 225‐246,1972
cites work: Q4168970
cites work: The length spectra as moduli for compact Riemann surfaces
description: scientific article; zbMATH DE number 3957573