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Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space. (English) Zbl 1154.57016

The simple length spectrum of a Riemann surface of genus \(g\) with \(n\) totally geodesic boundary components is the set of lengths of simple closed geodesics counted with multiplicities. The authors are interested in three questions: Is there a surface for which all the multiplicities are 1? How big is the set of such surfaces? Is it possible to deform a surface such that the multiplicity stays 1 for all simple geodesics?
Theorem 1.1 The set of surfaces with simple length spectrum is dense and its complement is Baire meagre. If \(A\) is a path in Teichmüller space \(T\) then there is a surface on \(A\) which has at least two distinct simple closed geodesics of the same length. Theorem 1.2 The sets \(E(\alpha, \beta)\) (the nonempty subsets of Teichmüller space where a pair of distinct simple closed geodesics \(\alpha, \beta\) have the same length) are connected analytic submanifolds of Teichmüller space. The set \(E\) (the set of all surfaces with at least one pair of simple closed geodesics of equal length) is connected.

MSC:

57M50 General geometric structures on low-dimensional manifolds
58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
30F60 Teichmüller theory for Riemann surfaces
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