## 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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