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Quasi-projections in Teichmüller space. (English) Zbl 0848.30031
Let \({\mathcal T}(S)\) be the Teichmüller space of a surface \(S\) of finite type and \(d\) the Teichmüller metric on \({\mathcal T}(S)\). For a closed geodesic \(L\) the closest-point-projection \(\pi_L: {\mathcal T}(S)\to {\mathcal P}(L)\) is defined by \[ \pi_L(\sigma):= \{\alpha \in L;\;d(\sigma, \alpha)= d(L, \alpha)\}. \] It is shown that for all \(\varepsilon> 0\) there is a constant \(b\) such that \[ \text{diam} (\bigcup \{\pi_L(\alpha);\;d(\alpha, \sigma)< d(\sigma, L)\})\leq b \] for all \(\varepsilon\)-precompact geodesics \(L\) and all \(\sigma\in {\mathcal T}(S)\).
Conversely, if \(L\) is a non-precompact geodesic, then this contraction property does not hold for any \(b\).
The author also gives some consequences of the contraction theorem which are directly analogous to well-known properties of hyperbolic space.

30F60 Teichmüller theory for Riemann surfaces
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