# zbMATH — the first resource for mathematics

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.

##### MSC:
 30F60 Teichmüller theory for Riemann surfaces
##### Keywords:
geodesics; Teichmüller space; contraction
Full Text: