## Institute for Mathematical Sciences

## Preprint ims94-16

** Y. Minsky**
* Quasi-Projections in Teichmuller Space*

Abstract: We consider a geometric property of the closest-points projection to a geodesic in Teichm\"uller space: the projection
is called contracting if arbitrarily large balls away from the
geodesic project to sets of bounded diameter. (This property
always holds in negatively curved spaces.) It is shown here to
hold if and only if the geodesic is precompact, i.e. its image
in the moduli space is contained in a compact set. Some
applications are given, e.g. to stability properties of certain
quasi-geodesics in Teichm\"uller space, and to estimates of
translation distance for pseudo-Anosov maps.

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