Quasi-projections in Teichmuller space


We consider a geometric property of the closest-points projection to a geodesic in Teichmuller space: the projection is called contracting if arbitrarily large balls disjoint from the geodesic project to sets of bounded diameter. (This property always holds in sufficiently 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 Teichmuller space, and to estimates of translation distance for pseudo-Anosov maps.

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