On Teichmüller contraction

Gardiner, Frederick P.

doi:10.1090/S0002-9939-1993-1152277-1

On Teichmüller contraction
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Proc. Amer. Math. Soc. 118 (1993), 865-875 Request permission

Abstract:

Universal Teichmüller space is the space of quasi-symmetric homeomorphisms $QS$ of a circle factored by those Möbius transformations that preserve the circle. Another Teichmüller space, which also has universal properties, is $QS$ factored by the closed subgroup $S$ of symmetric homeomorphisms. Teichmüller’s metric for $QS \bmod S$ is the boundary dilatation metric. Sullivan’s coiling property for Beltrami lines and the Hamilton-Reich-Strebel necessary and sufficient condition for extremality are proved for $QS\bmod S$. The coiling property implies a contraction principle for certain types of self-mappings of Teichmüller space. It is also shown that the boundary dilatation metric has an infinitesimal form and that this metric is the integral of its infinitesimal form.

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