Extremal lengths on Denjoy domains

Abstract

We consider the problem of computing the extremal lengths of certain homotopy classes of curves in certain symmetric surfaces. Specifically, we concentrate on plane domains which are conformal to the Riemann sphere with a collection of slits in the real axis removed; such a conformal type is called a Denjoy domain. Using Jenkins-Strebel forms, the extremal length of any sufficiently symmetric homotopy class of curves is computed in terms of the endpoints of the slits. One can then choose a symmetric pants decomposition of the surface and invert the formulas derived, which are a set of coupled quadratic equations. In this way, one obtains a coordinatization of the space of all marked Denjoy domains of a fixed topological type.