Abstract
In this note we want to relate the Weil-Petersson metric on Teichm¨uller space to the boundary correspondence between the actions on the boundary of Fuchsian groups. Consider the space of Riemann metrics g on a compact surface V with negative Euler characteristic. This can be endowed with a number of natural Riemannian metrics. Of particular interest is the Weil-Petersson metric, whose definition was proposed by Weil in 1958 based on earlier work of Petersson, cf. [18]. There is a particularly intuitive equivalent definition of the Weil-Petersson metric using the second derivative of lengths of typical (closed) geodesics due to Thurston and Wolpert [19].
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