Classification of Weil-Petersson isometries

Abstract

This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmüller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the Thurston classification. The second result concerns the asymptotic behavior of these geodesics. It is shown that geodesics that are equivariant with respect to independent pseudo-Anosov's diverge. It follows that subgroups of the mapping class group which contain independent pseudo-Anosov's act in a reductive manner with respect to the Weil-Petersson geometry. This implies an existence theorem for equivariant harmonic maps to the metric completion.