Classification of Weil-Petersson Isometries

G. Daskalopoulos and R. Wentworth

This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in
Teichm\"uller 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.