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.