Classification of Weil-Petersson isometries

By Georgios Daskalopoulos, Richard A. Wentworth. 2002

77 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/76e11082ff39933109a946e492b1608827335172

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In Applied Mathematics

In Geometry and Topology

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Citations in the corpus

Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows (2008, background): “In particular, a mapping class y is pseudo-Anosov if no non-zero power of y preserves any isotopy class of simple closed curves on S. As in the setting of the Teichm¨ uller metric, y preserves an invariant Weil-Petersson geodesic axis Ay Teich(S) (see [ DW , Wol5]) on which it acts by translation.”
Behavior of geodesic-length functions on Teichmüller space (2007, methodology): “Earlier expansions indicate that the metric tangent cone at the augmentation is singular, [ DW03 , Wol03, Yam04].”
Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity (2007, background): “Œ0;T / have endpoints converging in the completion and thus these limiting endpoints are the endpoints of a geodesic segment whose interior lies in the maximal stratum containing its endpoints (see Daskalopoulos and Wentworth [9] and Wolpert [25]).”
An analytic construction of the Deligne-Mumford compactification of the moduli space of curves (2013, background): “(This formula is also established in [11], [48] and [50]).”
Geometry of the Weil-Petersson completion of Teichm\ (2005, methodology): “For irreducible mapping classes, the positive translation length is realized on a unique geodesic within T , [13, 47].”
On the Geometry of Weil-Petersson Completion of Teichmüller Spaces (2004, unknown): “It should be also remarked that there appeared since two preprints [6] and [22] which contain results partly motivated on the material presented in the previous versions of this paper.”