There is a well-known correspondence between Weil-Petersson geodesic loops in the moduli space of a surface S and hyperbolic 3-manifolds fibering over the circle with fibre S. Much is unknown, however, about the detailed relationship between geometric features of the loops and those of the 3-manifolds.
In work with Leininger-Souto-Taylor we study the relation between WP length and 3-manifold volume, when the length (suitably normalized) is bounded and the fiber topology is unbounded. We obtain a WP analogue of a theorem proved by Farb-Leininger-Margalit for the Teichmuller metric. In work with Modami, we fix the fiber topology and study connections between the thick-thin decomposition of a geodesic loop and that of the corresponding 3-manifold. While these decompositions are often in direct correspondence, we exhibit examples where the correspondence breaks down. This leaves the full conjectural picture somewhat mysterious, and raises many questions.
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- Topology Seminar