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A characterization of short curves of a Teichmüller geodesic. (English) Zbl 1082.30037
In this paper, the author obtains a combinatorial condition to characterize short curves along a Teichmüller geodesic. This condition is similar to the one given by Minsky for a hyperbolic 3-manifold. He shows that short curves in a hyperbolic manifold homeomorphic to \(S\times R\) are also short in the corresponding Teichmüller geodesic, and he provides examples demonstrating that its converse is false.

MSC:
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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References:
[1] J Brock, D Canary, Y Minsky, The classification of Kleinian surface groups II: the ending lamination conjecture, in preparation · Zbl 1253.57009 · doi:10.4007/annals.2012.176.1.1
[2] Y Minsky, The classification of Kleinian surface groups I: models and bounds · Zbl 1193.30063 · doi:10.4007/annals.2010.171.1 · annals.princeton.edu
[3] K Rafi, Hyperbolic 3-manifolds and geodesics in Teichmüller space, PhD thesis, SUNY at Stony Brook (2001)
[4] M Rees, The geometric model and large Lipschitz equivalence direct from Teichmüller geodesic, preprint · www.liv.ac.uk
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