Geodesic
A combinatorial model of the Lipschitz metric for surfaces with punctures
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 910-929.

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The zipped word length function introduced by Ivan Dynnikov in connection with the word problem in the mapping class groups of punctured surfaces is considered. We prove that the mapping class group with the metric determined by this function is quasi-isometric to the thick part of the Teichmüller space equipped with the Lipschitz metric.
Keywords: Mapping class group, Teichmüller space, Teichmüller metric, Thurston's asymmetric metric. 
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V. A. Shastin. A combinatorial model of the Lipschitz metric for surfaces with punctures. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 910-929. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a46/

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