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Following the work of Rosendal and Mann and Rafi, we try to answer thefollowing question: when is the mapping class group of an infinite-type surfacequasi-isometric to a graph whose vertices are curves on that surface? With theassumption of tameness as defined by Mann and Rafi, we describe a necessary andsufficient condition, called translatability, for a geometrically nontrivialbig mapping class group to admit such a quasi-isometry. In addition, we showthat the mapping class group of the plane minus a Cantor set is quasi-isometricto the loop graph defined by Bavard, which we believe represents the firstexample of a mapping class group known to be non-elementary hyperbolic.

Graphs of curves and arcs quasi-isometric to big mapping class groups