Graphs of curves and arcs quasi-isometric to big mapping class groups
Anschel Schaffer-Cohen
Universidad Nacional Autónoma de México, Morelia, Mexico
Abstract
Following the works of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinite-type surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically non-trivial big mapping class group to admit such a quasi-isometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe represents the first known example of a hyperbolic mapping class group that is not virtually free.
Cite this article
Anschel Schaffer-Cohen, Graphs of curves and arcs quasi-isometric to big mapping class groups. Groups Geom. Dyn. 18 (2024), no. 2, pp. 705–735
DOI 10.4171/GGD/751