Graphs of curves and arcs quasi-isometric to big mapping class groups
Abstract
Following the work 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 nontrivial 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 example of a mapping class group known to be non-elementary hyperbolic.
Keywords
Cite
@article{arxiv.2006.14760,
title = {Graphs of curves and arcs quasi-isometric to big mapping class groups},
author = {Anschel Schaffer-Cohen},
journal= {arXiv preprint arXiv:2006.14760},
year = {2022}
}
Comments
33 pages, 8 figures; version accepted for publication in Groups, Geometry, and Dynamics. Added a corollary about coarsely bounded presentations, a consequence of hyperbolicity