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

TL;DR

This paper characterizes when big mapping class groups are quasi-isometric to curve graphs, introducing the translatability condition, and provides the first example of a non-elementary hyperbolic mapping class group.

Contribution

It establishes a necessary and sufficient translatability condition for big mapping class groups to be quasi-isometric to curve graphs under tameness assumptions.

Findings

Introduces the translatability condition for quasi-isometry.

Shows the mapping class group of the plane minus a Cantor set is hyperbolic.

First example of a non-elementary hyperbolic mapping class group.

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.