Graphs of curves and arcs quasi-isometric to big mapping class groups (Q6594734)

Let \(\Sigma\) be an orientable surface (\(2\)-manifold) without boundary. The surface \(\Sigma\) is said to have finite type if its fundamental group is finitely generated. Otherwise, it is said to have infinite type. The mapping class group of \(\Sigma\) is the group of isotopy classes of orientation-preserving self homeomorphisms of \(\Sigma\). If \(\Sigma\) has infinite type, then its mapping class group is called big.\N\NThis paper deals with the following question: when is the mapping class group of an infinite-type surface \(\Sigma\) quasi-isometric to a graph whose vertices are curves on \(\Sigma\)? A necessary and sufficient condition, called translatability, is given for a geometrically non-trivial big mapping class group to admit such a quasi-isometry. Using this, it is shown that for many graphs whose vertices are curves, the orbit map from the big mapping class group to the graph is not a quasi-isometry. It is also shown that the mapping class group of the plane minus a Cantor set is quasi-isometric to, so called, the loop graph.