CONSTRUCTING GROUP ACTIONS ON QUASI-TREES

Abstract

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary hyperbolic groups, CAT(0) groups with rank 1 elements, mapping class groups and the outer automorphism groups of free groups. As an application, we show that mapping class groups act on finite products of Gromov-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. It implies that mapping class groups have finite asymptotic dimension.