Constructing group actions on quasi-trees and applications to mapping class groups

TL;DR

This paper develops a general method for constructing group actions on quasi-trees, including key groups like mapping class groups, and demonstrates their properties such as finite asymptotic dimension and embeddings into hyperbolic spaces.

Contribution

It introduces a new construction for group actions on quasi-trees applicable to various important groups, expanding understanding of their geometric properties.

Findings

Mapping class groups act on finite products of hyperbolic spaces with quasi-isometric orbit maps.

Mapping class groups have finite asymptotic dimension.

The construction applies to hyperbolic, CAT(0), and Out(Fn) groups.

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 (relatively) hyperbolic groups, rank 1 CAT(0) groups, mapping class groups and Out(Fn). As an application, we show that mapping class groups act on finite products of {\delta}-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.