Constructing group actions on quasi-trees and applications to mapping class 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 δ-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.

The significant mathematical change is the statement and proof of Proposition 3.23 has been corrected. The introduction has been expanded and there has been a general improvement of the exposition following comments from the referees

References in corpus (4)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• The free splitting complex of a free group I: Hyperbolicity
• The mapping class group from the viewpoint of measure equivalence theory
• Median structures on asymptotic cones and homomorphisms into mapping class groups

Cited by in corpus (14)

• On metric relative hyperbolicity
• Growth tightness for groups with contracting elements
• Large scale rank of Teichmuller space
• Relative free splitting and free factor complexes I: Hyperbolicity
• Algebraic K-theory of Geometric Groups
• On the geometry of a proposed curve complex analogue for $Out(F_n)$
• On constructions preserving the asymptotic topology of metric spaces
• Genericity of contracting elements in groups
• Uniform local finiteness of the curve graph via subsurface projections
• A finiteness property on monodromies of holomorphic families
• Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic
• Combinatorics of tight geodesics and stable lengths
• Decomposition complexity growth of finitely generated groups
• An obstruction to embedding right-angled Artin groups in mapping class groups