Median structures on asymptotic cones and homomorphisms into mapping class groups

Abstract

The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median subspace. One of the applications is that a group with Kazhdan's property (T) can have only finitely many pairwise non-conjugate homomorphisms into a mapping class group. We also give a new proof of the rank conjecture of Brock and Farb (previously proved by Behrstock and Minsky, and independently by Hamenstaedt).

final version, to appear in Proc. LMS

References in corpus (8)

• Poc Sets, Median Algebras and Group Actions
• Injections of mapping class groups
• Mapping Class Groups do not have Kazhdan's Property (T)
• Geometry of the mapping class groups III: Quasi-isometric rigidity
• Actions of finitely generated groups on R-trees
• Limits of (certain) CAT(0) groups, II: The Hopf property and the shortening argument
• Copies of a one-ended group in a Mapping Class Group
• Group actions on median spaces

Cited by in corpus (10)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Constructing group actions on quasi-trees and applications to mapping class groups
• Geometry of the mapping class group II: A biautomatic structure
• Rigidity of high dimensional graph manifolds
• Superrigidity of actions on finite rank median spaces
• Coarse median structures and homomorphisms from Kazhdan groups
• A finiteness property on monodromies of holomorphic families
• The Asymptotic Cone of Teichmüller Space: Thickness and Divergence
• Embedding theorems for actions on generalized trees, I
• Universal tree-graded spaces and asymptotic cones