Median structures on asymptotic cones and homomorphisms into mapping class groups

TL;DR

This paper investigates the asymptotic cones of mapping class groups, showing they embed into products of real trees with median structures, and applies these results to homomorphisms and the rank conjecture.

Contribution

It provides a detailed analysis of asymptotic cones of mapping class groups, revealing median structures and embedding properties, and applies these to homomorphism finiteness and the rank conjecture.

Findings

Asymptotic cones embed into products of real trees with median structures.

Groups with Kazhdan's property (T) have finitely many non-conjugate homomorphisms into mapping class groups.

A new proof of the rank conjecture of Brock and Farb.

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).