Median structures on asymptotic cones and homomorphisms into mapping class groups

Journal article

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).

Authors: Behrstock, J; Drutu, C; Sapir, M

• Publication status: Published
• Journal: PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
• Volume: 102
• Issue: 3
• Pages: 503-554
• Publication date: 2008-10-29
• DOI: 10.1112/plms/pdq025
• EISSN: 1460-244X
• ISSN: 0024-6115
• Language: English
• Keywords: math.GT; 20F65, 20F69, 20F38, 22F50; math.GR