We study the coarse geometry of the mapping class group of a compact orientable
surface. We show that, apart from a few low-complexity cases, any quasi-isometric
embedding of a mapping class group into itself agrees up to bounded distance with a
left multiplication. In particular, such a map is a quasi-isometry. This is a
strengthening of the result of Hamenstädt and of Behrstock, Kleiner, Minsky and
Mosher that the mapping class groups are quasi-isometrically rigid. In the course of
proving this, we also develop the general theory of coarse median spaces and median
metric spaces with a view to applications to Teichmüller space, and related
spaces.
Keywords
mapping class group, quasi-isometry, rigidity, median
Mathematical Subject Classification 2010
Primary: 20F65
Milestones
Received: 8 April 2016
Revised: 4 January 2017
Accepted: 7 August 2017
Published: 3 November 2017
Authors
Brian H. Bowditch
Mathematics Institute
University of Warwick
Coventry, CV4 7AL
United Kingdom
