On the large-scale geometry of mapping class groups

Abstract

We provide two new routes for studying the geometry of mapping class groups, and of colourable hierarchically hyperbolic groups more generally.

Firstly, we show that they are quasimedian quasiisometric to finite-dimensional CAT(0) cube complexes, which are nonpositively-curved spaces with a particularly rich structure. Being quasimedian means that much of this additional structure is coarsely preserved, rather than just the metric.

Secondly, we show that they act geometrically on injective metric spaces. This lets us use facts from the theory of injective spaces to deduce properties of mapping class groups, such as semihyperbolicity.

Date of Award	21 Jun 2022
Original language	English
Awarding Institution	University of Bristol
Supervisor	Mark F Hagen (Supervisor)