Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture (Matthew Durham)

I will discuss recent work with Minsky and Sisto, in which we prove that mapping class groups of finite-type surfaces---and more generally, colorable hierarchically hyperbolic groups (HHGs)---are asymptotically CAT(0).  This is a simple but powerful non-positive curvature property introduced by Kar, roughly requiring that the CAT(0) inequality holds up to sublinear error in the size of the triangle.

We use the asymptotically CAT(0) property to construct visual compactifications for colorable HHGs that provide Z-structures in the sense of Bestvina and Dranishnikov.  It was previously unknown that mapping class groups are asymptotically CAT(0) and admit Z-structures.  As an application, we prove that many HHGs satisfy the Farrell-Jones Conjecture, providing a new proof for mapping class groups (Bartels-Bestvina) and establishing the conjecture for extra-large type Artin groups.

To construct asymptotically CAT(0) metrics, we show that every colorable HHG admits a manifold-like family of local approximations by CAT(0) cube complexes, where transition maps are cubical almost-isomorphisms.