Geometric Group Therapy Seminar

Speaker: Martina Jørgensen

Title: Injective hulls and Helly complexes of dimension two

Abstract: We introduce the notions of asymptotic rank and injective hulls before investigating a coarse version of Dress’ 2(n+1)-inequality characterising metric spaces of combinatorial dimension at most n. This condition, referred to as (n,δ)-hyperbolicity, reduces to Gromov's quadruple definition of δ-hyperbolicity for n=1. The ℓ∞ product of n δ-hyperbolic spaces is (n,δ)-hyperbolic and, without further assumptions, any (n,δ)-hyperbolic space admits a slim (n+1)-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. Using tools from recent developments in geometric group theory, we look at some examples and show that every Helly group and every hierarchically hyperbolic group of asymptotic rank n acts geometrically on some (n,δ)-hyperbolic space. Finally, we relate (2,0)-hyperbolicity to the Nagata dimension, a variation of Gromov's asymptotic dimension with important applications related to extendibility of Lipschitz maps, via locally finite Helly graphs of combinatorial dimension at most two. Joint work with Urs Lang.

Seminar website: https://sites.google.com/view/ggtcopenhagen/home/seminar