Year: 2022
Type: preprint
Abstract: We investigate a coarse version of a $2(n+1)$-point inequality characterizing metric spaces of combinatorial dimension at most $n$ due to Dress. This condition, experimentally called $(n,δ)$-hyperbolicity, reduces to Gromov's quadruple definition of $δ$-hyperbolicity in case $n = 1$. The $l_\infty$-product of $n$ $δ$-hyperbolic spaces is $(n,δ)$-hyperbolic. Every $(n,δ)$-hyperbolic metric space, without any further assumptions, possesses a slim $(n+1)$-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. In connection with recent work in geometric group theory, we show that every Helly group and every hierarchically hyperbolic group of (asymptotic) rank $n$ acts geometrically on some $(n,δ)$-hyperbolic space.
Source: arXiv (Cornell University)
Authors: Jørgensen, Martina, Lang, Urs
Institution: ETH Zurich
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: Higher rank hyperbolicity
OpenAlex ID: W4283076765