A combinatorial higher-rank hyperbolicity condition

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 $(n,\delta )$-hyperbolicity, reduces to Gromov’s quadruple definition of $\delta$-hyperbolicity in case $n=1$. The $l_{\infty }$-product of $n$ $\delta$-hyperbolic spaces is $(n,\delta )$-hyperbolic. Every $(n,\delta )$-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 $(n,\delta )$-hyperbolic space.