A combinatorial higher-rank hyperbolicity condition

TL;DR

This paper introduces a new higher-rank hyperbolicity condition called $(n,)$-hyperbolicity, generalizing Gromov's hyperbolicity, and explores its properties and applications to geometric group theory.

Contribution

It defines the $(n,)$-hyperbolicity condition, analyzes its properties, and connects it to actions of Helly and hierarchically hyperbolic groups.

Findings

$(n,)$-hyperbolic spaces exhibit a slim simplex property.

Product spaces of hyperbolic spaces are $(n,)$-hyperbolic.

Helly and hierarchically hyperbolic groups act on $(n,)$-hyperbolic spaces.

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.