Logarithmically Slim Polygons

Abstract

We study the relationship between the large-scale geometry of a geodesic metric space and the slimness of its geodesic polygons. We show that if there is a uniform constant $δ$ such that every geodesic polygon in an unbounded geodesic metric space is $δ$-slim, then the space has asymptotic Assouad--Nagata dimension one. We further consider graphs of bounded valency and allow the slimness constant of an $n$-gon to grow logarithmically with $n$. More precisely, under an explicit logarithmic bound on the slimness of geodesic $n$-gons, we prove that the graph has asymptotic Assouad--Nagata dimension one. Thus, sufficiently strong logarithmic control on the slimness of geodesic polygons forces one-dimensional large-scale geometry. We also obtain two criteria for graphs, outside the hyperbolic setting, to have asymptotic Assouad--Nagata dimension one. In particular, these criteria show that one-dimensional large-scale geometry can be detected from quantitative restrictions on the slimness of geodesic polygons, without requiring the space to be hyperbolic.