An alternative definition of coarse structures

Abstract

John Roe \cite{Roe lectures} introduced coarse structures for arbitrary sets $X$ by considering subsets of $X\times X$. That definition, while natural for analysts, is a bit more difficult to digest for topologists and geometers. In this paper we introduce large scale structures on $X$ via the notion of uniformly bounded families and we show their equivalence to coarse structures on $X$. That way all basic concepts of large scale geometry (asymptotic dimension, slowly oscillating functions, Higson compactification) have natural definitions and basic results from metric geometry carry over to coarse geometry.

13 pages

References in corpus (1)

• A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory