On Large Scale Inductive Dimension of Asymptotic Resemblance Spaces

TL;DR

This paper introduces a new large scale inductive dimension for asymptotic resemblance spaces, establishing its equivalence with the asymptotic dimensiongrad in r-convex metric spaces, including geodesic spaces and finitely generated groups.

Contribution

It defines the large scale inductive dimension for asymptotic resemblance spaces and proves its equality with the asymptotic dimensiongrad in r-convex metric spaces, answering a previously open question.

Findings

Large scale inductive dimension is introduced for asymptotic resemblance spaces.

Proves equivalence with asymptotic dimensiongrad in r-convex metric spaces.

Answers a question by E. Shchepin regarding these dimensions.

Abstract

We introduce the notion of large scale inductive dimension for asymptotic resemblance spaces. We prove that the large scale inductive dimension and the asymptotic dimensiongrad are equal in the class of r-convex metric spaces. This class contains the class of all geodesic metric spaces and all finitely generated groups. This leads to an answer for a question asked by E. Shchepin concerning the relation between the asymptotic inductive dimension and the asymptotic dimensiongrad, for r-convex metric spaces.