Asymptotic dimension and small subsets in locally compact topological groups

TL;DR

This paper investigates the relationship between small subsets and asymptotic dimension in locally compact topological groups, establishing conditions under which these concepts coincide, especially in Euclidean and Abelian groups.

Contribution

It proves that for spaces coarsely equivalent to Euclidean space, the ideal of small subsets matches the ideal of subsets with lower asymptotic dimension, and characterizes when this holds in locally compact Abelian groups.

Findings

Small subsets coincide with lower asymptotic dimension subsets in Euclidean-like spaces.

In locally compact Abelian groups, equality of ideals holds if and only if the group is compactly generated.

Provides a characterization of small subsets in the context of asymptotic dimension.

Abstract

We prove that for a coarse space $X$ the ideal $S(X)$ of small subsets of $X$ coincides with the ideal $D_<(X)$ of subsets $A\subset X$ of asymptotic dimension $asdim(A)<asdim(X)$ provided that $X$ is coarsely equivalent to an Euclidean space $R^n$. Also we prove that for a locally compact Abelian group $X$, the equality $S(X)=D_<(X)$ holds if and only if the group $X$ is compactly generated.