On asymptotic dimension of countable abelian groups

TL;DR

This paper calculates the asymptotic dimension of countable abelian groups, specifically the rationals and torsion groups, revealing their large-scale geometric properties.

Contribution

It provides explicit computations of the asymptotic dimension for the rationals and torsion abelian groups with invariant proper metrics, advancing understanding of their geometric structure.

Findings

The asymptotic dimension of the rationals is computed.

Countable torsion abelian groups have asymptotic dimension zero.

Results clarify large-scale geometry of these groups.

Abstract

We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also, we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.