On a question of Dikranjan and Zava

TL;DR

This paper proves that for certain countable metrizable abelian groups with a coarse structure generated by compact sets, the asymptotic dimension is infinite, answering a specific question posed by Dikranjan and Zava.

Contribution

It establishes that the asymptotic dimension of these groups is infinite, providing a significant answer to an open question in the field.

Findings

asdim (G, C) = ∞ for the specified groups

Answers Dikranjan and Zava's question about the circle subgroup

Highlights properties of coarse structures on topological groups

Abstract

Let $G$ be a non-discrete countable metrizable abelian topological group endowed with the coarse structure $ \mathcal{C} $ generated by compact subsets of $G$. We prove that $asdim (G, \mathcal{C} ) = \infty$. For an infinite cyclic subgroup $G$ of the circle, this answers a question of Dikranjan and Zava [3].