# Sequential coarse structures of topological groups

## TL;DR

This paper introduces a new coarse structure on topological groups based on converging sequences and explores its properties, showing that in abelian groups with non-trivial converging sequences, the asymptotic dimension is infinite.

## Contribution

It defines a novel coarse structure on topological groups using converging sequences and analyzes its asymptotic dimension in abelian groups.

## Key findings

- If G is discrete, the coarse structure is finitary.
- In abelian groups with non-trivial converging sequences, the asymptotic dimension is infinite.

## Abstract

We endow a topological group $(G, \tau)$ with a coarse structure defined by the smallest group ideal $S_{\tau} $ on $G$ containing all converging sequences with their limits and denote the obtained coarse group by $(G, S_{\tau})$. If $G$ is discrete then $(G, S_{\tau})$ is a finitary coarse group studding in Geometric Group Theory. The main result: if a topological abelian group $(G, \tau)$ contains a non-trivial converging sequence then $asdim \ (G, S_{\tau})= \infty $.