  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 $.
