SEQUENTIAL COARSE STRUCTURES OF TOPOLOGICAL GROUPS.

We endow a topological group (G,τ) with a coarse structure defined by the smallest group ideal Sτ on G containing all converging sequences with their limits and denote the obtained coarse group by (G,Sτ). If G is discrete then (G,Sτ) is a finitary coarse group studding in Geometric Group Theory. The main result: if a topological abelian group (G,τ) contains a non-trivial converging sequence then asdim (G,Sτ)=∞. We study metrizability, normality and functional boundedness of sequential coarse groups and put some open questions. [ABSTRACT FROM AUTHOR]