  We introduce the group-compact coarse structure on a Hausdorff topological
group in the context of coarse structures on an abstract group which are
compatible with the group operations. We develop asymptotic dimension theory
for the group-compact coarse structure generalizing several familiar results
for discrete groups. We show that the asymptotic dimension in our sense of the
free topological group on a non-empty topological space that is homeomorphic to
a closed subspace of a Cartesian product of metrizable spaces is 1.
