The authors define the group-compact coarse structure on a topological Hausdorff group \(G\) as the coarse structure on \(G\) generated by the family \(C(G)\) of all compact subsets of \(G\), i.e., as the family \(\mathcal{E}_{C(G)}=\{E\subseteq G\times G: \exists K\in C(G)\) \(E\subset G(K\times K)\}\) where \(GA=\{(ga,gb):(a,b)\in A\) and \(g\in G\}\) for \(A\subseteq G\times G\). They develop the asymptotic dimension theory for these structures and obtain generalizations of some results familiar for discrete groups. A free topological group on a topological space \(X\) is a pair \((F,i)\) where \(F\) is a Hausdorff topological group and \(i:X\to F\) a continuous mapping satisfying the following universal property: For every Hausdorff topological group \(H\) and every continuous mapping \(f:X\to H\) there exists a unique continuous homomorphism \(h:F\to H\) such that \(f=h\circ i\). The asymptotic dimension of a free topological group on a non-empty topological space that is homeomorphic to a closed subspace of a product of metrizable spaces is proved to be \(1\).