We study topological structure of the direct limit $glim G_n$ of an
increasing sequence of Polish ANR-groups $(G_n)_n$ in the category of
topological groups and find conditions under which the group $glim G_n$ is
(locally) homeomorphic to one of the following LF-spaces: $\IR^m$,
$\IR^\infty$, $l_2$ or $l_2\times\IR^\infty$.