Subgroups of continuous groups acting differentiably on the half-line
We consider groups of diffeomorphisms of the closed half-line which fix only the end point. When the group is a Lie group it is isomorphic to a subgroup of the affine group. On the other hand, when the group is isomorphic to a discrete subgroup of a solvable Lie group it is topologically equivalent to a subgroup of the affine group.