lipschitz groups and lipschitz maps

This contribution mainly focuses on some aspects of lipschitz groups, i.e., metrizable groups with lipschitz multiplication and inversion map. in the main result it is proved that metric groups, with a translation-invariant metric, may be characterized as particular group objects in the category of metric spaces and lipschitz maps. moreover, up to an adjustment of the metric, any metrizable abelian group also is shown to be a lipschitz group. finally we present a result similar to the fact that any topological nilpotent element x in a banach algebra gives rise to an invertible element 1-x, in the setting of complete lipschitz groups.