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 ۱-x‎, ‎in the setting of complete Lipschitz groups‎.

Keywords:

‎Lipschitz maps‎ , ‎group object in a category‎ , ‎translation-invariant metric