Lipschitz groups and Lipschitz maps

Abstract

‎‎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.

Keywords

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