On Polish groups admitting a compatible complete left-invariant metric

Abstract

We prove that the set of all Polish groups admitting a compatible complete left-invariant metric (called CLI) is coanalytic non-Borel as a subset of a standard Borel space of all Polish groups. As an application of this result, we show that there does not exist a weakly universal CLI group. This, in particular, answers in the negative a question of H.Becker.