Lacunary sets for actions of tsi groups

Abstract

Under a mild definability assumption, we characterize the family of Borel actions $\Gamma \curvearrowright X$ of tsi Polish groups on Polish spaces that can be decomposed into countably-many actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of $1_{\Gamma }$. In the special case that $\Gamma$ is non-archimedean, it follows that there is such a decomposition if and only if there is no continuous embedding of ${\mathbb{E}}_0^{\mathbb{N}}$ into $E_{\Gamma }^X$.