On the Properties of Plesio-Uniform Subgroups in Lie Groups

The paper is devoted to the study of properties of a class of subgroups $H$ in Lie groups $G$ that was recently introduced by the author. A closed subgroup $H$ in a Lie group $G$ is said to be plesio-uniform if there is a closed subgroup $P$ of $G$ that contains $H$ and for which $P$ is uniform in $G$ and $H$ is quasi-uniform in $P$. In the paper we give answers to several natural questions concerning plesio-uniform subgroups. It is proved that one obtains the same notion of plesio-uniformity when transposing the conditions of uniformity and quasi-uniformity in the definition of plesio-uniformity of a subgroup. If a closed subgroup $H$ of $G$ contains a plesio-uniform subgroup, then $H$ is also plesio-uniform. Other properties of plesio-uniform subgroups are also considered.