Pro-Lie Groups

Abstract

A topological group $G$ is pro-Lie if $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group. A locally compact group $G$ is an $L$-group if, for every neighborhood $U$ of the identity and compact set $C$, there is a neighborhood $V$ of the identity such that $gH{g^{ - 1}} \cap C \subset U$ for every $g \in G$ and every subgroup $H \subset V$. We obtain characterizations of pro-Lie groups and make several applications. For example, every compactly generated $L$-group is pro-Lie and a compactly generated group which can be embedded (by a continuous isomorphism) in a pro-Lie group is pro-Lie. We obtain related results for factor groups, nilpotent groups, maximal compact normal subgroups, and generalize a theorem of Hofmann, Liukkonen, and Mislove [4].