Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups

TL;DR

This paper investigates the conditions under which certain Lie groups have dense sets of compact elements and explores the properties of dense subgroups, correcting previous misconceptions and providing new examples in the theory of Lie groups.

Contribution

It extends Wu's results to a broader class of disconnected Lie groups and characterizes groups with large generating sets where the derived subgroup is not finitely generated.

Findings

Dense compact elements correspond to fixed-point-free actions.

Provides examples of almost-connected Lie groups lacking dense compact elements.

Characterizes Lie groups with dense generating sets where the derived subgroup is not finitely generated.

Abstract

The semidirect product $\mathbb{G}=\mathbb{L}\rtimes \mathbb{K}$ attached to a compact-group action on a connected, simply-connected solvable Lie group has a dense set of compact elements precisely when the $s\in \mathbb{K}$ operating on $\mathbb{L}$ fixed-point-freely constitute a dense set. This (along with a number of alternative equivalent characterizations) extends the Wu's analogous result for connected Lie $\mathbb{K}$, and also provides ample supplies of examples of almost-connected Lie groups $\mathbb{G}$ which do not have dense sets of compact elements, even though their identity components $\mathbb{G}_0$ do. This corrects prior literature on the subject, claiming the property equivalent for $\mathbb{G}$ and $\mathbb{G}_0$. In a related discussion we characterize those connected Lie groups $\mathbb{G}$ with large sets of $d$-tuples generating dense subgroups $\Gamma\le \mathbb{G}$ for which the derived subgroup $\Gamma^{(1)}$ fails to be finitely-generated: $\mathbb{G}$ must either be non-trivial topologically perfect or have non-nilpotent maximal solvable quotient.