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Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups
Alexandru Chirvasitu
https://arxiv.org/abs/2507.04065

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