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

2025/07/05 by Chirvasitu, Alexandru
#11R04 #17B30 #20F16 #20F18 #22C05 #22D05 #22D12 #22E25 #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.04065

Abstract

The semidirect product \mathbbG=\mathbbL\rtimes \mathbbK 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∈ \mathbbK operating on \mathbbL 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 \mathbbK, and also provides ample supplies of examples of almost-connected Lie groups \mathbbG which do not have dense sets of compact elements, even though their identity components \mathbbG₀ do. This corrects prior literature on the subject, claiming the property equivalent for \mathbbG and \mathbbG₀. In a related discussion we characterize those connected Lie groups \mathbbG with large sets of d-tuples generating dense subgroups Γ≤ \mathbbG for which the derived subgroup Γ⁽¹⁾ fails to be finitely-generated: \mathbbG must either be non-trivial topologically perfect or have non-nilpotent maximal solvable quotient.

Citations

• Pervasive ellipticity in locally compact groups
• On dense free subgroups of Lie groups

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• Pointwise-relatively-compact subgroups and trivial-weight-free representations
• Connected components of definable groups, and o-minimality II
• Solvable Subgroups of Locally Compact Groups
• On density of infinite subsets II: dynamics on homogeneous spaces