Pointwise-relatively-compact subgroups and trivial-weight-free representations

2025/06/23 by Chirvasitu, Alexandru
#17B05 #22D05 #22D12 #22E15 #22E25 #22E60 #54D30 #54E52 #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2506.18861

Abstract

A pointwise-elliptic subset of a topological group is one whose elements all generate relatively-compact subgroups. A connected locally compact group has a dense pointwise-elliptic subgroup if and only if it is an extension by a compact normal subgroup of a semidirect product \mathbbL\rtimes \mathbbK with connected, simply-connected Lie \mathbbL, compact Lie \mathbbK, with the commutator subgroup \mathbbK' acting on the Lie algebra Lie(\mathbbL) with no trivial weights. This extends and recovers a result of Kabenyuk's, providing the analogous classification with \mathbbG assumed Lie connected, topologically perfect, with no non-trivial central elliptic elements.

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