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

TL;DR

This paper characterizes when a connected locally compact group contains a dense pointwise-elliptic subgroup, linking it to a specific semidirect product structure and the action of the compact subgroup on the Lie algebra.

Contribution

It extends and generalizes Kabenyuk's classification by providing a detailed structure theorem for groups with dense pointwise-elliptic subgroups, involving Lie group decompositions and weight conditions.

Findings

Characterization of dense pointwise-elliptic subgroups in connected locally compact groups.

Identification of the semidirect product structure involving Lie groups.

Extension and recovery of Kabenyuk's classification under new conditions.

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 $\mathbb{L}\rtimes \mathbb{K}$ with connected, simply-connected Lie $\mathbb{L}$, compact Lie $\mathbb{K}$, with the commutator subgroup $\mathbb{K}'$ acting on the Lie algebra $Lie(\mathbb{L})$ with no trivial weights. This extends and recovers a result of Kabenyuk's, providing the analogous classification with $\mathbb{G}$ assumed Lie connected, topologically perfect, with no non-trivial central elliptic elements.