Pervasive ellipticity in locally compact groups

TL;DR

This paper classifies connected locally compact groups that are almost-elliptic, showing they have compact semisimple quotients and specific actions on Euclidean quotients, revealing structural conditions for ellipticity.

Contribution

It provides a precise classification of almost-elliptic locally compact groups based on their quotient structures and actions, extending understanding of group ellipticity properties.

Findings

Classification of almost-elliptic connected locally compact groups.

Characterization of extensions with non-trivial weights.

Conditions for ellipticity in terms of group actions.

Abstract

A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those with compact maximal semisimple quotient and whose maximal compact subgroups act trivial-weight-freely on the Euclidean quotients of the closed derived series' successive layers. In particular, an extension of a compact connected group $\mathbb{K}$ by a connected, simply-connected solvable Lie group $\mathbb{L}$ is (openly) almost-elliptic precisely when the weights of the $\mathbb{K}$-action on $Lie(\mathbb{L})$ afforded by the extension are all non-trivial.