Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups

TL;DR

This paper characterizes when automorphisms of connected Lie groups act distally on the space of one-parameter subgroups, linking this property to the structure of the group and the automorphism's behavior.

Contribution

It provides new characterizations of distal actions of automorphisms on the space of one-parameter subgroups, extending previous results and relating distality to subgroup compactness.

Findings

Distal action on Sub^p_G relates to the automorphism's behavior on the maximal central torus.

A connected Lie group acts distally on Sub^p_G iff it is compact or a product of a compact group and a vector group.

Extension of results to subgroup actions of Aut(G) and their distality properties.

Abstract

For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We relate the distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$, and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Some of our results extend those of Shah and Yadav.