Distal Actions of Automorphisms of Lie Groups $G$ on $\rm Sub_{G}$

TL;DR

This paper investigates how automorphisms of Lie groups act distally on the space of their closed subgroups, providing characterizations based on group structure and automorphism properties.

Contribution

It offers new criteria for the distality of automorphism actions on subgroup spaces of Lie groups, linking it to compactness and unipotency conditions.

Findings

Distality of automorphism actions relates to the compactness of generated subgroups.

Connected Lie groups act distally on subgroup spaces iff they are compact or a product of compact and vector groups.

Results extend to actions on abelian subgroup spaces.

Abstract

For a locally compact metrizable group $G$, we study the action of $\rm Aut(G)$ on $\rm Sub_G$, the set of closed subgroups of $G$ endowed with the Chabauty topology. Given an automorphism $T$ of $G$, we relate the distality of the $T$-action on $\rm Sub_G$ with that of the $T$-action on $G$ under a certain condition. If $G$ is a connected Lie group, we characterise the distality of the $T$-action on $\rm Sub_G$ in terms of compactness of the closed group generated by $T$ in $\rm Aut(G)$ under certain conditions on the center of $G$ or on $T$ as follows: $G$ has no compact central subgroup of positive dimension or $T$ is unipotent or $T$ is contained in the connected component of the identity in $\rm Aut(G)$. Moreover, we also show that a connected Lie group $G$ acts distally on $\rm Sub_G$ if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on $\rm Sub^a_G$, a subset of $\rm Sub_G$ consisting of closed abelian subgroups of $G$.