Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups

TL;DR

This paper investigates when automorphisms of nilpotent and solvable groups act distally on the space of closed subgroups, providing characterizations, examples, and applications to lattices in Lie groups.

Contribution

It offers new criteria for distality of automorphisms on subgroup spaces, especially for nilpotent and solvable groups, and explores their implications for lattices in Lie groups.

Findings

Automorphisms act distally iff some power is the identity in certain groups.

Distality on subgroup spaces relates to automorphism groups being compact.

Characterizations of groups acting distally on their subgroup spaces.

Abstract

For a locally compact group $G$, we study the distality of the action of automorphisms $T$ of $G$ on ${\rm Sub}_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. For a certain class of discrete groups $G$, we show that $T$ acts distally on ${\rm Sub}_G$ if and only if $T^n$ is the identity map for some $n\in{\mathbb N}$. As an application, we get that for a $T$-invariant lattice $\Gamma$ in a simply connected nilpotent Lie group $G$, $T$ acts distally on ${\rm Sub}_G$ if and only if it acts distally on ${\rm Sub}_\Gamma$. This also holds for any closed $T$-invariant co-compact subgroup $\Gamma$. For a lattice $\Gamma$ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on ${\rm Sub}_\Gamma$. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group from that in a nilpotent Lie group. For torsion-free compactly generated nilpotent (metrizable) groups $G$, we obtain the following characterisation: $T$ acts distally on ${\rm Sub}_G$ if and only if $T$ is contained in a compact subgroup of ${\rm Aut}(G)$. Using these results, we characterise the class of such groups $G$ which act distally on ${\rm Sub}_G$. We also show that any compactly generated distal group $G$ is Lie projective. As a consequence, we get some results on the structure of compactly generated nilpotent groups.