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

Abstract

For a locally compact group $G$, we study the distality of the action of automorphisms $T$ of $G$ on ${\mathrm{Sub}}_G$, 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 ${\mathrm{Sub}}_G$ 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 ${\mathrm{Sub}}_G$ if and only if it acts distally on ${\mathrm{Sub}}_{\Gamma }$. 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 ${\mathrm{Sub}}_{\Gamma }$. 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 ${\mathrm{Sub}}_G$ if and only if $T$ is contained in a compact subgroup of $\mathrm{Aut}(G)$. Using these results, we characterise the class of such groups $G$ which act distally on ${\mathrm{Sub}}_G$. 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.

29 pages, new results added

References in corpus (1)

• The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth

Cited by in corpus (1)

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