Distal Actions of Automorphisms of Lie Groups G on Sub_G

Abstract

For a locally compact metrisable 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 subgroup 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.

Expansive actions of automorphisms of locally compact groups $${\varvec{G}}$$ on $$\hbox {Sub}_{{\varvec{G}}}$$

“…Note that for homeomorphisms of compact infinite metric spaces, distality and expansivity are two opposite phenomena [7,Theorem 2]. Shah and Yadav in [23] have discussed the distality of the actions of automorphisms on Sub G and Sub a G . The study of expansivity of these actions in the current paper contributes to and enhances the understanding of the dynamics of actions of automorphisms of G on Sub G .…”

“…Note that Aut(G) with the modified compact-open topology is a topological group [21, 9.17], and by Lemma 2.4 of [23] it acts continuously on Sub G by homeomorphisms. Hence Lemma 2.6 holds for the case when X = Sub G and H = Aut(G) with the modified compact-open topology, where G is a locally compact metrizable group.…”