Distal Actions of Automorphisms of Lie Groups G on SubG

Abstract

For a locally compact metrisable group G, we study the action of Aut(G) on SubG , 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 SubG 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 SubG in terms of compactness of the closed subgroup generated by T in 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 Aut(G) . Moreover, we also show that a connected Lie group G acts distally on SubG 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 SubaG , a subset of SubG consisting of closed abelian subgroups of G.