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 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 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 Γ in a simply connected nilpotent Lie group G, T acts distally on Sub G if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on Sub G if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on Sub G . We also show that any compactly generated distal group G is Lie projective.

Distal Actions of Automorphisms of Lie Groups G on SubG

“…We conclude with a mention of the following: using many results in this paper, further study of class (NC) and the distality of actions of automorphisms of G on have been carried out where G is a certain type of locally compact group or a lattice in a Lie group [35, 37]. Expansivity of the action of automorphisms of G on has also been studied for locally compact groups and lattices in Lie groups recently [36, 37].…”