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.

For a locally compact Hausdorff group G and the compact space \mathrm{Sub}_G of closed subgroups of G endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of G on \mathrm{Sub}_G in terms of distality and expansivity. We prove that an infinite discrete group G , which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on \mathrm{Sub}^c_G , the space of cyclic subgroups of G , while only the finite order automorphisms of G act distally on \mathrm{Sub}^c_G . For an automorphism T of a connected Lie group G which keeps a lattice \Gamma invariant, we compare the behaviour of the actions of T on \mathrm{Sub}_G and \mathrm{Sub}_\Gamma in terms of distality. Under certain necessary conditions on the Lie group G , we show that T acts distally on \mathrm{Sub}_G if and only if it acts distally on \mathrm{Sub}_\Gamma . We also obtain certain results about the structure of lattices in a connected Lie group.