Distal Actions of Automorphisms of Lie Groups G on Sub G

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.

Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups

For a locally compact group 𝐺, we write S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) for the space of closed subgroups of 𝐺 endowed with the Chabauty topology. For any positive integer 𝑛, we associate to 𝐺 the function δ G , n \delta_{G,n} from G n G^{n} to S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) defined by δ G , n ⁢ ( g 1 , … , g n ) = gp ¯ ⁢ ( g 1 , … , g n ) , \delta_{G,n}(g_{1},\ldots,g_{n})=\overline{\mathrm{gp}}(g_{1},\ldots,g_{n}), where gp ¯ ⁢ ( g 1 , … , g n ) \overline{\mathrm{gp}}(g_{1},\ldots,g_{n}) denotes the closed subgroup topologically generated by g 1 , … , g n g_{1},\ldots,g_{n} . It would be interesting to know for which groups 𝐺 the function δ G , n \delta_{G,n} is continuous for every 𝑛. Let [ HW ] [\mathtt{HW}] be the class of such groups. Some interesting properties of the class [ HW ] [\mathtt{HW}] are established. In particular, we prove that [ HW ] [\mathtt{HW}] is properly included in the class of totally disconnected locally compact groups. The class of totally disconnected locally compact locally pronilpotent groups is included in [ HW ] [\mathtt{HW}] . Also, we give an example of a solvable totally disconnected locally compact group not contained in [ HW ] [\mathtt{HW}] .

ON THE CONNECTEDNESS OF THE CHABAUTY SPACE OF A LOCALLY COMPACT PRONILPOTENT GROUP

Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$ . More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$ .

DISTAL ACTIONS OF AUTOMORPHISMS OF NILPOTENT GROUPS G ON SUB G AND APPLICATIONS TO LATTICES IN LIE GROUPS

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.

Locally compact groups with totally disconnected space of subgroups

The closed subsets {\mathcal{F}(X)} of any topological space can be given a topology, called the Chabauty–Fell topology, in which {\mathcal{F}(X)} is quasi-compact. If X is a locally compact space, then {\mathcal{F}(X)} is Hausdorff. If {X=G} is a locally compact group, then the subset {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} of closed subgroups is closed in {\mathcal{F}(G)} , and the set of closed subgroups inherits its own compact topology, called the Chabauty topology. We develop some of the important properties of this topology. More precisely, the results contained in this paper deal with the following question: Given a locally compact topological group, when is {\mathcal{F}(G)} (resp. {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} ) a totally disconnected space?