Distal Actions of Automorphisms of Lie Groups $G$ on $\rm Sub_{G}$

Abstract

For a locally compact metrizable group G, we study the action of Aut(G) on 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 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 Sub G in terms of compactness of the closed group 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 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 Sub a G , a subset of Sub G consisting of closed abelian subgroups of G.

Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups

“…We give conditions on discrete groups G under which Sub c G is closed in Sub G and study the distality of the action of automorphisms of G on Sub c G . We also prove certain results for the automorphisms in the class (NC) introduced in [31], which contains those that act distally on Sub a G or on the closure of Sub c G . Throughout, let G be a locally compact (Hausdorff) group with the identity e. For a subgroup H of G, let H 0 denote the connected components of the identity e in H. For any T ∈ Aut(G), T 0 is the identity map of G.…”

“…for some n 2 ∈ N. This implies that for x ∈ Γ, T n 2 (x) = xy for some nontrivial y ∈ Γ ∩ N. Let n = lcm(n 1 , n 2 ) and let T n = S. Now suppose x ∈ Γ is such that S(x j ) = x j for all j ∈ N. Then S(x) = xy for some y ∈ Γ ∩ N. As Γ ∩ N is torsion-free, by Lemma 3.12 of [31], we get that S ∈ (NC). Hence T ∈ (NC), which contradicts the hypothesis.…”

“…Therefore, H is cyclic. The class (NC) of automorphisms is defined in [31]. An automorphism T of a locally compact metrizable group G belongs to class (NC) if given any closed cyclic subgroup A of G, T n k (A) → {e}, for any unbounded sequence {n k } ⊂ Z.…”

“…Each T ∈ Aut(G) defines a homeomorphism of Sub G and the corresponding map from Aut(G) → Homeo(Sub G ) is a group homomorphism. For automorphisms T of connected Lie groups G, Shah and Yadav in [31] have studied and characterised the distality of the T -action on Sub G under certain conditions on T or on G, e.g. T is unipotent or the maximal connected central subgroup of G is torsion-free.…”

“…This also holds for any compact totally disconnected metrizable group (see Theorem 4.3). We also characterise metrizable locally compact compactly generated nilpotent groups G whose inner automorphisms act distally on Sub G (see Theorem 4.6); this is an analogue of Corollary 4.5 of [31].…”