Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups

Abstract

For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub${}_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub${}_G^p$, the closure in Sub${}_G$ of the set of closed one-parameter subgroups of $G$. We relate the distality of the $T$-action on Sub${}_G^p$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$, and equate the distal action of any closed subgroup $\mathcal{H}$ on Sub${}_G^p$ with that of every element in $\mathcal{H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub${}_G^p$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Some of our results extend those of Shah and Yadav.

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References in corpus (3)

• The Space of Geometric Limits of One-generator Closed Subgroups of PSL2(R)
• Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub${}_G$ and Applications to Lattices in Lie Groups
• On the distality and expansivity of certain maps on spheres