On the existence of automorphisms acting expansively on the space of one-parameter subgroups of connected Lie groups

Himanshu Lekharu; Riddhi Shah

paper

On the existence of automorphisms acting expansively on the space of one-parameter subgroups of connected Lie groups

arXiv:2611.09352

Abstract

For a connected Lie group $G$ , we consider the action of Aut $(G)$ on Sub $_{G}$ , the set of all closed subgroups of $G$ endowed with the Chabauty topology. We study the action of automorphisms of $G$ on Sub $_{G}^{p}$ , the closure of the set of all closed one-parameter subgroups of $G$ in Sub $_{G}$ , in terms of expansivity. We show that a nontrivial connected Lie group $G$ with dim $(G) \geq 2$ does not admit any automorphism which acts expansively on Sub $_{G}^{p}$ . We also show that a certain class of subgroups of Aut $(G)$ , which include all one-parameter groups of automorphisms in Aut $(G)$ , do not act expansively on Sub $_{G}^{p}$ . These results generalise a results of Chatterjee, Lekharu and Shah on tori and those of Prajapati and Shah on connected Lie groups of dimension greater than $1$ .