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 $^p_G$ , 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 $^p_G$ 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 $^p_G$ with that of every element in ${\mathcal H}$ . Moreover, we show that a connected Lie group G acts distally on Sub $^p_G$ by conjugation if and only if G is either compact or is isomorphic to a direct product of a compact group and a vector group. Some of our results generalise those of Shah and Yadav.