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CHARACTERISATION OF DISTAL ACTIONS OF AUTOMORPHISMS ON THE SPACE OF ONE-PARAMETER SUBGROUPS OF LIE GROUPS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 15 August 2025, pp. 1-24
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- Article
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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.
