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Distal Actions of Automorphisms of Lie Groups G on SubG
Published online by Cambridge University Press: 21 December 2021
Abstract
For a locally compact metrisable group G, we study the action of
${\rm Aut}(G)$
on
${\rm Sub}_G$
, 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
${\rm Sub}_G$
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
${\rm Sub}_G$
in terms of compactness of the closed subgroup generated by T in
${\rm Aut}(G)$
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
${\rm Aut}(G)$
. Moreover, we also show that a connected Lie group G acts distally on
${\rm Sub}_G$
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
${\rm Sub}^a_G$
, a subset of
${\rm Sub}_G$
consisting of closed abelian subgroups of G.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 2 , September 2022 , pp. 457 - 478
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society
References
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