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Distal Actions of Automorphisms of Lie Groups G on SubG

Part of: Locally compact groups and their algebras Lie groups Topological dynamics

Published online by Cambridge University Press:  21 December 2021

RIDDHI SHAH  and
ALOK KUMAR YADAV
RIDDHI SHAH
Affiliation:
School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India, e-mail: riddhi.kausti@gmail.com, rshah@jnu.ac.in
ALOK KUMAR YADAV*
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research, Mohali 140306, India, e-mail: alokmath1729@gmail.com
*
†Corresponding author: Alok Kumar Yadav
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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.

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 22D45: Automorphism groups of locally compact groups 22E25: Nilpotent and solvable Lie groups 22E15: General properties and structure of real Lie groups
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

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