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CHARACTERISATION OF DISTAL ACTIONS OF AUTOMORPHISMS ON THE SPACE OF ONE-PARAMETER SUBGROUPS OF LIE GROUPS

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

Published online by Cambridge University Press:  15 August 2025

DEBAMITA CHATTERJEE Open the ORCID record for DEBAMITA CHATTERJEE [Opens in a new window]  and
RIDDHI SHAH Open the ORCID record for RIDDHI SHAH [Opens in a new window]
DEBAMITA CHATTERJEE
Affiliation:
School of Physical Sciences, https://ror.org/0567v8t28 Jawaharlal Nehru University , New Delhi 110 067, India e-mail: debamita.math@gmail.com
RIDDHI SHAH*
Affiliation:
School of Physical Sciences, https://ror.org/0567v8t28 Jawaharlal Nehru University , New Delhi 110 067, India e-mail: rshah@jnu.ac.in
*
e-mail: riddhi.kausti@gmail.com
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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.

Keywords

distal actions of automorphismsLie groupsone-parameter subgroupsChabauty topology

MSC classification

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

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by G. Willis

The authors would like to acknowledge the support in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program—ICTS Ergodic Theory and Dynamical Systems (code: ICTS/etds-2022/12).

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