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DISTAL ACTIONS OF AUTOMORPHISMS OF NILPOTENT GROUPS G ON SUBG AND APPLICATIONS TO LATTICES IN LIE GROUPS

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

Published online by Cambridge University Press:  23 June 2020

RAJDIP PALIT  and
RIDDHI SHAH
RAJDIP PALIT
Affiliation:
School of Physical Sciences, Jawaharlal Nehru University, New Delhi110 067, India, e-mails: [email protected]; [email protected]; [email protected]
RIDDHI SHAH
Affiliation:
School of Physical Sciences, Jawaharlal Nehru University, New Delhi110 067, India, e-mails: [email protected]; [email protected]; [email protected]
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Abstract

For a locally compact group G, we study the distality of the action of automorphisms T of G on SubG, the compact space of closed subgroups of G endowed with the Chabauty topology. For a certain class of discrete groups G, we show that T acts distally on SubG if and only if Tn is the identity map for some $n\in\mathbb N$ . As an application, we get that for a T-invariant lattice Γ in a simply connected nilpotent Lie group G, T acts distally on SubG if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on SubG if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on SubG. We also show that any compactly generated distal group G is Lie projective.

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 22E25: Nilpotent and solvable Lie groups 22E40: Discrete subgroups of Lie groups 22D45: Automorphism groups of locally compact groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 343 - 362
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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