Distality of Certain Actions on -Adic Spheres

Shah, Riddhi; Yadav, Alok Kumar

doi:10.1017/s1446788719000272

Cited by 5 publications

(3 citation statements)

References 12 publications

(28 reference statements)

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“…Note that Proposition 4.3 also holds for a non-compact totally disconnected (additive) group G = Q n p , (n ∈ N), a p-adic vector space, and T ∈ GL(n, Q p ) (where p is a prime). This follows from Lemma 2.1 of [31] and Lemma 4.2 above together with the fact that GL(n, Q p ) is a (metrisable) topological group and its topology is the same as the (modified) compact-open topology.…”

Section: Distal Actions Of Automorphisms On Submentioning

confidence: 85%

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

Palit¹,

Shah²

2020

Glasgow Math. J.

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For a locally compact group G, we study the distality of the action of automorphisms T of G on Sub G , 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 Sub G if and only if T n 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 Sub G 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 Sub G 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 Sub G . We also show that any compactly generated distal group G is Lie projective.

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Section: Distal Actions Of Automorphisms On Submentioning

confidence: 85%

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

Palit¹,

Shah²

2020

Glasgow Math. J.

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“…Note that Theorem 4.3 also holds for a non-compact totally disconnected (additive) group G = Q n p , (n ∈ N), a p-adic vector space, and T ∈ GL(n, Q p ) (where p is a prime). This follows from Lemma 2.1 of [30] and Lemma 4.2 above together with the fact that GL(n, Q p ) is a (metrizable) topological group and its topology is the same as the (modified) compact-open topology.…”

Section: Distal Actions Of Automorphisms On Sub G For Certain Compact...mentioning

confidence: 86%

Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups

Palit,

Shah

2019

Preprint

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For a locally compact group G, we study the distality of the action of automorphisms T of G on Sub G , 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 Sub G if and only if T n is the identity map for some n ∈ 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 Sub G if and only if it acts distally on Sub Γ . This also holds for any closed T -invariant cocompact subgroup Γ. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on Sub Γ . We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group from that in a nilpotent Lie group. For torsion-free compactly generated nilpotent (metrizable) groups G, we obtain the following characterisation: T acts distally on Sub G 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 Sub G . We also show that any compactly generated distal group G is Lie projective. As a consequence, we get some results on the structure of compactly generated nilpotent groups.

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“…Here, T = T s T u = T u T s and T s ∈ K and T u = T 1 is unipotent. By Lemma 2.2 of [32], T u acts distally on Sub a G . By Theorem 4.3, T u = Id.…”

mentioning

confidence: 96%

Distal Actions of Automorphisms of Lie Groups $G$ on $\rm Sub_{G}$

Shah,

Yadav

2019

Preprint

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For a locally compact metrizable group G, we study the action of Aut(G) on 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 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 Sub G in terms of compactness of the closed group generated by T in 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 Aut(G). Moreover, we also show that a connected Lie group G acts distally on 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 Sub a G , a subset of Sub G consisting of closed abelian subgroups of G.

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Distality of Certain Actions on -Adic Spheres

Cited by 5 publications

References 12 publications

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

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

Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups

Distal Actions of Automorphisms of Lie Groups $G$ on $\rm Sub_{G}$

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Distality of Certain Actions on -Adic Spheres

Cited by 5 publications

References 12 publications

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

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

Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups

Distal Actions of Automorphisms of Lie Groups $G$ on $\rm Sub_{G}$

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

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