The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

Bridson, Martin R.; Harpe, Pierre de la; Kleptsyn, Victor

doi:10.2140/pjm.2009.240.1

Cited by 31 publications

(32 citation statements)

References 17 publications

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“…If G = T d , d ≥ 2, then its lattices are finite and any automorphism of G keeps the finite group G n = {g ∈ G | g n = e} invariant for any fixed n ∈ N. Note that Aut(G) is isomorphic to GL(d, Z) and, by Selberg's Lemma, it admits a subgroup of finite index which is torsion-free. Hence for such a G, (6) above holds for any lattice but (10) or (11) can not hold in general.…”

Section: Distal Actions Of Automorphisms Of Lattices γ Of Lie Groups mentioning

confidence: 99%

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 Of Lattices γ Of Lie Groups mentioning

confidence: 99%

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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“…Proof of Theorem 5.4. Recall ( [BHK09]) that for a compact group K, given a neighborhood U ⊂ K of identity, we obtain a neighborhood N U (K 1 ) of K 1 ∈ Sub(K) in the Chabauty topology by…”

Section: Topology Of Crs E (â)mentioning

confidence: 99%

Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

Bowen

Grigorchuk

Kravchenko

2016

Trans. Amer. Math. Soc.

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We show that if G is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then G has 2 ℵ 0 many continuous ergodic invariant random subgroups. If G is a nonabelian free group then G has 2 ℵ 0 many continuous G-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary p-groups of countably infinite rank.

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“…It is currently unclear whether these spaces inform on representation varieties associated to fundamental groups of Riemann surfaces, but it seems likely that these methods will inform on representation varieties for braid groups. Furthermore, the space Comm(G) maps naturally by evaluation onto the space of closed, finitely generated abelian subgroups of G topologized by the Chabauty topology as in [14]. It is natural to ask whether this map is a fibration with fiber J (T ).…”

Section: Proposition 2•8 Let T H and T G Be Maximal Tori Of Compact mentioning

confidence: 99%

On spaces of commuting elements in Lie groups

Cohen

Stafa

2016

Math. Proc. Camb. Phil. Soc.

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Abstract. The main purpose of this paper is to introduce a method to "stabilize" certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(Z n , G). We show that this stabilized space of homomorphisms decomposes after suspending once with "summands" which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(Z n , G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces Hom(Z n , G), with the order of the Weyl group inverted.

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The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

Cited by 31 publications

References 17 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

Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

On spaces of commuting elements in Lie groups

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The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

Cited by 31 publications

References 17 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

Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

On spaces of commuting elements in Lie groups

Contact Info

Product

Resources

About

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