The closed subsets ℱ ( X ) {\mathcal{F}(X)} of any topological space can be given a topology, c... more The closed subsets ℱ ( X ) {\mathcal{F}(X)} of any topological space can be given a topology, called the Chabauty–Fell topology, in which ℱ ( X ) {\mathcal{F}(X)} is quasi-compact. If X is a locally compact space, then ℱ ( X ) {\mathcal{F}(X)} is Hausdorff. If X = G {X=G} is a locally compact group, then the subset 𝒮 𝒰 ℬ ( G ) {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} of closed subgroups is closed in ℱ ( G ) {\mathcal{F}(G)} , and the set of closed subgroups inherits its own compact topology, called the Chabauty topology. We develop some of the important properties of this topology. More precisely, the results contained in this paper deal with the following question: Given a locally compact topological group, when is ℱ ( G ) {\mathcal{F}(G)} (resp. 𝒮 𝒰 ℬ ( G ) {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} ) a totally disconnected space?
Abstract A topological group G is said to be approximable by discrete subgroups, if there exists ... more Abstract A topological group G is said to be approximable by discrete subgroups, if there exists a sequence of discrete subgroups ( H n ) n ∈ N of G such that, for any non-empty open set O of G, there exists an integer k such that O ∩ H n ≠ ∅ , for every n ≥ k . In this paper we shall prove that the identity component of a locally compact group approximable by discrete subgroups is nilpotent. For connected nilpotent Lie groups, explicit approximating sequences of discrete subgroups are given. As an application, we extend Jessen's theorem on Riemann sums for torus to the case of nilmanifolds.
Let G be a locally compact group. We denote by SUB(ℝ × G) the hyperspace of closed subgroups of ℝ... more Let G be a locally compact group. We denote by SUB(ℝ × G) the hyperspace of closed subgroups of ℝ × G endowed with the Chabauty topology. In this paper, some general results about the connectedness of the hyperspace SUB(ℝ × G) are established. The main result proved here is that if G is a prosolvable group, then SUB(ℝ × G) is connected. We also prove a partial converse to the result just mentioned.
A locally compact group G is said to be approximable by discrete subgroups if there exists a sequ... more A locally compact group G is said to be approximable by discrete subgroups if there exists a sequence of discrete subgroups ( H n ) n ∈ ℕ $(H_n)_{n\in \mathbb {N}}$ of G such that for any nonempty open set O of G, there exists an integer k such that O ∩ H n ≠ ⌀ $O\cap H_n \ne \varnothing $ , for every n ≥ k. It has been proved that the identity component of a Lie group approximable by discrete subgroups is nilpotent. In this paper we shall extend this result to pro-Lie groups. In particular, every connected locally compact group approximable by discrete subgroups is nilpotent. An application of discrete approximation is given.
Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup o... more Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up-down representation (Ind G 1)| A . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/ in the case when is a lattice subgroup of G and A is a one-parameter subgroup of G.
The closed subsets ℱ ( X ) {\mathcal{F}(X)} of any topological space can be given a topology, c... more The closed subsets ℱ ( X ) {\mathcal{F}(X)} of any topological space can be given a topology, called the Chabauty–Fell topology, in which ℱ ( X ) {\mathcal{F}(X)} is quasi-compact. If X is a locally compact space, then ℱ ( X ) {\mathcal{F}(X)} is Hausdorff. If X = G {X=G} is a locally compact group, then the subset 𝒮 𝒰 ℬ ( G ) {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} of closed subgroups is closed in ℱ ( G ) {\mathcal{F}(G)} , and the set of closed subgroups inherits its own compact topology, called the Chabauty topology. We develop some of the important properties of this topology. More precisely, the results contained in this paper deal with the following question: Given a locally compact topological group, when is ℱ ( G ) {\mathcal{F}(G)} (resp. 𝒮 𝒰 ℬ ( G ) {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} ) a totally disconnected space?
Abstract A topological group G is said to be approximable by discrete subgroups, if there exists ... more Abstract A topological group G is said to be approximable by discrete subgroups, if there exists a sequence of discrete subgroups ( H n ) n ∈ N of G such that, for any non-empty open set O of G, there exists an integer k such that O ∩ H n ≠ ∅ , for every n ≥ k . In this paper we shall prove that the identity component of a locally compact group approximable by discrete subgroups is nilpotent. For connected nilpotent Lie groups, explicit approximating sequences of discrete subgroups are given. As an application, we extend Jessen's theorem on Riemann sums for torus to the case of nilmanifolds.
Let G be a locally compact group. We denote by SUB(ℝ × G) the hyperspace of closed subgroups of ℝ... more Let G be a locally compact group. We denote by SUB(ℝ × G) the hyperspace of closed subgroups of ℝ × G endowed with the Chabauty topology. In this paper, some general results about the connectedness of the hyperspace SUB(ℝ × G) are established. The main result proved here is that if G is a prosolvable group, then SUB(ℝ × G) is connected. We also prove a partial converse to the result just mentioned.
A locally compact group G is said to be approximable by discrete subgroups if there exists a sequ... more A locally compact group G is said to be approximable by discrete subgroups if there exists a sequence of discrete subgroups ( H n ) n ∈ ℕ $(H_n)_{n\in \mathbb {N}}$ of G such that for any nonempty open set O of G, there exists an integer k such that O ∩ H n ≠ ⌀ $O\cap H_n \ne \varnothing $ , for every n ≥ k. It has been proved that the identity component of a Lie group approximable by discrete subgroups is nilpotent. In this paper we shall extend this result to pro-Lie groups. In particular, every connected locally compact group approximable by discrete subgroups is nilpotent. An application of discrete approximation is given.
Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup o... more Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up-down representation (Ind G 1)| A . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/ in the case when is a lattice subgroup of G and A is a one-parameter subgroup of G.
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