• Hatem Hamrouni

Articles written in Proceedings – Mathematical Sciences

• On Approximation of Lie Groups by Discrete Subgroups

A locally compact group 𝐺 is said to be approximated by discrete sub-groups (in the sense of Tôyama) if there is a sequence of discrete subgroups of 𝐺 that converges to 𝐺 in the Chabauty topology (or equivalently, in the Vietoris topology). The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete subgroups is nilpotent. The converse, in general, does not hold. For example, a connected simply connected nilpotent Lie group is approximated by discrete subgroups if and only if 𝐺 has a rational structure. On the other hand, if 𝛤 is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group 𝐺 then 𝐺 is approximated by discrete subgroups $\Gamma_n$ containing 𝛤. The proof of the above result is by induction on the dimension of 𝐺, and gives an algorithm for inductively determining $\Gamma_n$. The purpose of this paper is to give another proof in which we present an explicit formula for the sequence $(\Gamma_n)_{n\geq 0}$ in terms of 𝛤. Several applications are given.

• A characterization of totally disconnected compactly ruled groups

A locally compact group $G$ is called compactly ruled if it is a directed union of compact open subgroups. We denote by $\mathcal{SUB}(G)$ the space of closed subgroups of $G$ equipped with the Chabauty topology. In this paper, we show that the subspace $\mathcal{SUB}_{\rm co} (G)$ of $\mathcal{SUB}(G)$ consisting of compact open subgroups is dense in $\mathcal{SUB}(G)$ if and only if $G$ is totally disconnected compactly ruled.

• Closed subgroups of compact groups having open Chabauty spaces

Given a locally compact group $G$, we denote by $\mathcal{SUB}(G)$ the set of closed subgroups of $G$ equipped with the Chabauty topology, which is a compact Hausdorff. For a closed subgroup $H$ of $G$, the Chabauty topology on $\mathcal{SUB}(H)$ is equivalent to the subspace topology on $\{L \in \mathcal{SUB}(G) \mid L \subseteq H\}$ inherited from $\mathcal{SUB}(G)$, and so $\mathcal{SUB}(H)$ becomes a closed subset of $\mathcal{SUB}(G)$. In some cases, the subspace $\mathcal{SUB}(H)$ is open in $\mathcal{SUB}(G)$, but it can also fail to be open as the example of the trivial subgroup $\{0\}$ of the additive group $\mathbb{Z}$ shows. In this paper, we are interested to determine the subset $\mathcal{T} (G)$ of closed subgroups $H$ of a compact group $G$ such that $\mathcal{SUB}(H)$ is open in $\mathcal{SUB}(G)$. An explicit description of $\mathcal{T} (G)$ is given. The paper also contains some topological properties of $\mathcal{T} (G)$. As an application, we show that if $G$ is a compact group and if the lattice $\mathcal{SUB}(G)$ is isomorphic to the lattice $\mathcal{SUB}(\mathbb{Z}_p)$ of the group $\mathbb{Z}_p$ of $p$-adicintegers, then $G$ is topologically isomorphic to $\mathbb{Z}_l$, for some prime $l$.

• Proceedings – Mathematical Sciences

Volume 133, 2023
All articles
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• Editorial Note on Continuous Article Publication

Posted on July 25, 2019