On Approximation of Lie Groups by Discrete Subgroups
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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.
Hatem Hamrouni^{1} ^{} Salah Souissi^{2} ^{}
Volume 132, 2022
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