Abstract
A locally compact group G is said to be approximated by discrete subgroups (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 Γ n containing Γ. The proof of the above result is by induction on the dimension of G, and gives an algorithm for inductively determining Γ n . The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (Γ n ) n ≥ 0 in terms of Γ. Several applications are given.
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Acknowledgements
The authors would like to thank the referee for the helpful comments and suggestions which improved this paper, in particular for shortening the proof of Lemma 4.4. The second author is supported by laboratory mathematical physics, special functions and applications: LR11.ES35, University of Sousse.
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HAMROUNI, H., SOUISSI, S. On approximation of Lie groups by discrete subgroups. Proc Math Sci 124, 37–55 (2014). https://doi.org/10.1007/s12044-013-0153-4
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DOI: https://doi.org/10.1007/s12044-013-0153-4
Keywords
- Nilpotent Lie group
- rational structure
- discrete uniform subgroup
- lattice subgroup
- Chabauty topology
- Vietoris topology