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On Approximation of Lie Groups by Discrete Subgroups Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 1, February 2014, pp. 37–55. c Indian Academy of Sciences On approximation of Lie groups by discrete subgroups HATEM HAMROUNI1 and SALAH SOUISSI2 1Department of Mathematics, Faculty of Sciences at Sfax, University of Sfax, Route Soukra, B. P. 1171, 3000 Sfax, Tunisia 2Department of Mathematics, Faculty of Sciences at Monastir, University of Monastir, Avenue de l’environnement, 5019 Monastir, Tunisia Email: [email protected]; [email protected] MS received 11 August 2012; revised 27 January 2013 Abstract. 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 topol- ogy). 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 is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group G then G is approximated by discrete subgroups n containing . The proof of the above result is by induction on the dimension of G, and gives an algo- rithm for inductively determining n. The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (n)n≥0 in terms of .Several applications are given. Keywords. Nilpotent Lie group; rational structure; discrete uniform subgroup; lattice subgroup; Chabauty topology; Vietoris topology. 1991 Mathematics Subject Classification. 22E40. 1. Introduction and statement of results The following definition is introduced by Tôyama in [24]. DEFINITION 1.1 A locally compact group G is said to be approximated by discrete subgroups if there is a sequence of discrete subgroups (Hn)n≥0 of G satisfying the following condition: (A) For any open set O of G, there exists an integer k such that O ∩ Hn =∅, for every n ≥ k. 37 38 Hatem Hamrouni and Salah Souissi Let X be the class of locally compact groups approximated by discrete subgroups. A locally compact group belonging to the class X will often be called X-group. The first results in this direction were obtained by Tôyama. Theorem 1.2 (Theorem 1 of [24]). Any connected Lie group with discrete center is not a X-group. Theorem 1.3 (Theorem 2 of [24]). Every non-commutative compact Lie group is not a X-group. In 1951, Kuranishi proved the following remarkable theorem (Corollary, page 64 of [13]). Theorem 1.4. Let G be a Lie group approximated by discrete subgroups. Then G is nilpotent. Remark 1.5. A version of Theorem 1.4 was also noted earlier in Theorem 2 of [26]. Let S (G) denote the space of discrete co-compact subgroup of a Lie group G.We have the following result of Kuranishi (Theorem 2 of [13]). Theorem 1.6 (The approximation theorem). Let G be a connected, simply connected nilpotent Lie group such that S (G) =∅. Then the following statements hold: (1) GisaX-group. (2) Let ∈ S (G). Then G is approximated by discrete subgroups (n)n≥0 containing . The next theorem can be easily deduced from Theorem 4 of [13], which was announced without complete proof. This paper gives a simple proof of this result. Theorem 1.7. Let G be a connected simply connected nilpotent Lie group. If G is a X-group, then S (G) =∅. On the other hand, the proof of statement (2) of Theorem 1.6 is by induction on the dimension of G, and gives an algorithm for inductively determining n. In this paper, we are interested in the following problem. Problem 1.8. Given a connected simply connected nilpotent Lie group with discrete uni- form subgroup , construct an explicit sequence (n)n∈N which satisfies the condition of (2) of the approximation theorem. The main result of this paper is the following: Theorem 1.9. Let G be a connected simply connected nilpotent Lie group and adis- ∗ 1 crete uniform subgroup of G. For every n ∈ N , let n be the subgroup of G generated by the set {x ∈ G : xn ∈ }. Then ∗ 1 (1) for every n ∈ N , n ∈ S (G); 1 (2) the group G is approximated by ( n )n≥1. In the case when the group G is not simply connected, we obtain the following result. Approximation of Lie groups by discrete subgroups 39 Theorem 1.10. Let G be a connected nilpotent Lie group. Let G˜ be the universal covering group of G, and Z be a discrete normal subgroup of G˜ such that G =∼ G˜ /Z. Let π be the canonical projection of G˜ onto G. Let ∈ S (G˜ ) such that Z ∩ ∈ S (Z). Then 1 (1) for every n ≥ 1, π( n ) ∈ S (G); 1 (2) the group G is approximated by the sequence (π( n ))n≥1. Remark 1.11. The hypothesis of the existence of in S (G˜ ) such that Z ∩ ∈ S (Z) is necessary for the group G to be X-group (see Theorem 4 of [13]). Notations. In this paper we adopt the following notation: Let G be a group. We use the notation H ≤ G to mean that H is a subgroup of G. The index of a subgroup H in G is denoted by [G : H]. The subgroup of G generated by a subset A is denote by A .Let H, K be two subgroups of G. We denote by [H, K ] the subgroup of G that is generated by {hkh−1k−1}. The derived series G = D0(G) ≥ D1(G) ≥··· of G is defined inductively + by setting Dn 1(G) =[Dn(G), Dn(G)], for every n ≥ 0. Let D(G) denote D1(G). 2. Reformulation of Definition 1.1 2.1 Topologies of closed subgroups This subsection follows from [5, 10, 11] and [19]. Various topologies on closed subgroups of a locally compact topological group are considered. For the interrelationships between these topologies, see [19]. 2.1.1 Chabauty–Fell topology. Let X be a locally compact topological space. We denote by F (X) the space of closed subsets of X equipped with the Chabauty–Fell topology: the open are any meetings of finite intersections of parts of the form: O1(K ) ={H ∈ F (X) : H ∩ K =∅}, O2(U) ={H ∈ F (X) : H ∩ U =∅}, where K ⊆ X is compact and U is an open set of X. This is a compact space. PROPOSITION 2.1 (Lemma E.1.1 of [1]) If X is a locally compact topological space with a countable basis of open sets, the space F (X) has a countable basis too and moreover it is metrizable. PROPOSITION 2.2 Let X1 and X2 be two locally compact topological spaces. Let φ : X1 −→ X2 be an open continuous mapping. Then the mapping −1 F (X2) −→ F (X1), F −→ φ (F) is continuous. 40 Hatem Hamrouni and Salah Souissi As a consequence of Proposition 2.2 we have the following corollary, where we use as notation for ‘is homeomorphic to’. COROLLARY 2.3 If X1 X2, then F (X1) F (X2). More precisely, if φ : X1 −→ X2 is a homeomorphism of topological groups, then ∗ φ : F (X1) −→ F (X2), F −→ φ(F), is a homeomorphism. If X = G is a locally compact group, the space C (G) of closed subgroups of G is closed in F (G); in this specific case this topology was introduced by Chabauty [5]. Moreover we can define the topology of geometric convergence on C (G) (Definition 9.1.1, p. 225 of [23]). DEFINITION 2.4 (Geometric convergence) A sequence (n)n≥0 of closed subgroups converges geometrically to ∈ C (G) if the two conditions below are satisfied: (GC1) Let ϕ : N −→ N be an increasing map and let (xϕ(n))n≥0 be a sequence in G such that xϕ(n) ∈ ϕ(n) for any n ≥ 0. If (xϕ(n))n≥0 converges to some x ∈ G, then x ∈ . (GC2) Any point in is the limit of a sequence (xn)n≥0 with xn ∈ n for each n ≥ 0. For the proof of the next result, we refer to Lemma 2 of [12] or Proposition E 1.2 of [1]. Theorem 2.5. The convergence in the Chabauty topology is equivalent to the geometric convergence. The following elementary lemma will be of use in the sequel. Lemma 2.6 (Squeeze theorem for sequences). Let G be a locally compact group. Let (An)n≥0,(Bn)n≥0 and (Cn)n≥0 be three sequences of C (G). Suppose that for all n sufficiently large An ≤ Bn ≤ Cn.IfAn −→ F and Cn −→ F, then Bn −→ F. 2.1.2 Vietoris topology. Let G be a locally compact group, C (G) the space of closed subgroups of G. The Vietoris topology on C (G) is the topology on C (G) which has families of the form D1(U) ={H ∈ C (G) : H ⊂ U}, D2(V ) ={H ∈ C (G) : H ∩ V =∅}, where U and V run over the open sets of G, as a prebase. Approximation of Lie groups by discrete subgroups 41 2.1.3 Topological limits. DEFINITION 2.7 Let X be a topological space and (Ai )i≥0 be a sequence of closed subsets of X.
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