A Note on Pro-lie Groups

We give a short proof of the theorem that a closed subgroup of a countable product of second countable Lie groups is pro-Lie. The point of this note is to give a short and self-contained, modulo well known results, proof of a theorem of Hofmann and Morris [3] (see also [4, Theorem 3.35] and [5]) in the case of second countable groups. Another simple proof of the result of Hofmann and Morris was found by Glöckner [2]. All the ideas in the argument which we present here come from the proof of Lemma 2.3 in our paper [6]. We thank Isaac Goldbring for pointing out to us possible connection between our considerations in [6] and pro-Lie groups. All groups below are assumed to be second countable. 1. Common knowledge background In Proposition 1.1 we collect some well known properties of Lie groups that will be used. Important to us will be the notion of dimension of a Lie group, which can be understood as the dimension of the underlying manifold. Proposition 1.1. (i) Connected components of a Lie group are open and the connected component of the identity is a Lie group. (ii) If M is a Lie group and N a closed subgroup of M , then N is a Lie group; if, additionally, N is normal, then M/N is a Lie group. (iii) Let M, N be Lie groups and let f : M → N be a continuous homomorphism. If f is injective, then dim(M) ≤ dim(N); if f is surjective, then dim(M) ≥ dim(N). (iv) Let M, N be Lie groups with dim(M) = dim(N) and with N connected. If f : M → N is a continuous injective homomorphism, then f is surjective. Proof. Point (i) is clear. For point (ii) see [7, Theorem 3.42] for the proof that N is Lie and [7, Theorem 3.64] for the proof that M/N is Lie. Point (iii) follows from [7, Theorem 3.32]. As for point (iv), by [7, Theorem 3.32], f(M) is an open, so closed and open, subgroup of N . Since N is connected, f(M) = N . ¤ Date: April 7, 2009. 1 2 ALEKSANDRA KWIATKOWSKA AND SÃ LAWOMIR SOLECKI Each second countable groups admits a metric generating its topology. If one can find a complete such metric, it is customary to call such a group a Polish group. One can check that a second countable group is Polish precisely when it is complete in the sense of [4]. All locally compact second countable groups are Polish. In the proposition below, we collect some basic and well known facts about Polish groups. Proposition 1.2. Let G be a Polish group. (i) If there is a continuous homomorphism f : H → G such that H is a Lie group and f(H) has countable index in G, then G is Lie. (ii) If H is a closed normal subgroup of G, then G/H with the quotient topology is a Polish group. (iii) If H is a Polish group and f : G → H is a Borel measurable homomorphism, then f is continuous. In particular, if K is another Polish group and f : G → K and g : H → K are continuous homomorphisms with g being injective and with f(G) ⊆ g(H), then g−1 ◦ f : G → H is a continuous homomorphism. Proof. Point (i) follows from [1, Theorem 2.3.3] after noticing that f(H) is nonmeager in G as countably many of its translates cover G. Point (ii) is [1, Theorem 2.2.10]. The first part of point (iii) is a particular case of [1, Theorem 2.3.3]. The second part follows from the first one after observing that g−1 is a Borel measurable function. ¤ 2. Theorem Recall that a second countable group G is pro-Lie if it is Polish and each neighborhood of 1 contains a normal subgroup N such that G/N is Lie. Theorem 2.1. A closed subgroup of a countable product of Lie groups is pro-Lie. Proof. Let Li, i ∈ N, be Lie groups, and let G < ∏ i Li be closed. Let πn : ∏ i Li → ∏ i≤n Li and πn,N : ∏ i≤N Li → ∏ i≤n Li, for N ≥ n, be projections. The closure in the Lie group i≤N Li of the subgroup πN (G) is itself a Lie group, and we let AN = the connected component of 1 of πN (G). Let Bn,N = ker (πn,N 1 AN ) . A NOTE ON PRO-LIE GROUPS 3 Note that since πN,N+1(AN+1) is a connected subgroup of πN (G), we have (1) πN,N+1(AN+1) ⊆ AN . Claim 1. For every n there is in ≥ n such that for N ≥ in dim (Ain/Bn,in) = dim (AN/Bn,N ) . Proof. Let N ≥ n. Inclusion (1) induces an injective continuous homomorphism AN+1/(π−1 N,N+1(Bn,N ) ∩AN+1) → AN/Bn,N . It follows by Proposition 1.1(iii) that (2) dim(AN+1/(π−1 N,N+1(Bn,N ) ∩AN+1)) ≤ dim(AN/Bn,N ). Note that π−1 N,N+1(Bn,N ) ∩AN+1 ⊆ Bn,N+1, and therefore, by the second part of Proposition 1.1(iii), dim(AN+1/Bn,N+1) ≤ dim(AN+1/(π−1 N,N+1(Bn,N ) ∩AN+1)). From this inequality and from (2) we get dim(AN+1/Bn,N+1) ≤ dim(AN/Bn,N ). We conclude that the natural number valued function N → dim(AN/Bn,N ) is nonincreasing, and the claim follows. ¤ For n ∈ N, in ≥ n will denote the natural number from Claim 1. Claim 2. Let n ∈ N. For N ≥ in, πn,N+1(AN+1) = πn,N (AN ). Proof. The homomorphisms πn,N 1 AN and πn,N+1 1 AN+1 induce injective continuous homomorphisms π̂n,N : AN/Bn,N → ∏ i≤n Li and π̂n,N+1 : AN+1/Bn,N+1 → ∏ i≤n Li. Furthermore, from (1), we see that π̂n,N+1(AN+1/Bn,N+1) = πn,N+1(AN+1) ⊆ πn,N (AN ) = π̂n,N (AN/Bn,N ). (3)