Seminar Sophus Lie 2 (1992) 123–134

Finitely generated connected locally compact groups

Karl H. Hofmann and Sidney A. Morris

Introduction

Hofmann and Morris [6] proved that a locally compact connected group G has a finite subset generating a dense subgroup if and only if the weight w(G) of G does not exceed 𝔠, the cardinality of the continuum. The minimum cardinality of such a topological generating set is an invariant of the group, is denoted by σ(G), and is called the topological rank of G. For compact abelian groups of weight ≤ 𝔠, this number is 1. It was also shown there that for any compact connected group G of weight ≤ 𝔠, the invariant σ(G) ≤ 2. Cleary and Morris [3] observed that σ(ℝ^{n}) = n+1, for n ≥ 1 and proved the surprising result that for any compact connected group G with w(G) ≤ 𝔠, the invariant σ(G × ℝ^{n}) = n + 1 for n ≥ 1. Here we extend their result significantly. For example, if G is a compact connected group of weight ≤ 𝔠 and L is a nonsingleton connected Lie group, then σ(G × L) = σ(L). For more general locally compact groups we bound the topological rank of a group in terms of an associated Lie group.

Partial results

We consider here Hausdorff topological groups only. For a subset X ⊆ G we let ⟨X⟩ denote the group generated by X.

Lemma 1. Let P and A denote topological groups, X a topological generating set of A and π: X → P a function such that π(X) is a topological generating set of P. Set G = P × A, graph(π) = \{(π(x), x) : x ∈ X\}, and H = \overline{⟨graph(π)⟩}. Let pr_{P} and pr_{A} denote the projections. Then

(∗)  \overline{pr_{P} (H)} = P and \overline{pr_{A}(H)} = A.

Proof. Since X, resp. π(X), is a topological generating set of A, resp., P, then pr_{P} (π(X)), resp. pr_{A}(X) topologically generates the group P, resp. A. Thus pr_{P} (H) and pr_{A}(H) are dense in P and A, respectively.

Lemma 2. Let P and A denote topological groups, and H a closed subgroup of P × A such that (∗) is satisfied. In the notation of Lemma 1:

(i) If P is compact, then pr_{A} |H: H → A is an open surjective morphism of topological groups.

(ii) N =^{def} ((P × {1}) ∩ H) × (({1} × A) ∩ H) is normal in G.

(iii) Set N_{P} = {p ∈ P : (p, 1) ∈ H} and A_{N} = {a ∈ A : (1, a) ∈ H}. Then G/N can be identified with P/N_{P} ×A/N_{A} and H/N with a closed subgroup thereof. Further, H = G iff N = G.

(iv) If N = {1} and P is compact, then there is a unique injective morphism f: A → P with dense image extending π such that H = {(f(a), a) : a ∈ A}.

Proof. (i) Since P is compact, the projection pr_{A} is a proper map. Hence it maps the closed group H onto its image in such a fashion that the induced map H/((P × {1}) ∩ H) → A is an isomorphism of topological groups.

(ii) The normalizer of (P ×{1})∩H contains H (since P ×{1} is normal in G) and {1} × A (since {1} × A centralizes P × {1}). Since pr_{P} (H) is dense in P, then H({1} × A) is dense in G. Hence (P × {1}) ∩ H is normal in G. Likewise ({1} × A) ∩ H. Thus N is normal.

(iii) is straightforward.

(iv) Since H meets the factor P trivially, H is the graph of a morphism f: A → P. Since H is closed and P is compact, it is continuous. Since H meets A trivially, f is injective. By (i) the image of f is dense in P. Since (π(x), x) ∈ H for all x, the morphism f extends π. Since X is a topological generating set, the extension is uniquely determined by this condition.

Lemma 3. Let P be a compact connected group and A a locally compact group. Assume that H is a closed subgroup of G = P × A satisfying (∗) and that a maximal compact connected abelian subgroup T of P × {1} is contained in H. Then H = G.

Proof. We consider N as in Lemma 2. By Lemma 2(ii) we have T ⊆ N. Set T = T_{1} ×{1} and N ∩(P ×{1}) = N_{1} . The maximal compact connected abelian subgroup T_{1} of P is contained in N_{1} , which by Lemma 2(ii) is a normal subgroup of P. Since all maximal compact connected abelian subgroups are conjugate and their union covers P we conclude that N_{1} = P. Hence P ×{1} ⊆ N ⊆ H. Thus H contains the kernel of pr_{A}. Then from (∗) it follows that H is dense in G. As H is closed, H = G follows.

Lemma 4. Let P be any compact connected group and T a maximal compact connected abelian subgroup. Let A denote a locally compact group with a topological generating set X. Assume that X = Y ∪Z ⊆ A and π: X → P are such that T ∪ π(Z) is a topological generating set of P and that {π(y), y) : y ∈ Y } topologically generates a subgroup containing T × {1}. Then graph(π) = {(π(x), x) : x ∈ X} is a topological generating set of P × A.

Proof. Let H = \overline{⟨X⟩}. We must show H = P × A. By Lemma 3 this is the case if the conditions of Lemma 3 hold. Since X is a topological generating set of A then pr_{A}(H) is dense in A, and since pr_{P} (H) contains T and π(Z) it is dense in P. Thus (∗) holds. The remaining condition of Lemma 3 is satisfied by hypothesis.

We note that every locally compact connected group G contains a characteristic subgroup G_{k} such that G_{k} contains the radical R and has no nontrivial compact simple homomorphic image, and is such that G_{k}/R contains no compact connected normal subgroup. We have G = G_{k} iff G has no nontrivial compact connected simple group as a homomorphic image. We then say that G is without compact factors. Indeed if G is semisimple, then there is a closed semisimple Lie subgroup G_{k} containing all simple noncompact factors and no compact one. Then G is the product of G_{k} and the product of all compact simple factors. If G is not semisimple, then G_{k} is the full inverse image of (G/R)_{k} in G.

We note that, in particular, if f: G → G^{∗} is a quotient homomorphism, then f(G_{k}) = G_{k}^{∗}.

Proposition 5. Assume that P is any compact group whose identity component P_{0} is contained in \overline{P′}, and A is a locally compact connected group. Let X be a topological generating set of A.

(i) If π: X → P is any function such that π(X) is a topological generating set of P then graph(π) = {(π(x), x) : x ∈ X} is a topological generating set of a subgroup H ⊆ P × A containing {1} × A_{k}.

(ii) If A is without compact factors, then graph(π) topologically generates P × A for any π as in (i).

Proof. We let H be the closure of the group generated by graph(π) and denote P ×A by G. We claim G = H. Let N be as in Lemma 2. If H/N = G/N then H = G. Now (A/N_{A})_{k} = (A_{k}N_{A})/N_{A} and (P/N_{P})_{0} = (P_{0}N_{P})/N_{P} , the latter being contained in \overline{(P/N_{P})′} = (\overline{P′}N_{P})/N_{P} . Thus, without loss of generality we can assume N = {1}.

We claim that A is isomorphic to P and thus is compact. By Lemma 2, there is a unique injective morphism f: A → P extending π with dense image. Thus A is maximally almost periodic and is therefore V ×C with a vector group V and a compact connected group C. Also, P is now a compact, connected group satisfying P = \overline{P′}. This last property implies that the identity component of the center of P is trivial. Hence pr_{P} (V) is a point and thus V is singleton. Hence A = C is compact, and is mapped isomorphically onto P. Now A ≅ P is a compact connected semisimple group and thus A_{k} = {1}. This proves (i).

In order to prove (ii) we note that if A_{k} = A, then after factoring N we have A = {1}. After having factored N, we have P = {1}. We conclude that N = G and thus H = G by Lemma 2(iii).

Certain information obtained in the proof may be of independent interest: If N = {1} then P ≅ A.

We shall say that a topological group G is (topologically) perfect if it satisfies \overline{G′} = G. (This holds for example for all locally compact connected groups without radical.)

As a corollary of Proposition 5 we have:

Proposition 6. Let P be a compact group such that P_{0} is perfect. If A is locally compact connected group without compact factors, and if σ(P) ≤ σ(A), then σ(P × A) = σ(A).

We shall now make a few observations about the abelian situation which we shall use presently.

If X is a subset of a locally compact abelian group A, let F (X) denote the free (discrete) abelian group on X and f: F (X) → A the morphism extending the inclusion X → A. The image is dense iff the adjoint morphism f̂: Â → F̂ (X) = 𝕋^{X} is injective. Thus σ(A) is the smallest among the cardinals of those sets X for which Â has a continuous injective image in 𝕋^{X}. For instance, if A = ℝ^{n}, then Â ≅ ℝ^{n}, and ℝ^{n} has a continuous injective image in 𝕋^{n+1} but not in 𝕋^{n}. If A is compact and connected then Â is discrete and torsion free. It is (algebraically) isomorphic to a subgroup of 𝕋^{X} if (and only if) the rank of Â does not exceed 𝔠· card X, where 𝔠 is the cardinality of the continuum. Recall that the dimension of A in this case is the rank of Â.

Lemma 7. Assume that K is a compact connected abelian group of dimension ≤ ℵ_{0} and A = ℝ^{n} × K with n ≥ 1. Let X be a topological generating set of A and assume that C is a compact abelian group with σ(C) ≤ card X. Then there is a function π: X → C such that {(π(x), x) : x ∈ X} is a topological generating set of C × A.

Proof. Let f̂: Â → 𝕋^{X} be as in the remarks preceding the lemma. The image V = f̂(ℝ^{n} × {1}) has a compact connected closure, and since f̂ is injective the index of V in \overline{V} is of continuum cardinality. The rank of f̂(\{0\} × K̂) is countable. Hence its divisible hull K^{∗} is countable and torsion free. Since V is divisible, there is a subgroup W of 𝕋^{X} such that 𝕋^{X} = W ⊕ V ⊕ K^{∗} algebraically. The the rank of W is at least 𝔠, but also it cannot be bigger since X ⊆ A and card A = 𝔠. Since W contains the torsion group of 𝕋^{X} and has the same rank as 𝕋^{X} it is algebraically isomorphic to 𝕋^{X} under an algebraic isomorphism i: 𝕋^{X} → W. Now let φ: (𝕋^{X})_{d} ⊕Â → 𝕋^{X} be the injective morphism given by φ(α ⊕ β) = i(α) + f̂(β). Notice that the character group A(X) of (𝕋^{X})_{d} is the free compact abelian group on the set X. (See [4].) The dual morphism φ̂: F (X) → A(X) × A is given by φ̂(g) = (î(g), f(g)) and has dense image. It satisfies φ̂(x) = (î(x), x) where {î(x) : x ∈ X} is topologically generating in A(X). Since σ(C) ≤ card(X) there is a surjective homomorphism η: A(X) → C. If we set π = η ∘ (î|X) then we have the desired function.

We obtain at once the following variant of the preceding lemma:

Lemma 7′. Suppose that K is a compact connected abelian group of dimension ≤ ℵ_{0} and A = ℝ^{n} × K with n ≥ 1. Let {x_{j} : j ∈ J} be a topological generating family in A and suppose that C is a compact abelian group with σ(C) ≤ card J. Then there is a function π: J → C such that {(π(j), x_{j}) : j ∈ J} is a topological generating set of C × A.

Lemma 8. If P is a nonsingleton monothetic compact group, then σ(P ×ℝ^{n}) = n + 1.

Proof. For n = 0 the assertion is trivial. If n ≥ 1 then σ(ℝ^{n}) = n + 1 by the remarks preceding Lemma 7. (There are other arguments for this conclusion: See [3].) Now Lemma 7 implies the conclusion (with K = {1}).

Lemma 9. Let A denote a locally compact group such that A/\overline{A′} is second countable and connected. Assume that C is a compact abelian group satisfying σ(C) ≤ σ(A). Then for any generating set X of A there is a function π: X → P such that graph(π) is a generating set of P × A.

Proof. Let X denote a topological generating set of A and let f: A → A/\overline{A′} be the quotient map. Then {f(x) : x ∈ X} is a topological generating family of f(A). By Lemma 7′, there is a function π: X → C such that {(π(x), f(x)) : x ∈ X} topologically generates C × f(A). Let H be the closed subgroup generated by {(π(x), x) : x ∈ X} in C × A. We shall prove that H = G =^{def} C × A and thereby finish the proof.

Let h = (π(x_{1}), x_{1}) ⋯ (π(x_{m}), x_{m}) and h′ = (π(x′_{1}), x′_{1}) ⋯ (π(x′_{n}), x′_{n}). Set [a, b] = aba^{−1}b^{−1} . Then [h, h′] = (1, [x_{1} ⋯ x_{m}, x′_{1} ⋯ x′_{n}]) since C is abelian. Hence D = \{1\} × \overline{A′} ⊆ H. But if we identify C × f(A) with G/D, then H/D = C × f(A) because H/D contains all (π(x), f(x)), x ∈ X. Then H = G follows.

Lemma 10. Let P be any compact group with identity component P_{0} ⊆ \overline{P′} and assume w(P_{0}) ≤ 𝔠. Also assume that A is a locally compact connected group with a second countable factor group A/A_{k} such that σ(P) ≤ σ(A) and σ(P/P_{0}) ≤ σ(A/A_{k}) − 2. Then for any topological generating set X of A there is a function π: X → P such that graph(π) topologically generates P × A.

Proof. If A = A_{k} then Proposition 5 proves the assertion. Now assume A_{k} ≠ A. We let f: A → A/A_{k} denote the quotient map. If X is a topological generating set of A then f(X) is a topological generating set of A/A_{k}. Since A/A_{k} is nonabelian, there are at least two elements y and z such that f(y) ≠ f(z).

Let T denote a maximal compact connected abelian subgroup of P. Then there is a p ∈ P_{0} such that T ∪ {p} is a topological generating set of P_{0} (see [6], Corollary 2.5). Since w(P_{0}) ≤ 𝔠, then T is monothetic. If M is the closed subgroup generated by f(y) in A/A_{k}, then there is a t ∈ T such that (t, f(y)) topologically generates T × M by Lemma 7. Now the closed subgroup topologically generated by {(t, f(y)), (p, f(z))} ∪ ({1} × f(X_{0})) with X_{0} = X \ {y, z} has dense projections into A and P_{0} and thus generates P_{0} × A/A_{k} by Lemma 3.

Define π: X → P by π(y) = t, π(z) = p, and such that π(X_{0}) is a generating set of P modulo P_{0} . Such a choice is possible because σ(P/P_{0}) ≤ σ(A/A_{k})−2 ≤ |X_{0}|. Let H denote the closed subgroup generated in G = P ×A by graph(π). Let N be defined as in Lemma 2. Then {1} × A_{k} is contained in N by Proposition 5. Now H/N is topologically generated by {(π(x), f(x)) : x ∈ X} = {(t, f(y)), (k, f(z))} ∪ {(π(x), f(x)) : x ∈ X_{0}} and thus agrees with G/N by the preceding. Now G = H follows.

Lemma 11. Let θ: G → H be an open surjective morphism of locally compact groups such that ker θ is compact totally disconnected and contained in G_{0} . Then

(i) for each generating set X of H each subset X′ of G with θ(X′) = X is a generating subset of G, and

(ii) σ(G) = σ(H).

Proof. We begin by noting that (ii) follows from (i): If X is topological generating set of G with card X = σ(G), then θ(X) is a topological generating set of H, whence σ(H) ≤ card X = σ(G). The reverse inclusion, however, we deduce from (i) by taking a generating subset X of H with card X = σ(H) and considering a subset X′ ⊆ G such that θ|X′: X′ → X is bijective. Then σ(G) ≤ card X′ = card X = σ(H) by (i).

Now we prove (i). Let X be a topological generating set of H. Let X′ ⊆ G be any subset such that θ(X′) = X. Let K = \overline{⟨X′⟩}. Then θ(K) = H since N =^{def} ker θ is compact. It follows that θ(K_{0}) = H_{0} . Since N is compact, NK_{0} is closed, and since θ: G → H is a proper map, then θ|NK_{0}: NK_{0} → θ(K_{0}) is a proper map. Thus NK_{0}/N ≅ θ(K_{0}) = H_{0} ≅ G_{0}/N. We conclude G_{0} = NK_{0} . The homogeneous spaces G_{0}/K_{0} and N/(N ∩ N_{0}) are homeomorphic, and the latter is totally disconnected, as N is totally disconnected. The former is connected as G_{0} is connected. Thus they are singleton and K_{0} = G_{0} . Now let Θ: G/G_{0} → H/H_{0} be defined by Θ(gG_{0}) = θ(g)H_{0} . Then Θ is an isomorphism because G/G_{0} ≅ (G/N)/(G_{0}/N) ≅ H/H_{0} . Now Θ(K/G_{0}) = θ(K)/H_{0} = H/H_{0} = Θ(G/G_{0}). Since Θ is an isomorphism we conclude K/G_{0} = G/G_{0} and thus K = G. Hence X′ is a generating set of G.

The principal result

At last we are able to state and prove the main result.

Theorem 12. Let P be compact group and A a locally compact group satisfying the following hypotheses:

(a) A is connected,

(b) w(A) ≤ ℵ_{0} (that is, A is second countable),

(c) w(P) ≤ 𝔠,

(d) σ(P) ≤ σ(A), and

(e) σ(A) ≥ 2 and σ(P/P_{0}) ≤ σ(A) − 2. Then

(i) for every generating set X of A there is a function π: X → P such that {(π(x), x) : x ∈ X} is a generating set of P × A, and

(ii) σ(P × A) = σ(A).

Proof. (i) implies (ii): If we consider X with card X = σ(A) then σ(P ×A) ≤ card({(π(x), x) : x ∈ X}) = card(X) = σ(A). Conversely, if Y is a generating set of P × A with cardinality σ(P × A), then pr_{A}(Y) is a generating set of A of cardinality ≤ card(Y) = σ(P × A). Hence σ(A) ≤ σ(P × A).

Proof of (i):

Case 1: P is abelian. Then the assertion is true by Lemma 9 in view of (a,b,d).

Case 2: The identity component Z_{0}(P) of the center of P is trivial. Then by Theorem 1.3 in [5] we know that P_{0} = (\overline{P′})_{0} ⊆ \overline{P′}. Then the assertion follows from Lemma 10 in view of (a,b,c,d,e).

Case 3: P is arbitrary. By the Theorem of Lee [10] there is a compact zero dimensional group D of P such that P = P_{0}D. The set (P_{0})′D is a compact subgroup B since (P_{0})′ is a compact characteristic subgroup of P_{0} and thus is normal in P. We have P = Z_{0}(P)B (see [5], Theorem 1.3) and D =^{def} Z_{0}(P) ∩ B is a compact totally disconnected central subgroup of P contained in P_{0} . Now A, as a locally compact connected group, is σ -compact. Hence P × A is σ - compact. Hence Lemma 11 applies to the quotient map θ: (P × A) → \frac{P ×A}{D×\{1\}}. Assume that we have a generating set X of A and a function π_{0}: X → P/D such that graph(π_{0}) is a generating set for (P/D)×A ≅ \frac{P ×A}{D×\{1\}}. Now define π: X → P such that π(x)D = π_{0}(x) for all x ∈ X. Then graph(π) = {(π(x), x) : x ∈ X} maps onto graph(π_{0}). By Lemma 11 then graph(π) is a generating set of P ×A.

We therefore can replace P by P/D and thus assume that P is the direct product of the subgroups Z_{0}(P) and B. Thus G = Z_{0}(P) × B × A. Let X be a generating set of A. By Case(1) we find a function π_{1}: X → Z_{0}(P) such that {(π_{1}(x), 1, x) : x ∈ X} is a generating set of Z_{0}(P) × {1} × A. Now we notice that

B/B_{0} → \frac{Z_{0}(B) × B × A}{Z_{0}(B) × B_{0} × A} ≅ P/P_{0}.

Hence σ(B/B_{0}) = σ(P/P_{0}) and all hypotheses are satisfied which allow us to apply Lemma 10 with Z_{0}(P) × {1} × A in place of A and B in place of P. This yields a function π_{2}: X → B such that {(π_{1}(x), π_{2}(x), x) : x ∈ X} is a generating set for Z_{0}(P) × B × A. If we recall P = ℤ_{0}(P) × B and define π: X → P by π(x) = (π_{1}(x), π_{2}(x)) we see that this is the assertion of the theorem.

Note that if P is connected then the condition σ(P/P_{0}) ≤ σ(A) − 2 is trivially satisfied and by [6] conditions (c) and (e) imply (d). In particular, for the case that A is a connected Lie group, we obtain:

Corollary 13. Let P denote a compact connected group of weight ≤ 𝔠 and L a nonsingleton connected Lie group. Then

(i) for every generating set X of A there is a function π: X → P such that graph(π) is a generating set of P × A, and

(ii) σ(P × L) = σ(L).

Proof. (i) If L is compact abelian, then L and P × L are monothetic and the assertion is true. Otherwise σ(L) ≥ 2 and Theorem 12 applies. (ii) follows from (i)

We note that assumptions of the sort of conditions (d) and (e) are necessary in order for the conclusion to hold. The topological rank of the group ℤ(2)^{n} × ℝ is min{2, n} which is large if n is large while σ(R) = 2.

It is interesting to realize that condition (b) cannot be replaced by the condition w(A) ≤ 𝔠 without invalidating the theorem. In order to understand this let us consider the following example:

Let P be any nonsingleton compact connected group with w(P) ≤ 𝔠. Let K = ℝ̂_{d} denote the universal solenoid. Set A = ℝ × K and observe w(P) ≤ 𝔠 = w(K) = w(A) and σ(P) = 1, σ(A) = 2. Thus hypotheses (a,c,d,e) of Theorem 12 are satisfied. The prescription of a generating set X of two elements of A is equivalent to giving a morphism of locally compact groups e: ℤ^{2} → A with dense image and that is tantamount to giving an injective morphism ê: Â → 𝕋^{2} . But Â = ℝ × ℝ_{d}. Hence we can choose ê to be an isomorphism of Â onto a full torsion free complement of the torsion group (ℚ/ℤ)^{2} of 𝕋^{2} . The prescription of any π: X → P is the specification of two elements p, q ∈ P. Let α: ℤ^{2} → P denote the unique morphism sending (1, 0) to p and (0, 1) to q. Define δ: ℤ^{2} → P ×A by δ(t) = (α(t), e(t)). Now {(π(x), x) : x ∈ X} is a generating set of P × A iff δ has dense image. But that means exactly that

δ̂: P̂ × Â = P̂ × ℝ × ℝ_{d} → 𝕋^{2}, δ̂(r, s, t) = α̂(r) + ê(s, t)

is injective. But since ê(ℝ × ℝ_{d}) is a complement of the torsion subgroup of 𝕋^{2} and α(Â) would have to be a nonzero torsion free subgroup meeting ê(ℝ × ℝ_{d}) trivially, this is patently impossible. Notice that we can even take P = 𝕋.

We saw that σ(ℝ^{n}) = n + 1 for n ≥ 1. So Corollary 13 yields:

Corollary 14. (Cleary and Morris [3]) Let P be a compact connected group of weight ≤ 𝔠. Then σ(P × ℝ^{n}) = n + 1, if n ≥ 1.

Iwasawa Pairs

There is a “classical” theorem by Iwasawa [8] which says that a locally compact connected group contains a compact normal subgroup N and a local Lie group U commuting elementwise with N such that (n, u) ↦ nu: N × U → NU is a homeomorphism onto an identity neighborhood. Thus there is a Lie group L with an identity neighborhood isomorphic to U, and we obtain an injective morphism φ: L → G such that (n, g) ↦ nφ(g) : N × L → G is a surjective homomorphism which, by the Open Mapping Theorem, is also open. Let us formalize these remarks as follows:

Lemma 15. Let G denote a locally compact connected group. Then there is a compact normal subgroup N and a connected Lie group L and an injective morphism φ: L → G such that

(i) [N, φ(L)] = {1}, (ii) G = Nφ(L), and (iii) there is an identity neighborhood U in L such that (n, u) ↦ nφ(u) :

N × U → Nφ(U) is a homeomorphism onto an identity neighborhood of 1 such that [N, φ(U)] = {1}.

Proof. (i) and (ii) are consequences of (iii), and (iii) is the Iwasawa’s Local Product Theorem (see [8]).

Definition 16. If G is a locally compact connected group, then an Iwasawa pair is a pair (N, φ: L → G) with a morphism φ: L → G satisfying the conditions of Lemma 15.

Lemma 17. Let G denote a locally compact connected group with weight w(G) ≤ 𝔠. If (N, φ: L → G) is an Iwasawa pair, then σ(N_{0}) ≤ 2.

Proof. From w(N_{0}) ≤ w(G) ≤ 𝔠 we conclude σ(N_{0}) ≤ 2. (See [7], 4.13.)

We observe that G/N is a Lie group locally isomorphic to L. Let p: G → G/N denote the quotient map. The kernel of the covering Φ = p ∘ φ: L → G/N is K = φ^{−1}(N ∩ φ(L)).

For an abelian group A we set rank A = dim_{ℚ}(ℚ ⊗_{ℤ} A) and call this number the torsion free rank of A.

Lemma 18. Let θ: G → H be a surjective morphism of connected locally compact groups and write N = ker θ. Assume that S is a closed subgroup of G.

(i) If G = NS and N is countable, then S = G. (ii) If N is finite and θ(S) is dense, then S = G (iii) If N is finite, then σ(G) = σ(H). (iv) If θ is a covering homomorphism of connected Lie groups then σ(G) ≤

σ(H) + rank(ker θ).

Proof. (i) If ⋃_{n∈N} nS = NS = G, then the locally compact space G is a countable union of homeomorphic closed subsets nS. Then by the Baire Category Theorem, one of them has inner points. Thus the subgroup S has nonempty interior and so is open. Hence it is also closed. As G is connected, S = G follows.

(ii) If N is finite, then NS is a closed subgroup. Since θ(S) is denseit follows that NS is dense. Thus NS = G.

(iii) Let Y ⊆ H denote a generating set in the sense that H = \overline{⟨Y ⟩} and let X ⊆ θ^{−1}(Y) be any subset such that θ|X: X → Y is bijective. Define S = \overline{⟨X⟩}. Then θ(S) is dense in H. As N is finite, (ii) applies and proves the assertion.

(iv) Now N is finitely generated abelian. Its torsion group M is finite. By (iii) we have σ(G) = σ(G/M) and the covering morphism Θ: G/M → H, Θ(gM) = θ(g) has a free kernel of rank rank N which is also σ(N). Since always σ(G/M) ≤ σ(H) + σ(ker Θ) (cf. [3]), the assertion follows.

Conclusion (iii) also follows from Lemma 11.

For the quotient map f: ℝ → 𝕋 = ℝ/ℤ we have σ(ℝ) = 2, σ(𝕋) = 1 and rank(ℤ) = 1. This shows that the inequality in (iv) is sharp.

For an Iwasawa pair (N, φ: L → G) we write β(N, φ) = rank(ker Φ) and call it its Betti number.

Proposition 19. If G is a noncompact locally compact connected group of weight not exceeding the cardinality of the continuum, then for any Iwasawa pair (N, φ: L → G),

σ(G) ≤ σ(L) ≤ σ(G/N) + β(N, φ) ≤ σ(G) + β(N, φ).

Proof. Since G is not compact then L is not compact, in particular, L is nonsingleton. Then Corollary 13 shows σ(N_{0} × L) = σ(L). We consider the surjective morphism π: N ×L → G, π(n, g) = nφ(g). In Lemma 17 of [7] we show that the arc component G_{a} of the identity in G is N_{a}φ(L) where N_{a} is the arc component of 1 in N. But G_{a} is dense in G and G_{a} ⊆ N_{0}φ(L) = π(N_{0} × L). Thus π maps N_{0} × L onto a dense subgroup of G. Hence any topological generating set X of N_{0} × L is mapped onto a topological generating set π(X) of G. Therefore, σ(G) ≤ card π(X) ≤ card X = σ(N_{0} × L) = σ(L).

Notice that we always have σ(G/N) ≤ σ(G).

In order to prove the remaining inequality we consider the quotient map q: G → G/N and the morphism f =^{def} q ∘ φ: L → G → G/N. Now ker f = {g ∈ L : φ(g) ∈ N}. If U ⊆ L is as in Lemma 15(iii), then φ(U) ∩ N = {1} and thus U ∩ker f = {1}. Thus ker f is discrete. Because of G = Nφ(L) the morphism f is surjective, and since L is connected, hence σ -compact, it is open. Thus f is a covering homomorphism. Now Lemma 18(iv) applies and shows the assertion.

In order to illustrate the situation by an example we define N = ℚ̂/ℤ ≅ ∏_{p prime} ℤ_{p} where ℤ_{p} is the additive group of p-adic integers. Then q: ℚ/ℤ → 𝕋 = ℝ/ℤ gives an injective morphism q̂: ℤ → N with dense image. Write i: ℤ → N × ℝ, i(n) = (q̂(n), −n). Then i is an injective morphism with discrete image (since the projection of the image onto ℝ is discrete). Define G = (N × ℝ)/(im i) and let p: N × ℝ → G denote the quotient homomorphism. Since the compact space N × [0, 1] maps onto G under p we know that G is compact. The dual of the exact sequence

0 → ℤ →^{i} ℕ × ℝ →^{p} G → 0

is the exact sequence

0 → Ĝ \underrightarrow{_{p}} (ℚ/ℤ) × ℝ \underrightarrow{_{i}} ℝ/ℤ → 0

with î(q + ℤ, r) = q − r + ℤ. Thus im p̂ = ker î = {(q + ℤ, r) : q − r ∈ ℤ} = {(q + ℤ, q) : q ∈ ℚ} ≅ ℚ. It follows that G is the rational solenoid ≅ ℚ̂. The image N^{∗} = p(N × {0}) in G is an isomorphic copy of N, and (N^{∗}, φ: ℝ → G), φ(r) = p(0, r) is an Iwasawa pair for G. Since G = ℚ̂, then G is monothetic; that is, σ(G) = 1. We know σ(ℝ) = 2. We have G/N ≅ ℝ/ℤ and ker(ℝ → G → G/N) = {r ∈ ℝ : φ(r) = p(0, r) ∈ N^{∗}} = {r ∈ ℝ : (∃n ∈ ℤ) (0, r) ∈ (q̂(n), −n) + (N × {0}} = ℤ. Thus β(N^{∗}, φ) = 1. Thus the second inequality of Proposition 19 is sharp.

Remarks and Open Questions

There is a result by Kuranishi [9] saying that every connected semisimple Lie group has topological rank 2. This suggests the following question:

Question A. If G is a perfect connected nondegenerate Lie group is σ(G) = 2?

An analysis of the topological rank of a connected Lie group requires an answer to the following question:

Question B. What is the topological rank of a solvable connected Lie group?

In this context the following result of Cleary [1] is relevant:

Proposition 20. If G is a connected nilpotent Lie group then σ(G) = σ(G/\overline{G′}).

Note that in Proposition 20 G/\overline{G′} ≅ ℝ^{n} × 𝕋^{m} and thus σ(G) = n + 1.

Finally we mention that in [6,7] we discussed the cardinal invariant s(G) for a locally compact group defined as the minimum cardinal of a suitable subset X of G where X is a topological generating subset of G such that X is discrete and closed in G \ {1}. Trivially, σ(G) ≤ s(G). Further it was shown in [6] that for a locally compact connected group G of weight ≤ 𝔠, the cardinal s(G) is finite and so equals σ(G). This leaves open the question whether these two cardinals are always equal. They are not as the example below demonstrates.

Let X be a set of cardinality 2^{𝔠}. Now let G = A(X) be the free compact abelian group on the discrete set X [4]. Then Ĝ = (𝕋^{X})_{d} and thus w(G) = rank Ĝ = 2^{𝔠·2^{𝔠}} = 2^{2^{𝔠}}. In [6] we showed for a compact connected group H with w(H) > c that s(H)^{ℵ_{0}} = w(H)^{ℵ_{0}} . Thus s(G)^{ℵ_{0}} = 2^{2^{𝔠}}. On the other hand, X is a topological generating subset of G. Hence σ(G) ≤ card X = 2^{𝔠}. Since (2^{𝔠})^{ℵ_{0}} = 2^{𝔠} we conclude that σ(G) < s(G).

If σ(G) is infinite, then it coincides with the density d(G). The relation d(G) = log w(G) was proved by Comfort and Itzkowitz in [1].

References

[1] Comfort, W. W., and G. L. Itzkowitz, Density characters in topological groups, Math. An.. 226 (1977), 223–227.

[cl] Cleary, Joan, “On the structure of locally compact Hausdorff groups”, Ph.D. Thesis, La Trobe University, Australia, 1989, ix+110 pp.

[3] Cleary, Joan, and Sidney A. Morris, Generators for locally compact groups, submitted.

[4] Hofmann, Karl Heinrich and Sidney A. Morris, Free compact groups I: Free compact abelian groups, Topology Appl. 23 (1986), 41-64, and Errata, ibid 28 (1988), 101–102.

[5] —, Free compact groups II: The center, Topology Appl. 28 (1988), 215– 231.

[6]  —, Weight and c, J. Pure Appl. Algebra 68 (1990), 181–194.

[7] —, Generators on the arc component of compact connected groups, submitted.

[8] Iwasawa, K., On some types of topological groups, Ann. of Math. 50 1949, 507–558.

[9] Kuranishi, M., Two element generation on semi-simple Lie groups, Kǒdai Math. Sem. Rep. Nos. 5–6, Dec. 1949.

[10] Lee, Dong Hoon, Supplements for the identity component in locally compact groups,, Math. Z. 104 (1968), 28–49.

Fachbereich Mathematik Technische Hochschule Darmstadt Schlossgartenstr. 7 D-6100 Darmstadt, Germany hofmann@mathematik.th-darmstadt.de Faculty of Informatics University of Wollongong Locked Bag 8844, South Coast Mail Centre, NSW 2521, Australia sid@wampyr.cc.uow.edu.au

Received July 3, 1992