A Distinguished Subgroup of Compact Abelian Groups

: Here “group” means additive abelian group. A compact group G contains δ –subgroups, that is, compact totally disconnected subgroups ∆ such that G / ∆ is a torus. The canonical subgroup ∆ ( G ) of G that is the sum of all δ –subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of ∆ ( G ) when considering only ﬁnite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G , we prove the following: ∆ ( G ) contains tor ( G ) , is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G , and G / ∆ ( G ) is torsion-free and divisible; ∆ ( G ) is a functorial subgroup of G , it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G . The subgroup ∆ ( G ) appeared before in the literature as td ( G ) motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor, preserves proper exactness of short sequences of compact groups.


Introduction
The topological groups studied in this paper are mainly the Pontryagin duals of discrete abelian groups with some emphasis on the duals of torsion-free groups. The latter are exactly the compact connected abelian groups. Non-compact topological groups prominently appear in Section 7.
The result ( [1], Proposition 8.15, p. 416) deals with the existence of compact totally disconnected subgroups ∆ of a compact group G such that G/∆ is a torus. These δsubgroups enter into the Resolution Theorem for compact abelian groups ( [1], Theorem 8.20, p. 420, see also Section 6). The duals of the short exact sequences ∆ G T where G is a compact group, ∆ is a δ-subgroup of G and thus T is a torus, are precisely the exact sequences F A D where A is a discrete group, F is a free subgroup of A and D is a torsion group. This suggests the study of the full free subgroups F of A, i.e., the free subgroups of A with torsion quotient. Let F (A) denote the family of all full free subgroups of A and let D(G) denote the family of all δ-subgroups of the compact group G. In Theorem 1, a comprehensive description of F (A) is established, and by duality a similarly comprehensive description of D(G) is obtained (Theorem 6). In fact, there is an anti-isomorphism of semi-lattices δ : F (A) → D(G) where G = A ∧ (Theorem 5).
The canonical subgroup ∆(G) := ∑ D(G) of G, referred to as "Fat Delta", has interesting properties: (FD1) It contains tor(G), is dense in G, and G/∆(G) is torsion-free and divisible (Theorem 6(2),(4), (6) and Theorem 10 (2)). (FD2) If G is not totally disconnected, then ∆(G) is a proper subgroup of G, and hence is not locally compact (Proposition 6(1)). The group ∆(G) coincides with tor(G) if and only if G = T × E with T a finite dimensional torus and E a bounded group (Theorem 9). We obtain a "canonical" resolution theorem (Theorem 15) for a compact abelian group G where the canonical ∆(G) replaces a random δ-subgroup.
In [2], the case of connected compact groups of finite dimension was studied; here we generalize to arbitrary compact abelian groups of any dimension, but even in the case of finite dimension, our results on Fat Delta surpass by far those in [2].
In Section 7, we provide a different 'projective' characterization of td(G) (see Proposition 9(1)) and various applications of ∆(G) = td(G). It is proved that td, as a functor, preserves proper exactness of short sequences of compact groups (Corollary 4). The interest in the subgroup td(G) of compact groups (see Definition 4) was triggered by the intensive research on the Open Mapping Theorem since the early seventies of the last century [3][4][5][6][7][8][9][10][11][12][13][14][15] (see Definition 5 for the relevant properties and Theorem 17 for criteria for the inheritance of these properties from dense subgroups). Section 7.3 is focused on the topological p-Sylow subgroups td p (G) of td(G).
In Section 8, we discuss some open problems.
In a forthcoming paper [16], we extend the characterization (FD6) to larger classes of topological abelian groups (e.g., subgroups of LCA groups). To this end, we introduce there a new series of functorial subgroups in TAG, related to td(G) and td p (G), and consider alternative definitions of Fat Delta for non compact groups.

Notation and Background
Our reference on abelian groups is [17]. As a rule A, B, C, D, E, . . . denote discrete groups and G, H, K, L, . . . are used to denote topological groups. Unless otherwise stated, p is an arbitrary prime number. If C is a category of groups, then "A is a C-group" and "A ∈ C" means that A is an object of C. By A ≤ B we mean that A is a sub-object of B when A, B ∈ C. We will deal with the following categories: • The category AG of discrete abelian groups with morphisms algebraic homomorphisms, ∼ = denoting isomorphism in this category, also called algebraic isomorphism; • TAG is the category of topological abelian groups with morphisms continuous algebraic homomorphisms, ∼ =t denoting isomorphism in this category; • LCA is as usual the full subcategory of TAG consisting of locally compact Hausdorff groups.
We will use N := {1, 2, . . .} and N 0 := N ∪ {0} while P denotes the set of all prime numbers. Furthermore, R denotes the additive group of real numbers, Z the integers and T the additively written circle group R/Z equipped with the compact quotient topology. A torus is a topological group isomorphic with a power T m where m is any cardinal.
The torsion subgroup (p-torsion subgroup) of an abelian group G is denoted by tor(G) (tor p (G), respectively). We have tor(T) = Q/Z ≤ T with the subspace topology, and tor p (T) ≤ Q/Z with the subspace topology. We use Z(p ∞ ) := tor p (T) = {m/p n + Z | m, n ∈ N 0 } in agreement with ( [1], p. 27).
The m-socle of a group X is X[m] := {x ∈ X | mx = 0} and the socle of X is Soc(X) = p∈P X [p]. By µ X m we denote multiplication by m in X. For a subgroup, Y of X and m ∈ N, define This concept is used to construct larger full free subgroups from given full free subgroups.
The group D is a divisible hull of A if D is divisible and A is an essential subgroup of D, equivalently, if D/A is a torsion group and p∈P D[p] ⊆ A. Divisible hulls exist for any group and divisible groups are direct sums of copies of Q and of Z(p ∞ ), p ∈ P ( [17], p. 136).
The Z-adic topology of Z (having as a local base at 0 the filter base {nZ : n ∈ N}) will be denoted by ν Z . We denote by G ∧ the Pontryagin dual of a TAG-group G, while G is reserved for the completion of G. In particular, Z is the completion of (Z, ν Z ) and Z p is the completion of Z in the p-adic topology.
For topological groups G, H we will deal with cHom(G, H), the set of all continuous homomorphisms from G to H. Throughout, we assume that the groups of morphisms cHom(G, H) carry the compact-open topology. We will use the notation of ( [1], p. 337), so recall that the sets W(C, U) = { f ∈ cHom(G, H) | f [C] ⊆ U} where C is compact in G and U is open in H, form a basis for the topology of cHom(G, H).
By c(G) we denote the 0-component of G and by a(G) the arc component of 0 ∈ G. A Hausdorff topological group G is zero-dimensional if G has a base of clopen sets. Clearly, every linearly topologized group is zero-dimensional and every zero-dimensional group is totally disconnected. Recall that a group is linearly topologized if it possesses a neighborhood basis at 0 consisting of subgroups. Lemma 1 ([1], E8.6, p. 414). Let G be a locally compact abelian group. Then G is totally disconnected if and only if G is zero-dimensional.
A topological abelian group G is said to be precompact if its completion is compact. It is a well-known and deep fact that a topological abelian group G is precompact if and only if the topology of G is generated by its continuous characters, which means that the characters χ ∈ G ∧ separate the points of G and the injective (continuous) diagonal map Proposition 1. Let G be a topological abelian group and let G i , i ∈ I, be a family of topological groups. Then cHom(G, ∏ i∈I G i ) ∼ =t ∏ i∈I cHom(G, G i ).
To show that π is continuous, consider the generic open neighborhood V = ∏ i∈I V i , To show that π is open, we consider a basic open subset U of Hom(A, Let A be a discrete group and G any topological group. Then, the compact open topology on Hom(A, G) coincides with the subspace topology of Hom(A, G) ⊆ G A where G A carries the product topology (=topology of point-wise convergence). This is well-known and is easily seen noting that the compact subsets of A are exactly the finite subsets.
Let G and H be topological groups.
is compact and it is easy to show that Φ is a quotient map. Proposition 2. Let A i , i ∈ I, be a family of discrete abelian groups, G a topological abelian group. Then Proof. Let ins i : A i → i∈I A i be the insertions belonging to the direct sum. The map Φ is the standard algebraic isomorphism and We first show that Φ −1 is continuous. By definition of the product topology, is the projection belonging to the product. Let U be an open neighborhood of 0 ∈ G and let F be a finite subset of A i . Then W := W(F, U) is a generic neighborhood of 0 ∈ Hom(A i , G).
We show next that Φ is continuous. Let F be a finite subset of i∈I A i and U an open neighborhood of 0 ∈ G. Then W = W(F, U) is a generic open neighborhood of 0 ∈ Hom( i∈I A i , G). Then there is a finite subset J of I such that F ⊆ i∈J A i . Furthermore, for j ∈ J, there exist finite sets The following is surely well-known.

Lemma 2.
Let G and H be topological abelian groups and ϕ : G → H a surjective homomorphism with kernel K.
(1) Suppose that ϕ is continuous and K is dense in G. Then H is indiscrete.
(2) Suppose that ϕ is an open map and H is indiscrete. Then K is dense in G.
(3) Suppose that H is indiscrete and cHom(G, H) is endowed with the compact-open topology.
Then cHom(G, H) is indiscrete. ( 3) The open sets of cHom(G, H) are the sets of the form W : By hypothesis U = ∅ or U = H. Whatever C may be, in the first case W = ∅ and in the second case W = cHom(G, H).

) of Full Free Subgroups in AG
The following notation relating an arbitrary group A with its torsion-free quotient A 0 := A/ tor(A) will be used throughout.
Let A ∈ AG and let ϕ 0 : A → A 0 be the natural epimorphism. For future use we record the short exact sequence In the literature, the dimension of a compact abelian group is defined in several equivalent ways. The cardinal dim(G) = rk(G ∧ ) will serve for the purposes of this article.
For every prime p we define the p-rank of A by rk p (A) := dim Z/pZ (A[p]). A discrete divisible group D is determined up to isomorphism by the invariants rk p (D) counting the summands isomorphic to Z(p ∞ ) and rk(D/ tor(D)) counting the summands isomorphic to Q. See ( [17], Chapter 4) for details. Lemma 3. If A is a torsion-free group, then rk p (A/pA) ≤ rk(A).
Proof. It suffices to check that if {b 1 , . . . , b n } is a linearly independent subset of A/pA, where b i = a i + pA, a i ∈ A, then {a 1 , . . . , a n } is linearly independent in A. Assume that m 1 a 1 + · · · + m n a n = 0, with m i ∈ Z for i = 1, 2, . . . , n. As A is torsion-free, we can assume without loss of generality that gcd(p, m j ) = 1 for some j = 1, 2, . . . , n. After projecting in A/pA we obtain m 1 b 1 + · · · + m n b n = 0. By the choice of {b 1 , . . . , b n } this gives m i = pZ for all i. This contradicts gcd(p, m j ) = 1. Now we see that the p-ranks of a compact connected group G of finite dimension are bounded from above by dim(G). Corollary 1. Let A be a discrete torsion-free group of finite rank n. Then rk p (A ∧ ) = rk p (A/pA) ≤ n.
Proof. Clearly, G = A ∧ is a compact connected group with dim(G) = n. The socle G[p] of G is the kernel of µ G p , the multiplication by p in G, and hence closed and therefore compact.
We have the proper exact sequence G[p] G µ G p G which gives the proper exact sequence We first illuminate the abundance of full free subgroups in a group.

Lemma 4.
Let tor(A) = A ∈ AG. Then the following hold.
(1) {a i | i ∈ I} is a linearly independent set in A if and only if {a i + tor(A) | i ∈ I} is a linearly independent set in A 0 . Moreover, {a i | i ∈ I} is maximal linearly independent if and only if {a i + tor(A) | i ∈ I} is maximal linearly independent. (2) If {a i | i ∈ I} is a (maximal) linearly independent set in A and ∀ i ∈ I : t i ∈ tor(A), then {a i + t i | i ∈ I} is a (maximal) linearly independent subset of A. (3) Every linearly independent set extends to a maximal linearly independent set. In particular, every torsion-free element in A is contained in a maximal linearly independent subset. (4) If {a i | i ∈ I} is a maximal linearly independent subset of A, then F = i∈I Za i is a full free subgroup of A. Conversely, if F = i∈I Za i is a full free subgroup of A, then {a i | i ∈ I} is a maximal linearly independent subset of A.
Proof. Maximal linearly independent subsets exist by Zorn's Lemma.
The rest consists of easy and well-known observations.
We always assume that A 0 = {0}, i.e., we assume that A is not a torsion group. The dual T ∧ of a torsion group T is a compact totally disconnected group. Theorem 1. For A ∈ AG, the family F := F (A) has the following properties.
(1) Let F, F ∈ F . Then F ∩ F ∈ F . (2) If F ∈ F , F ≤ F and F/F is a torsion group, then F ∈ F . (3) If F ∈ F , then ∀ m ∈ N : mF ∈ F and m mF = {0}. (5) Follows from (1). (6.1) We first assume that A is torsion-free. Then the multiplication µ A m is injective, and µ A m : A → mA is an isomorphism. Thus, It remains to show that A/F is a torsion group. As A 0 /F 0 is torsion and Remark 1. In general, F (A) is not closed under finite sums, so F (A) may not be a lattice, and therefore, A = ∑ F (A) may not be the directed union (direct limit) of its members. However, for A = A 0 , using Theorem 1(6) (with tor(A) = {0}), given F ∈ F (A), also the larger m −1 A F is a full free subgroup, and as A/F is a torsion group, we obtain an ascending chain of full free subgroups of A whose union is A.
In the case of a torsion-free group A of finite rank, the quotients A/F for F ∈ F (A) are somewhat alike ( [2], Theorem 3.5(9)). For arbitrary rank there is a great variety of quotients A/F. Proposition 3. Let A be an abelian group of infinite rank m. Let F ∈ F (A). Then rk(F) = |F| = m. Let T be any torsion group that is m-generated. Then there is an epimorphism ϕ : F T with F ϕ := Ker(ϕ) ∈ F (A), and there is an exact sequence T Proof. Routine and simple.
In the case of infinite rank, the sum of two full free subgroups need not be free, as shown by Jim Reid (( [20], Theorem 2.2)): Theorem 2. Let A be a torsion-free group of infinite rank.
(a) (( [20], Theorem 2.2) and its proof) Given a free subgroup F of A with rk(F) = rk(A), there is a second free subgroup F 1 such that A = F + F 1 . (b) ( [20], Corollary 3.5) There exists a full free subgroup F 0 of A such that A/F 0 is divisible (A is "quotient divisible").
One can deduce from (a) that in a torsion-free group A of infinite rank every non-free subgroup of torsion index is the sum of two full free subgroups.

Definition 1. An abelian group
Theorem 2(a) yields: Theorem 3. A ∈ AG is F -summable if and only if A is either torsion-free of finite rank or is free of arbitrary rank.
Proof. If tor(A) = {0}, then there are full free subgroups whose sum contains torsion elements (Lemma 4(2)). So a summable group must be torsion-free.
Suppose that A is torsion-free and F -summable of infinite rank. Then A is the sum of two free subgroups and hence of two full free subgroups. As A is summable, it is free. The converse is clear.
If the torsion-free group A has finite rank, then full free subgroups are finitely generated and finitely generated torsion-free subgroups are free.

The Semi-Lattices F (A) F (A) F (A) and D(G) D(G) D(G)
Let A ∈ AG and G = A ∧ . Then G is compact, not necessarily connected. Let F ∈ F (A).
Then F ins A α A/F is exact where α is the natural epimorphism. Therefore, Let G be a compact group and A = G ∧ . Then A is a possibly mixed group. Let For a general topological abelian group G, the family Lat(G) of closed subgroups is a lattice with the operations C 1 ∧ C 2 = C 1 ∩ C 2 and C 1 ∨ C 2 = C 1 + C 2 . There also exist greatest lower bounds and least upper bounds for infinite families: Let C be a family of closed subgroups of G. Then C is a closed subgroup of G and C = C. The subgroup ∑ C is closed and ∑ C = C. See ([1], p. 361).
We will establish that F (A) and D(A ∧ ) are anti-isomorphic semi-lattices. To do so, we use results of ([1], p. 351) where we find annihilators H ⊥ defined as follows.
For G ∈ LCA, we have the pairing Note that X ⊥⊥ ⊆ G is not the same as (G ∧∧ , X ⊥ ) = (G ∧∧ , (G ∧ , X)). However, they are topologically isomorphic: Lemma 6. Let A ∈ LCA. Then, for X ⊆ A, the natural evaluation isomorphism η A : A → A ∧∧ restricts to an isomorphism X ⊥⊥ → (A ∧∧ , X ⊥ ), η A X ⊥⊥ = (A ∧∧ , X ⊥ ). In particular, X ⊥⊥ is a full free subgroup of A if and only if (A ∧∧ , X ⊥ ) is a full free subgroup of A ∧∧ .
Theorem 5. Let A ∈ AG and G = A ∧ . The lattice anti-isomorphism H → H ⊥ of Theorem 4 restricts to an anti-isomorphism of semi-lattices δ : F (A) → D(G). In particular we have: We now establish, for a compact group G = A ∧ , the properties of D(G) corresponding to the properties of F (A). Recall that for any m ∈ N and any subgroup Y of X, we have Definition 2. For a compact abelian group G set ∆(G) := ∑ D(G).
We collect here some properties of the subgroup ∆(G), "Fat Delta".
The family D := D(G) has the following properties.
Proof. (1) Theorem 5 establishes the semi-lattice property. As D is closed under finite sums, we have ∑ D = D.
(2) By ( (3) We have the following commutative diagram with natural maps and exact rows We conclude that Furthermore, δ(F) and G are both compact and hence, so are mδ(F) and mG, therefore closed, and equal to the closures.
The fact that linearly independent sets can be enlarged to maximal linearly independent sets has the following dual.

Proposition 4. Let G = A ∧ be a compact abelian group of infinite dimension.
Suppose that Θ is a subgroup of G such that G/Θ is a torus of dimension m. Then Θ contains some ∆ ∈ D(G) and m ≤ dim(G).
T where T is a torus of dimension m, we conclude the exact sequence As T ∧ is free of rank m as the dual of a torus, so is E and m ≤ rk(A). Let F be a full free subgroup containing E. Then ∆ := F ⊥ ∈ D(G) and Θ = E ⊥ ⊇ F ⊥ = ∆.
Let G = A ∧ . We next study the connection between D(G), D(c(G)), ∆(G), and ∆(c(G)). Given A, let T = tor(A) and let F ∈ F (A). Then, we obtain the following commutative diagram with exact rows and its dual.
We now set We have the following easy consequences.
(6) With the established notation ∆(c(G)) is divisible and hence algebraically a direct summand of ∆(G).
(2) c(G) is divisible as the dual of a torsion-free group. The rest is evident.
(5) Follows immediately from (3) and (4). (6) c(G) is divisible and ∆(c(G)) is pure in c(G). Hence, ∆(c(G)) is divisible. (7) We have the following commutative diagram with exact row and natural maps: A number of results on free subgroups are worth dualizing. Theorem 8. Let G = A ∧ be a compact abelian group of infinite dimension.
(1) Suppose that D is a closed subgroup of G such that G/D is a torus. Then there exists By enlarging F 1 if necessary we may assume that F 1 is maximal, i.e., full free. There exists Corollary 2. Let G = A ∧ be a compact connected abelian group of infinite dimension, i.e., A is torsion-free of infinite torsion-free rank.
(1) Suppose that D is a subgroup of G such that G/D is a torus. Then there exists a subgroup D of G such that D ∩ D = 0 and G/D is a torus. In particular, for every There exists a torsion-free ∆ ∈ D(G).
We can easily settle the question when ∆(G) is as small as possible, i.e., ∆(G) = tor(G).
Proof. We only need to consider the consequences of ∆(G) being a torsion group. As ∆(G) = ∑ D(G), this occurs if and only if every ∆ ∈ D(G) is a torsion group. Since ∆ is compact, it must be bounded torsion. Furthermore, we use that for every F ∈ F (A), the dual (A/F) ∧ is topologically isomorphic to some ∆ ∈ D(G), so a bounded torsion group.
(a) Assume first that A is torsion-free. By Corollary 2(2) we have n := rk(A) < ∞. Now pick an arbitrary F ∈ F (A). Since (A/F) ∧ is a bounded torsion group, so is A/F, hence, mA ⊆ F for some m ∈ N, so A ∼ = mA is free of rank n and G ∼ =t T n .
(b) In the general situation, by Theorem 7(3), ∆(c(G)) must be a torsion group and hence by (b), A/ tor(A) must be free of finite rank. So A = F ⊕ tor(A) for some finite For the latter group to be torsion, it must be bounded. For any A ∈ AG, and a short exact sequence K As for F (A) and D(G) it follows that κ is a bijective map satisfying κ( Remark 3. One can ask further whether a compact group has other connected factors of dimension 1 (so-called solenoids of which T is an example). For finite dimensional connected compact groups this leads to the "Main Decomposition" that was derived in [22].

The Fat Delta of Compact Groups
So far we know from Theorem 6 that for any compact abelian group G, In this section, we will establish further properties of Fat Delta. We start with a preliminary observation.

Lemma 7.
Let G and H be compact abelian groups and let α : G → H be a continuous epimorphism. Then we have: (1) If G is totally disconnected, then so is H.
(2) If G is a torus, then so is H.
(2) Now suppose G is a torus. Then G ∧ is free, so since subgroups of free groups are free, H ∧ is free. Thus, H is a torus.
The significance of Proposition 5 is that it shows that for a compact group G Fat Delta ∆(G) coincides with the subgroup td(G) that is defined and motivated by totally different considerations (see Definition 4 and Proposition 9(2)). For the sake of easy reference, we list the results that could be proved easily in the present context but are proved in greater generality in the exhaustive study of td(G) in Section 7. Proposition 6. Let G be a compact abelian group. Then the following are true.
(1) ∆(G) is zero-dimensional, in particular totally disconnected (Theorem 19). Consequently, if G is not totally disconnected, then G = ∆(G) and hence ∆(G) is not a locally compact subgroup of G. (2) Any countable extension of ∆(G) is zero-dimensional (in particular totally disconnected) as well (Proposition 11).

Lemma 8.
Let G, H, K be topological abelian groups.
(1) Suppose that H is a topological subgroup of K such that for all f ∈ cHom(G, K) we have f [G] ⊆ H. Let ins : H → K be the insertion. Then ins * : cHom(G, H) → cHom(G, K) : ins * ( f ) = ins • f is a topological isomorphism.
(2) Suppose that H α K β L is a short exact sequence in TAG, α is proper, and G is some other topological group. Then is an exact sequence in TAG and α * is proper. The map β * is not claimed to be surjective.
C be a short exact sequence of discrete groups and let G be a divisible topological group. Then is an exact sequence of topological groups. In addition, β * is proper. (4) For a discrete torsion group T = p∈P tor p (T), we have cHom(T, Q/Z) ∼ =t T ∧ , the topological isomorphism being ins * , and T ∧ ∼ =t ∏ p∈P (tor p (T)) ∧ where (tor p (T)) ∧ ∼ =t Hom(tor p (T), Z(p ∞ )).

Proof. (1) It is evident that ins * is bijective and maps
(2) By standard discrete homological algebra We show next that our sequence is exact at cHom(G, K). As β * • α * = (β • α) * = 0 we have Im(α * ) ⊆ Ker(β * ). To show that Ker(β * ) ⊆ Im(α * ), let f ∈ Ker(β * ). By the discrete exactness there exist g ∈ Hom(G, H) such that f = α • g. To conclude, we need to show that g is continuous. To do so let U be open in H. By assumption α is proper, hence, there is an open set It remains to show that α * is proper. Let Then there is f ∈ cHom(G, H) such that g = α • f . We show that f ∈ W(C, U). In fact, As α is injective it follows that f [C] ⊆ U, i.e., f ∈ W(C, U).

Lemma 9.
Let K, X, Y, K , X , Y be topological abelian groups. It is assumed that the diagram is commutative, all maps are continuous, its rows are exact, ξ X is proper, β is a quotient map, i.e., β is open, and ξ K is an isomorphism. Then α is a quotient map.

Proof.
Let U be open in X. As ξ is proper, there is an open set V of X such that There exists x ∈ X such that α(x) = y. Hence, β(ξ X (x)) = η(α(x)) = β(v), and thus, v − ξ X (x) ∈ Ker(β). It follows that there exists k ∈ K such that ξ K (k) = v − ξ X (x) and so ξ X (k + As ξ is injective it follows that k + x ∈ U and α(k + x) = α(x) = y.
We have the proper short exact sequence of topological groups where, as usual, T is the quotient group of R, Q/Z the subgroup of T, and R/Q carries the quotient topology which is indiscrete as Q is dense in R (Lemma 2(1)).
Let A be a discrete group and F a full free subgroup of A of rank m := rk(A). We We obtain a diagram as follows.
(1) By standard discrete homological algebra the diagram is commutative and rows and columns are exact. (2) All the domains of the Hom groups carry the discrete topology, hence cHom = Hom in all cases.

Hom(A/F, T)) is a delta subgroup of G. As F was arbitrary it follows that ∆(G) ⊆ ins * [Hom(A, Q/Z)]. It remains to show that ∆(G) ⊃ ins * [Hom(A, Q/Z)].
Let f ∈ Hom(A, Q/Z) and set K = Ker( f ). Then f [A] ⊆ Q/Z is a torsion group, so A/K is a torsion group and any full free subgroup F of K is a full free subgroup of A. Let F be so given. Then g : A/F → Q/Z : g(a + F) = f (a) is a well-defined homomorphism and f = g • ϕ F = ϕ * F (g), so f ∈ ∆ ⊆ ∆(G).
We will use the following well-known result below. Theorem 11 ([24], page 86, Corollary 8.48). Let G, C be Hausdorff abelian groups, assume that C is complete, H is a dense subgroup of G. Then every morphism f : H → C has a unique extension f : G → C.
which is also extended by id G , hence, by uniqueness we have Theorem 12 and Corollary 3 imply that G → ∆(G) is a category equivalence on the category of compact abelian groups to the category of all ∆(G). This calls for a useful characterization of the class of topological groups that appear as ∆(G) for some compact abelian group G. So far we can say the following. If D is a topological group such that D ∼ =t ∆(G) for some compact group G, then the following are true.
(1) D is totally disconnected and zero-dimensional (Proposition 11). Given a group D with all the required properties, we would have ∆( D) ∼ =t D, i.e., the completion functor is the inverse of the functor ∆.
Theorem 12 and the preceding discussion suggest to study the structure of ∆(G) for a given compact group G. We will attempt this below in the simplest possible case of solenoids. A solenoid is a compact connected group of dimension 1, i.e., the dual of a torsion-free group of rank 1. To do so, we will use a simple result on divisible hulls of discrete groups and Lemma 10 on divisible hulls of certain products of groups.
Lemma 10. Let P be a set of prime numbers, X p be discrete groups and X = ∏ p∈P X p . For each p ∈ P, let D p be a divisible hull of X p . Let D := ∏ p∈P D p . Assume that each D p /X p is a pprimary group. Let D(X) be a subgroup of D containing X such that D(X)/X = tor(D/X). Then (1) D(X) is a divisible hull of X, is torsion-free, hence D(X) is pure in D and therefore divisible. It remains to show that X is essential in D(X). For any prime q, we have (2) Let (d p ) ∈ D(X). Then m(d p ) ∈ X for some m = 0 which requires that ∀ p ∈ P : md p ∈ X p . Our hypotheses imply that d p ∈ X p for all those p that do not divide m. So D(X) ⊆ ∏ loc p∈P (D p , X p ) and equality is evident. (3) The map ξ : D(X) → p∈P D p /X p : ξ((d p )) = ∑ p∈P d p + X p is evidently welldefined, surjective, and Ker(ξ) = X.
Torsion-free groups A with rk(A) = 1, rank-one groups for short, are discussed and classified in ( [17], Chapter 12, Section 1). These are exactly the groups isomorphic with additive subgroups of Q containing Z. Types are equivalence classes [(h p ) p∈P ] of "height sequences" (h p ) p∈P where 0 ≤ h p ≤ ∞. Two height sequences are equivalent if they differ only at finitely many places where both sequences have finite entries. For the precise definition of type see Lemma 11 (1) or ( [17], p. 409, 411).
Two rank-one groups are isomorphic if and only if their types are equal. Lemma 11 displays a representative rank-one group, its type, and dual solenoid. For a prime p, we define 1 Proof. (1) Given p ∈ P either A contains every fraction 1/p k (in which case h p = ∞) or A contains a smallest fraction 1/p h p . These fractions generate A and determine the type of A.
We illustrate the situation with some special cases.
The next theorem deals with the general case. The relevance of the final assertion will become clear in Section 7.3 (see Definition 9 and Example 4, see also Problem 2). Theorem 13. Let A = ∑ p∈P 1 p hp Z. Define Σ = A ∧ and P ∞ as above, and let P fin := {p | 0 < h p < ∞} and P 0 := {p | h p = 0}.
Moreover, Soc(Σ) is dense in Σ if and only if P 0 is infinite. Recall that Soc(Σ) = p∈P 0 Z(p) ⊕ p∈P fin Z(p), where Z(p) = Σ[p] when the latter is non-trivial. Let φ : for some x ∈ Σ and p ∈ P 0 , then h p = 0 and pt = 0 in Σ/∆, so px ∈ ∆. It follows from the above description of ∆ that ∆ is p-divisible for p ∈ P 0 .

Resolutions
The Resolution Theorem, a structure theorem for compact abelian groups, first appeared in [25] and later in an extended form in ( [1], Theorem 8.20, p. 420), where it got its name.
We can now recall the original Resolution Theorem.
(2) Ker(ϕ) is algebraically and topologically isomorphic to Γ := exp −1 [∆], and Γ is a closed totally disconnected subgroup of L(G). In particular, it does not contain any nonzero vector spaces.
, the identity component of G. We first revisit the classical Resolution Theorem for compact connected groups of finite dimension with substantial additions as we determine the kernel of the resolution map ϕ explicitly up to topological isomorphism (see (4)).

Theorem 14 (Resolution Theorem). Let G be a compact abelian group of finite dimension n
(1) ϕ is surjective, continuous, and open.
(4) Γ ∼ =t Z n where Z n carries the discrete topology, i.e., the subspace topology in R n .
(5) It is routine to verify that exp . We obtain the exact sequence  1(a), p. 28), and in particular totally disconnected. This is possible only when the quotient ∆/Z ∆ is trivial. Therefore, Z ∆ = ∆.

Remark 5. (a)
For the torus G = T n one has L(G) = R n , so the Resolution theorem applied to G is simply the covering homomorphism ϕ : R n → T n if one takes ∆ = 0 (in general ∆ must be a finite subgroup of T n ). (b) Using the fact that exp[L(G)] = a(G), the covering map ϕ could be replaced by the surjective, continuous, and open map ψ : ∆ × a(G) → G : ϕ(d, x) = d + x, for (d, x) ∈ ∆ × a(G) which has the advantage that now both groups ∆ and a(G) are subgroups of G. One has to take into account that the map exp G : L(G) → a(G) need not be injective. More precisely, K(G) = Ker(exp) is trivial precisely when G is torus-free. However, even when G is torus-free, this map is only a continuous isomorphism that need not be a homeomorhism.
(c) As an application of Theorem 14 we obtain a nice presentation of the solenoid Σ 2 = Q ∧ from Example 1 (2). As shown there, Σ 2 has a delta subgroup ∆ = Z = ∏ p∈P Z p and Σ 2 /∆ ∼ = T. So by Definition 3 (iii), L(Σ 2 ) ∼ =t R. Hence, Theorem 14 gives a resolution ϕ : The same representation can also be obtained directly by standard use of Pontryagin duality. Indeed, let 1 = (1 p ) p∈P ∈ ∆ and u = (1, −1) ∈ ∆ × R. Then u ∼ =t Z and K = (∆ × R)/ u is a compact connected torsion-free group of dimension one, so its dual K ∧ is a discrete divisible torsion-free group of rank one. Therefore, K ∧ ∼ = Q and K ∼ =t Q ∧ .
We also obtain a "canonical resolution", where the arbitrary ∆ ∈ D(G) is replaced by the canonical subgroup ∆(G).
By (2) Γ is torsion-free. We will show that Γ is divisible and Γ/Γ ∆ is a torsion group. This says that Γ is the usual algebraic divisible hull of Γ ∆ ∼ =t Ze 1 ⊕ · · · ⊕ Ze n ⊂ L(G) (see the proof of item (4) of Theorem 14).
In the next example, we apply the canonical resolution theorem 15 to two solenoids. The first one is T = Z ∧ and its canonical resolution adds nothing essentially new.  Remark 5(c)). Theorem 15 gives the canonical resolution ϕ : Denote by Q the group ∆(Σ 2 ) equipped with the finer topology obtained by taking ∆ as an open topological subgroup of Q. Then Q is a locally compact ring and A := Q × R is the adele ring of Q. Composing ϕ with the identity A → ∆(Σ 2 ) × R we obtain a continuous surjective homomorphism ϕ : A → Σ 2 which is again open by the Open Mapping Theorem (as A is σ-compact). Hence, Σ 2 is a quotient of A.

Fat Delta Through the Looking Glass of Quasi-Torsion Elements
Fat Delta existed previously in the literature in a rather different form and in greater generality. In Section 7.1 we recall the definition of quasi-torsion element and the subgroup td(G) of quasi-torsion elements, showing that td(G) = ∆(G) for compact groups (Proposition 9).

Quasi-Torsion Elements
Definition 4 (( [3], p. 127), [4]). Let G be a Hausdorff abelian topological group. Define td(G) to be the set of all quasi-torsion elements of G, where x ∈ G is quasi-torsion if x is either finite or its subspace topology is non-discrete and linear.
This definition was given by [4] for arbitrary, not necessarily abelian, topological groups. Then td(G) need not be a subgroup of G, as the following example shows. Example 3. Take the compact group G = SL 3 (R) of rotations of R 3 . Then td(G) = tor(G) is the set of all torsion elements of G, while the subgroup td(G) generated by td(G) is the whole G since td(G) is invariant under conjugations and G is a simple group. A geometric proof of the equality td(G) = G is based on the well-known fact that every rotation can be presented as a composition of two symmetries (known to have order 2).

Remark 6.
If every convergent sequence is eventually constant in a topological abelian group G, then td(G) = tor(G) (the assumption td(G) = tor(G) leads to a contradiction: if x ∈ td(G) \ tor(G), then the group x is non-discrete and metrizable, so x has convergent sequences that are not eventually constant).
Infinite compact groups always have convergent sequences that are not eventually constant (since they contain copies of the Cantor set {0, 1} ω ). An example of an infinite precompact abelian group where every convergent sequence is eventually constant can be obtained as follows. For a TAG-group (G, τ) the Bohr topology of (G, τ) is the initial topology τ + of all χ ∈ (G, τ) ∧ (that can be obtained by the diagonal embedding G → T G ∧ ). For the sake of brevity we also write G + for (G, τ + ). In case τ is discrete, G + is usually denoted by G # . It is a well-known fact that in G # every convergent sequence is eventually constant ( [3]), so td(G # ) = tor(G # ). Proposition 9. Let G be a topological abelian group.

1.
If x ∈ G, then x ∈ td(G) if and only if there exists a continuous homomorphism f : td(G) is a subgroup of G containing every compact totally disconnected subgroup of G; 3.
If G is complete (in particular, locally compact), then td(G) coincides with the union of all compact, totally disconnected subgroups of G.
Proof. (1) Assume that x ∈ td(G). If x is finite, then x is isomorphic to a quotient group of (Z, ν Z ), so the desired homomorphism f is easy to obtain. If x is infinite and carries a non-discrete linear topology, then the homomorphism f : (Z, ν Z ) → G with f (1) = x is obviously continuous. On the other hand, if there exists a continuous homomorphism f : (Z, ν Z ) → G with f (1) = x, then the subgroup x is either finite or has linear precompact topology, so x ∈ td(G).
If N is a compact, totally disconnected subgroup of G, then N has a linear topology. Therefore, for every x ∈ N, the subgroup x is either finite or its subspace topology is linear and non-discrete (as otherwise x it would be a closed (so compact) discrete subgroup of N, a contradiction). Therefore, x ∈ td(G).
(3) Assume now that G complete and x ∈ td(G). Then x is quasi-torsion and x is either finite or its subspace topology is non-discrete and linear. Hence, its closure x is the completion of x , and thus, compact and totally disconnected.
For a compact group G = A ∧ , by Proposition 5 and Proposition 9(2), we have td(G) = ∆(G), and by Theorem 10 ∆(G) = Hom(A, Q/Z). We summarize: We quote from previous papers reconfirming foregoing results.    (3) showing that it remains valid in the larger category TAG. Now we use item (1) from Proposition 10 to show that the subgroup td(G) is zerodimensional when G is precompact, i.e., a subgroup of a compact group. We shall see in [16] that this remains true under the weaker assumption that G is locally precompact, i.e., a subgroup of a locally compact group. Proposition 11. Let G be a precompact abelian group. Then every subgroup H of G with [H : (H ∩ td(G))] < c is zero-dimensional. In particular, td(G) is zero-dimensional.
Proof. The following folklore fact will be needed in the sequel: Proof. H is either finite of dense. If H is finite then it is clearly zero-dimensional. If H is dense, then for any fixed a ∈ T \ H also a + H is dense and disjoint from H. Hence, First, we show that χ[td(G)] ⊆ Q/Z for any χ ∈ G ∧ . Assume that x ∈ td(G), to check that χ(x) ∈ Q/Z pick an arbitrary χ ∈ G ∧ . Then χ(x) ∈ td(T), by Proposition 10(1). The first supply of non-compact (totally) minimal groups was obtained by means of the following notions of "strong" density: Clearly, totally dense subgroups are dense (while Z(p ∞ ) is dense in T, but not totally dense). Obviously, the totally dense subgroups have the following weaker property: Definition 7 ([5,11,15]). A subgroup H of a topological abelian group G is topologically essential if N ∩ H = {0} for every non-trivial closed subgroup N of G.
The term used for this property in [5,11,15] and in the remaining literature on the Open Mapping Theorem is "essential", but we prefer the more precise term "topologically essential" to avoid possible confusion. Theorem 17. Let H be a dense subgroup of a compact abelian group G.
(a) ( [11,15]) H is minimal if and only if H is topologically essential in G. (b) ( [28]) H is totally minimal if and only if H is totally dense in G.
Banaschewski [5] found the following general criterion: if H is a dense subgroup of a topological abelian group G, then H is minimal if and only if G is minimal and H is topologically essential in G. These criteria match perfectly the following remarkable result of Prodanov and Stoyanov [14] proved at a later stage, but conjectured by Prodanov in 1972 (see [13] for an earlier partial result in the totally minimal case): Theorem 18 (Prodanov-Stoyanov Theorem). Minimal abelian groups are precompact.
This theorem allows one to use exclusively the form of the criteria given in Theorem 17, so to reduce the study of the (totally) minimal abelian groups to that of the dense topologically essential (resp., totally dense) subgroups of the compact abelian groups. This explains the interest in topologically essential or totally dense subgroups of the compact abelian groups. Proposition 12 ([11]). The minimal topologies on Z are precisely the p-adic topologies.
It was proved in [9] that the 2-adic topology of Z is minimal.
Proof. Assume that τ is a minimal topology on Z and let K be the completion of (Z, τ). By the Prodanov-Stoyanov Theorem the group K is compact. By Theorem 17(a), Z is essential in K, hence K is torsion-free. Therefore, the dual of K is a discrete divisible group [1,3,23,29], hence a direct sums of copies of Q and of Z(p ∞ ), p ∈ P. Therefore, K = (Q ∧ ) α × ∏ p Z β p p . Again by Theorem 17(a), Z must be essential in this product, hence only one of these cardinals α, β p can be non-zero, and it must be equal to 1. Since Q ∧ has a Delta subgroup isomorphic to ∏ p Z p , again Theorem 17(a) implies that α = 0. In other words, K ∼ = Z p for some prime p, therefore, τ coincides with the p-adic topology on Z. To conclude, the minimality of the p-adic topology follows from Theorem 17(a), since Z is essential in K = Z p , as all non-trivial closed subgroups of K are open.
A similar argument shows that Q n admits no minimal topologies for 0 < n < ∞. The functorial subgroup td(G) of a compact abelian group G is not only dense in G (Theorem 6(2)), but it is totally dense in G, as the next proposition shows. Proposition 13. Let G be a compact abelian group. Then td(G) is totally dense in G.
Proof. Let N be a closed subgroup of G. Then N ∩ td(G) = td(N) by Proposition 10. Therefore, it suffices to check that td(G) is dense in G for every compact group G. This follows from Theorem 6, but we prefer to give an independent proof here.
Let N := td(G). Applying to the closed subgroup N of G the exactness of td in the sense of Proposition 10(2), we deduce that td(G/N) = {0}. To see that this implies G/N = {0} and so N = G, consider the discrete dual X = (G/N) ∧ and assume by way of contradiction that X = {0}. Then there exists a subgroup Y of X such that X/Y = {0} is torsion. Then Y ⊥ ∼ = (X/Y) ∧ is a non-trivial compact totally disconnected subgroup of G/N, so td(G/N) = {0}, a contradiction.
We obtain the following theorem which, among other things, reconfirms that ∆(G) is dense in G when G is compact. Theorem 19. Let G be a compact abelian group. Then td(G) is a dense totally minimal zerodimensional subgroup of G.
Proof. Proposition 13 ensures the total density (hence, density as well) of td(G). Total minimality of td(G) is then an immediate consequence of Theorem 17. To prove that td(G) is zero-dimensional, apply Proposition 11.
Since td(G) = G when G is not totally disconnected, this theorem provides a universal example of a non-compact totally minimal (and zero-dimensional) abelian group. This explains why it is not surprising that most of the first known examples of non-compact totally minimal groups known in the seventies were just Q/Z = td(T) ( [15]), (Q/Z) n = td(T n ) ( [9]), (Q/Z) N = td(T N ) ([10]), and (Q/Z) α = td(T α ) ( [27,30]).

Sylow Subgroups of td(G) for G ∈ TAG
The characterization in Theorem 9 of the compact abelian groups G with td(G) = tor(G) gives a very narrow class (practically rather close to the class of Lie groups). This shows that the restraint td(G) = tor(G) is too stringent, or from another point of view, the subgroup td(G) is too large to be useful in certain circumstances. This is why here we recall a smaller subgroup of td(G) containing tor(G) that still keeps the advantages of td(G), but it is closer to tor(G). This subgroup is simply the subgroup generated by all topologically p-Sylow subgroups td p (G) of td(G) defined as follows: Definition 8 ( [3,31]). An element x of a topological abelian group G is topologically p-torsion if p n x → 0. Let G p := {x ∈ G | x is topologically p-torsion} and let td p (G) := (td(G)) p .
Then G p is a subgroup of G. In case G is a profinite group, G p is usually called the topological p-Sylow subgroup of G. We shall also keep this terminology when G is not necessarily profinite. Clearly, H p = G p ∩ H for a subgroup H of G.
Obviously, tor p (G) ≤ td p (G) ≤ G p for every G. The notation td p (G) used in Definition 8 is borrowed from [4,27], where td p (G) denotes the subgroup of all elements x ∈ G (called quasi-p-torsion in [4]) such that x is either a cyclic p-group, or x is isomorphic to Z equipped with the p-adic topology.
The equivalence of both definitions follows from: if x ∼ = Z is equipped with a Hausdorff linear topology such that p n x → 0, then this linear topology necessarily coincides with the p-adic topology.
The sum ∑ p td p (G) is direct ( [4]). Following [4], we write wtd(G) = p∈P td p (G) in the sequel. Clearly, tor(G) ≤ wtd(G) ≤ td(G), but these subgroups need not coincide in general. It is proved in [4] that, when G is compact, even the smaller subgroup wtd(G) is still totally dense in G. Since both total density and topological essentiality are transitive properties, a dense subgroup G of a compact abelian group K is totally dense (resp., topologically essential) in K if and only if td(G) = G ∩ td(K) is totally dense (resp., topologically essential) in td(K) if and only if wtd(G) is totally dense (resp., topologically essential) in wtd(K). The next theorem from [29] shows that one can characterize the totally disconnected compact abelian groups in the class of all compact abelian groups G by specifying whether the subgroups td p (G) of G are closed (compact) or not: Theorem 20 ([3,29]). For a compact abelian group G and every prime p the subgroup td p (c(G)) is dense in c(G). In particular, the following conditions are equivalent: Following [12], call a compact group almost countable if it is the completion of countable minimal abelian group. This class of compact groups was described by Prodanov [12] as follows: a compact abelian group G is almost countable if and only if n = dim(G) < ∞ and there exists a homomorphism f : G → T n such that Ker f = ∏ p ( Z e p p × F p ), where F p is a finite p group and e p ∈ {0, 1} for every prime p. These are the compact abelian groups G such that td(G) has a countable essential subgroup.
The larger class K of compact abelian groups, that contain copies of the group Z N p for no prime p was studied in [6]. It is stable under extension and contains all almost countable compact groups, as well as all exotic tori. Its subclass of compact groups G that contain copies of the group Z 2 p for no prime p coincides with the completions of minimal abelian groups of countable rank, or equivalently, these are the compact abelian groups G such that td(G) has an essential subgroup of countable rank (see [3] or [6]).
∧ , so ∆ p ∼ =t Z p when p ∈ P ∞ and ∆ p is a cyclic p-group otherwise.
(2) It follows from (1) that there are c many pairwise non-isomorphic connected one-dimensional exotic tori G; they all have wtd(G) ∼ = Q/Z, according to Corollary 5. Nevertheless, for these exotic tori G the subgroups wtd(G) remain pairwise non isomorphic (since, similarly to Theorem 12, if wtd(G) ∼ =t wtd(H), then G ∼ =t H for every pair of compact abelian groups G, H). (3) According to Theorem 13, if G is an exotic torus, then Soc(G) is dense in G if and only if P 0 is infinite (see ( [32], Proposition 2.5) for a more general result in the case of connected exotic tori of arbitrary dimension). According to Theorem 21, in this case, Soc(G) is the smallest dense topologically essential subgroups of G. (4) The second assertion in (3) is related to the following more general fact proved in ( [33], Theorem 5.1) justifying the interest in dense socles: a connected compact abelian group G contains a smallest dense topologically essential (i.e., smallest dense minimal) subgroup of G if and only if G is an exotic torus with dense Soc(G).

Final Comments and Open Problems
One can deduce from Lemma 11(2) that for a solenoid Σ all delta subgroups ∆ of Σ have the property that all subgroups of finite index of ∆ are open. Problem 1. Classify the compact abelian groups whose delta subgroups have the property that all their subgroups of finite index are open.
If G = A ∧ is a finite-dimensional compact connected abelian group, one can easily extend the argument in the proof of Theorem 13 and prove that Soc(G) is dense in G if P 0 (G) is infinite, where P 0 (G) is defined in this more general case as follows (a different proof in case G is an exotic torus can be found in ( [32], Proposition 2.5)). Let n = dim G, then there exists a short exact sequence Z n A A/Z n , where A/Z n is torsion (actually, isomorphic to a subgroup of (Q/Z) n ). In this notation, P 0 (G) = {p ∈ P : rk p (A/Z n ) = 0}. Obviously, P 0 (G) = P 0 , as defined in Theorem 13, when n = 1. The following example shows that when dim G > 1, infinity of P 0 (G) is not a necessary condition for the density of Soc(G).
Example 5. Split P = π 1 π 2 in two disjoint infinite subsets π 1 , π 2 (e.g., take π 1 to be the set of all primes of the form 4k + 1). For i = 1, 2 define the rational group A i = 1/p : p ∈ π i and the solenoid Σ i = A ∧ i . Then both Σ 1 and Σ 2 have dense socles, by Theorem 13, so G = Σ 1 × Σ 2 has dense socle as well. Nevertheless, P 0 (G) = ∅. Problem 2. Find a criterion for density of Soc(G) for a finite-dimensional compact connected abelian group G.
Author Contributions: Each author contributed to every aspect of the research of, and the writing of, this paper. All authors have read and agreed to the published version of the manuscript.
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