Minimal pseudocompact group topologies on free abelian groups

MINIMAL PSEUDOCOMP A CT GR OUP TOPOLO GIES ON FREE ABELIAN GR OUPS DIKRAN DIKRAN JAN, ANNA GIORDANO BR UNO, AND D MITRI SHAKHMA TO V De dic a te d to R ob ert L owen on the o c c asi on of his 60th anniversary Abstra ct. A Hausdorff top ological gro up G is minimal if every con t inuous isomorphism f : G → H b etw een G and a Hausdorff topological group H is o p en. Significantly strengthening a 1981 result of Stoy anov, we prov e th e follo wing theorem: F or ever y infinite minimal ab elian group G th ere exists a sequ ence { σ n : n ∈ N } of cardinals such that w ( G ) = sup { σ n : n ∈ N } and sup { 2 σ n : n ∈ N } ≤ | G | ≤ 2 w ( G ) , where w ( G ) is th e weigh t of G . If G is an infinite minimal ab elian group, then either | G | = 2 σ for some cardinal σ , or w ( G ) = min { σ : | G | ≤ 2 σ } ; moreo ver, the equalit y | G | = 2 w ( G ) holds whenever cf ( w ( G )) > ω . F or a cardinal κ , we denote by F κ the free ab elian group with κ many generators. If F κ admits a pseudo compact group t op ology , th en κ ≥ c , where c is the cardinality of the conti nuum. W e show that th e existence of a minimal p seudo compact group top ology on F c is equiv alen t to the Lusin’s Hyp othesis 2 ω 1 = c . F or κ > c , we pro ve th at F κ admits a (zero-dimensional) minimal pseudo compact group top ology if and only if F κ has b oth a minimal group top ology and a pseudo compact group top ology . If κ > c , t hen F κ admits a connected m in imal pseudocompact group top ology of weig ht σ if and only if κ = 2 σ . Finally , w e establish that no infinite torsion-free ab elian group can b e equipp ed with a locally connected minimal group top ology . Thr oughout this p ap er al l top olo gic al gr oups ar e Hausdorff. W e denote by Z , P and N resp ectiv ely the set of in tegers, the set of primes and the set of natural n umbers. Moreo v er Q d enotes the group of rationals and R the group of r eals. F or p ∈ P the symb ol Z p is used for the group of p -adic int egers. The symb ol c stands for the cardinalit y of the conti nuum. F or a top olo gical g roup G t he s ym b ol w ( G ) s tands for the weigh t of G . The P ontry agin dual of a top ol ogical ab elian group G is d en oted b y b G . If H is a group and σ is a cardinal, then H ( σ ) is used t o denote the dir ect sum o f σ man y copies of the grou p H . If G and H are groups, then a map f : G → H is called a monomorphism pro v id ed that f is b oth a group h omomorphism and an injection. F or undefin ed terms see [16, 1 7]. Definition 0.1. F or a cardinal κ we use F κ to denote th e free ab elian group with κ man y generators. 1. In troduction The follo wing notion w as introd uced indep enden tly b y Cho qu et (see Do ¨ ıtc hin o v [14]) and S tephenson [24]. Key wor ds and phr ases. minimal group, pseudo compact group, free ab elian group, essential subgroup, connected topology , zero-dimensional top ology . T o app ear in: T opol ogy and its A pplications . The first author was partially supp orted by the SRA grants P1-0292-0101 and J1-9643-0101. The third auth or was partially sup p orted by the Grant-in-Aid for Scientific R esearch no. 19540092 by the Japan So ciety for th e Promotion of Science (JSPS). 1 2 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV Definition 1.1. A Hausdorff group top ol ogy τ on a group G is ca lled minimal pro vided that ev ery Hausdorff group top ology τ ′ on G su c h that τ ′ ⊆ τ sa tisfies τ ′ = τ . Equiv alentl y , a Hausdorff top ological group G is minim al if ev ery co nti nuous isomorphism f : G → H b et we en G and a Hausd orff topological group H is a top ological isomorphism. There exist ab elian group s whic h admit n o minimal group top ologies at all, e.g., the group of rational n u mb ers Q [21] or Pr ¨ ufer’s group Z ( p ∞ ) [11 ]. T his s u ggests the general problem to d etermine the algebraic str ucture of the m in imal ab elian groups, or equiv a- len tly , the follo wing Problem 1.2. [9 , Problem 4.1 ] Describ e the ab elian gr oups that ad mit minimal gr oup top olo gie s. Pro danov solv ed Problem 1.2 first for all free ab elian groups of fin ite rank [20], and later on he impro ved this result extending it to all cardinals ≤ c [21]: Theorem 1.3. [20, 21] F or every c ar dinal κ ≤ c , the gr oup F κ admits minimal gr oup top olo gie s. Since | F κ | = ω · κ for eac h c ardinal κ , un coun table free ab elian groups are determined up t o isomorphism by their cardinalit y . This suggests the problem of c haracterizing the cardinalit y of minimal ab elian groups. The follo wing set-theoretic d efinition is ultimate ly relev an t to this p roblem. Definition 1 .4. (i) F or infinite cardinals κ and σ the symb ol Min ( κ, σ ) denotes the follo wing statemen t: Th er e exists a sequence of cardinals { σ n : n ∈ N } su c h that (1) σ = sup n ∈ N σ n and sup n ∈ N 2 σ n ≤ κ ≤ 2 σ . W e s a y that the sequence { σ n : n ∈ N } as ab o ve witnesses Min ( κ, σ ). (ii) An infin ite cardinal n um b er κ satisfying Min ( κ, σ ) for some infinite card in al σ will b e called a Stoyanov cardinal. (iii) F or the sak e of conv enience, w e add to the class of Sto yano v cardinals also all fin ite cardinals. The cardinals fr om item (ii) in the ab o v e definition were first in tro du ced by Sto y ano v in [25] under the name “p erm iss ible cardinals”. Their imp ortance is evident from th e follo wing fun damen tal result of Stoy ano v provi ding a complete c haracterization of the p ossible cardinalities of m inimal ab elian groups, thereby solving Problem 1.2 for all free ab elian g roups : Theorem 1.5. [25] (a) If G is a minimal ab elian gr oup, then | G | is a Stoyano v c ar dinal. (b) F or a c ar dinal κ , F κ admits minimal gr oup top olo g ies if and only if κ is a Stoyanov c ar dinal. If κ is a finite cardinal satisfying (1), then κ = 2 n for some n ∈ N . On the other hand, ev ery finite group is compact and thus min imal. F urthermore, the group F n admits minimal group top ologies for ev er y n ∈ N by Theorem 1.3. It is in order to include also the case of fi nite group s in T heorem 1.5(a) and fi nitely generated grou p s in T heorem 1.5(b) that we decided to add item (iii) to De finition 1.4. It is worth noting that the co mmuta tivit y of the group i n Th eorem 1.5(b) is important b ecause all restrictions on the cardinalit y disapp ea r in the case of (non-ab elia n) free groups: Theorem 1.6. [23, 22] Every fr e e gr oup admits a minimal gr oup top olo gy. MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 3 F or free groups with infi n itely man y generators this theorem h as b een pr o ved in [23]. The r emaining case w as co ve red in [22]. A subgrou p H of a top ol ogical group G is essential (in G ) if H ∩ N 6 = { e } for ev ery closed normal subgroup N of G with N 6 = { e } , wh ere e is the identit y element of G [20, 24]. This notion is a crucial ingredien t of the so-called “minimalit y criterion”, du e to Pro danov and S tephenson [20, 24], describin g the dense min im al subgroups of co mpact groups. Theorem 1.7. ([20, 24]; see also [10, 12]) A dense sub gr oup H of a c omp act gr oup G is minimal if and only if H is essential in G . A top ological group G is pseudo c omp act if ev ery contin uous real-v alued function d efi ned on G is b ounded [18]. In the spirit of Theorem 1.5(b) c h aracterizing the free ab elian groups admitting minimal top ologies, one can also describ e the free ab elian groups that admit pseudo c omp act group top ol ogies ([5, 13]; see Theorem 4.4 ). The aim of this article is to pro vide simultane ous minimal and pseu d o compact top ologizatio n of free ab elian groups . T o ac hiev e this goal, w e n eed an alternativ e description of S to yano v cardinals obtained in Prop osition 3.5 as we ll as a mo re precise form of Theorem 1.5(a) giv en in Theorem 2 .1. W e fin ish this section with a fund amen tal restriction on the size of p s eudo compact groups d ue to v an Dou w en. Theorem 1.8. [26] If G is an infinite pseudo c omp act gr oup, then | G | ≥ c . 2. Ma in resu l ts 2.1. Cardinality and w eigh t of minimal ab elian groups. Let κ b e a cardinal. Recall that the c ofinality cf ( κ ) of κ is d efined to b e the smallest cardinal κ suc h that there exists a transfinite sequence { τ α : α ∈ κ } of cardin als suc h that κ = s u p { τ α : α ∈ κ } and τ α < κ for all α ∈ κ . W e sa y th at κ is exp onential if κ = 2 σ for some cardinal σ , and w e call κ non-exp onential otherwise. Recall that κ is called a str ong limit pr o v id ed that 2 µ < κ for ev er y cardinal µ < κ . When κ i s infinite, w e d efine log κ = min { σ : κ ≤ 2 σ } . W e start this section with a muc h sharp er v ersion of Theorem 1 .5(a) sho wing that the w eigh t w ( G ) of a m inimal ab elian group G can b e tak en a s the cardinal σ from Definition 1.4(ii) witnessing th at | G | is a Sto y ano v cardinal: Theorem 2.1. If G is an infinite minimal ab elian gr oup, then Min ( | G | , w ( G )) holds. This theorem, a long with the complete “int ernal” charac terization o f the Stoy ano v car- dinals obtained in Prop osit ion 3.5 p erm its us to establish some new imp ortan t relations b et we en the cardinalit y and the w eigh t of an arb itrary minimal ab elia n group. Theorem 2.2. If κ is a c ar dinal with cf ( κ ) > ω and G is a minimal ab elian gr oup such that w ( G ) ≥ κ , then | G | ≥ 2 κ . Let us recall that | G | = 2 w ( G ) holds for ev ery compact group G [3]. T aking κ = w ( G ) in Theorem 2.2 w e obtain the follo win g extension of this prop ert y to all minimal abelian groups: Corollary 2.3. L et G b e a minima l ab elian gr oup with cf ( w ( G )) > ω . Then | G | = 2 w ( G ) . Example 8 .3(a) b elo w and Theorem 1.6 sho w that neither cf ( w ( G )) > ω nor “ab elian” can b e r emo ved in C orollary 2.3. T aking κ = ω 1 in T h eorem 2.2 one obtains the follo wing surprisin g metrizabilit y criterion for “small” minimal ab elian groups: Corollary 2.4. A minimal ab elian gr oup of size < 2 ω 1 is metrizable. 4 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV The condition cf ( w ( G )) > ω pla ys a prominent role in the a b ov e results. In p articular, Corollary 2.3 implies that cf ( w ( G )) = ω for a minimal ab elian group w ith | G | < 2 w ( G ) . Our n ext theorem giv es a more p recise inf ormation in this direction. Theorem 2.5. L et G b e an infinite minimal ab elian g r oup su c h that | G | is a non-exp onential c ar dinal. Then w ( G ) = log | G | and cf ( w ( G )) = ω . Under the assumption of GCH, the equalit y w ( G ) = log | G | holds true for ev ery compact group. Theorem 2.5 establishes this p rop erty in Z FC f or all minimal ab elian groups of non-exp onentia l size. Let us note th at the restraint “non-exp onenti al” cannot b e omitted, ev en in the compact case. In deed, the equalit y w ( G ) = log | G | may fail f or compact ab elian groups: Und er the Lusin’s Hyp othesis 2 ω 1 = c , for the group G = Z (2) ω 1 one has w ( G ) = ω 1 6 = ω = log c = log | G | . Example 2.6. There exists a consisten t example of a compact ab elian group G such that cf ( w ( G )) = ω and w ( G ) > log | G | (see Example 3. 4 (b)). 2.2. Minimal pseudocompact group top olog ies on free ab elian groups. S ince pseudo compact metric sp aces are compact, fr om Corollary 2.4 we immediately get the follo wing: Corollary 2.7. L et G b e an ab elian gr oup such that | G | < 2 ω 1 . Then G admits a m inimal pseudo c omp act gr oup top olo gy if and only if G ad mits a c omp act metric gr oup top olo gy. By Theorem 1. 8, this corollary is v acuously true under the Lusin’s Hyp othesis 2 ω 1 = c . Corollary 2.7 sho ws that for ab elian group s of “small size” minimal and pseud o compact top ologizat ions are connected in some sense by compactness. W e shall see in Corollary 8.2 b elo w that the same phenomenon happ ens for divisible abelian groups, irresp ecti v ely of th eir size. Rather surprisingly , the mere existence of a minimal group topology on F κ quite often implies the existence of a group top ology on F κ that is b oth m inimal and pseudo co mpact. In other w ord s , one often gets p seudo compactness “ for free”. Theorem 2.8. L et κ and σ b e infinite c ar dinals. Assume also that σ is not a str ong limit. If F κ admits a minimal gr oup top olo gy of weight σ , then F κ also admits a zer o-dimensional minimal pseudo c omp act gr oup top olo g y of weight σ . Recall that the b eth c ar dinals i α are d efined by recursion o n α as follo ws. Let i 0 = ω . If α = β + 1 is a su ccessor ordinal, then i α = 2 i β . If α is a limit ordinal, then i α = sup { i β : β ∈ α } . The restriction on w eigh t in Th eorem 2.8 is necessary , as ou r next example demonstrates. Example 2.9. Let κ = i ω . Clearly , the sequence { i n : n ∈ N } w itnesses that κ is a Sto y ano v cardinal, so F κ admits a minimal gr oup top olo gy τ by Theorem 1 .5(b). On the other hand, since κ is a strong limit cardinal with cf ( κ ) = ω and | F κ | = κ , the group F κ do es not adm it any pseudo c omp act gr oup top olo gy b y the result of v an D ou we n [26]. Note that w ( F κ , τ ) = log | F κ | = log κ = κ by Theorem 2.5, so σ = w ( F κ , τ ) is a strong limit cardinal. “Going in the opp osite direction”, in Example 4.7 b elo w we will defi ne a cardinal κ su c h that F κ admits a pseu do compact group topology of w eigh t σ that is not a strong limit cardinal, and ye t F κ do es not admit any minimal group top ology . These t wo examples sho w that the existence of a minimal group top ol ogy and the existence of a pseudo compact group top olo gy on a free ab elia n group are “ind ep endent ev en ts”. F or a free group of size > c that admits b oth a min imal group top ology and a pseudo com- pact group top ology , the n ext theorem discov ers the surpr isin g possib ility of “sim ultaneous MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 5 top ologizat ion” with a top ology w hic h is b oth minimal and pseudo compact. Moreo ver, i t turns out that this to p ol ogy can also be chosen to be ze ro-dimensional. Theorem 2.10. F or every c ar dinal κ > c the fol lowing c onditions ar e e qu i valent: (a) F κ admits b oth a minimal gr oup top olo gy and a pseudo c omp act gr oup top olo gy; (b) F κ admits a minimal pseudo c omp act gr oup top olo gy; (c) F κ admits a zer o-dimensional minimal pseudo c omp act gr oup top olo gy. The free ab elian grou p group F c of cardinalit y c admits a minimal group top ology (Theorem 1.3) and a pseud o compact group top o logy [13]. O ur next theorem shows that the statement “ F c admits a minimal pseudo co mpact group top ology” is b oth consisten t with and indep endent of ZF C . Theorem 2.11. The fol lowing c onditions ar e e quivalent: (a) F c admits a minimal pseudo c omp act gr oup top olo gy; (b) F c admits a c onne cte d minimal pseudo c omp act gr oup top olo gy; (c) F c admits a zer o-dimensional minimal pseudo c omp act gr oup top olo g y; (d) the Lusin ’s Hyp othesis 2 ω 1 = c holds. Since ev ery infin ite pseud o compact group has cardinalit y ≥ c (Theorem 1.8), T heorems 2.10 and 2.11 provide a complete d escription of f ree ab elian groups th at ha v e a m in imal (zero-dimensional) pseu d o compact group top ology . The equiv alence of (a) and (b) in Theorem 2.10 (resp ec tiv ely , (a) and (d) in T heorem 2.11) w as announ ced without pro of in [9, Theorem 4.11]. Motiv ated b y Th eorem 2.10(c) and Theorem 2.11(c), wh ere the minimal pseudo co mpact top ology can b e additionally c hosen zero-dimensional (or connected, in Theorem 2.11(b)), w e arriv e at the follo w ing natural questio n: If κ is a c ar dinal su ch that F κ admits a minimal gr oup top olo gy τ 1 and a pseudo c omp act gr oup top olo gy τ 2 , and one of these top olo g i es is c onne cte d, do es then F κ admit a c onne cte d minimal pseudo c omp act g r oup top olo gy τ 3 ? Theorem 2. 11 answers this question in t he case o f F c . The next th eorem giv es an answ er for κ > c , sh o w ing a symmetric b eha vior, as far as conn ectedness is concerned. T his should b e compared w ith th e equiv alent items in Theorem 2.11 where item (a) conta ins n o restriction b ey ond min imalit y and p seudo compactness, whereas item (c) conta ins “zero- dimensional”. Theorem 2.12. L et κ and σ b e infinite c ar dinals with κ > c . The fol lowing c onditions ar e e quivalent: (a) F κ admits a c onne cte d minimal pseudo c omp act gr oup top olo gy (of weight σ ); (b) F κ admits a c onne cte d minimal gr oup top olo gy (of weight σ ); (c) κ is exp onential ( κ = 2 σ ). This theorem is “asymmetric” in some sen s e to ward minimalit y . Indeed, item (b) should b e co mpared with the fact that the existe nce of a connected pseudo compact group top ology on F κ need not necessarily imply that F κ admits a connected minimal group top ology (see Example 4 .8). If a free ab elian group admits a pseud o co mpact group top ology , then it admits also a pseudo compact group top ology which is b oth connected and locally co nnected [13, Theo- rem 5. 10]. When minimalit y is added to the mix, the situatio n b eco mes totally differen t. In Example 4.8 b elo w w e exhibit a free ab elian group F κ that admits a connected, lo- cally connected, pseudo compact group t op ology , and y et F κ do es not ha ve an y connected minimal g roup top olo gy . Ev en m ore striking i s t he fo llo wing Theorem 2.13. A lo c al ly c onne cte d minimal torsion-fr e e ab elian gr oup is trivial. 6 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV Theorem 2.13 strengthens significantl y [13, C orollary 8. 8] b y replacing “compact” in it with “minimal”. Corollary 2.14. No fr e e ab elian g r oup admits a lo c al ly c onne cte d, minimal gr oup top olo gy. The reader ma y wish to compare this corollary with Theorems 2 .11 and 2.12. The pap er is organized as follo ws. In S ection 3 we giv e some p rop erties of S to yano v cardinals, w hile Section 4 con tains all n ecessary facts concernin g pseud o compact top olo- gizatio n. The culmin ation here is Corollary 4.12 establishin g that, roughly sp ea king, if F κ admits a minimal group top ol ogy τ 1 and a pseud o compact group top ology τ 2 , then one can assum e, without loss of generalit y , that this pair satisfies w ( F κ , τ 1 ) = w ( F κ , τ 2 ). Sections 5 and 6 prepare the remainin g necessary tools for th e pro of of the main results, deferred to Section 7. Finally , in Section 8 w e discuss the counterpart of the s im ultaneous minimal and pseudo compact top ologization for other classes of ab elian groups suc h as divisible groups , torsion-fr ee groups and torsion groups, as well as the same problem for (non-comm utativ e) free groups. 3. Proper ties of Stoy anov car dinals W e s tart with an example of small S to yano v cardinals. Example 3.1. If ω ≤ κ ≤ c , then Min ( κ, ω ). In our next example w e discuss the connectio n b et we en Min ( κ, σ ) and the prop ert y of κ to b e exp onen tial. Example 3.2. Let κ b e a n infinite cardinal. (a) If κ = 2 σ , then Min ( κ, σ ) holds. In p articular, an exp onential c ar dinal is Stoyano v. (b) If { σ n : n ∈ N } is a se quenc e of c ar dinals witnessing Min ( κ, σ ) such that σ = σ m for some m ∈ N , then κ = 2 σ . Indeed, (1 ) and our a ssump tion yield 2 σ = 2 σ m ≤ s u p n ∈ N 2 σ n ≤ κ ≤ 2 σ . Hence κ = 2 σ . (c) If cf ( σ ) > ω , then Min ( κ, σ ) if and only if κ = 2 σ . If κ = 2 σ , then Min ( κ, σ ) b y item (a). Assu me Min ( κ, σ ), and let { σ n : n ∈ N } b e a sequ en ce of cardinals witnessing Min ( κ, σ ). F rom (1 ) and cf ( σ ) > ω we get σ = σ m for some m ∈ N . Applying item (b) giv es κ = 2 σ . Clearly , Min ( κ, σ ) implies σ ≥ log κ . W e sho w no w that this inequalit y b ecomes an equalit y in c ase κ is non-exponential . Lemma 3.3. L et κ b e a non-exp onential infinite c ar dinal. Then: (a) Min ( κ, σ ) if and only if cf ( σ ) = ω and log κ = σ ; (b) Min ( κ, log κ ) if a nd only if cf (lo g κ ) = ω . Pr o of. (a) T o prov e the “only if ” part, assume that Min ( κ, σ ) holds, and let { σ n : n ∈ N } b e a sequence of cardin als w itnessing M in ( κ, σ ). Since κ ≤ 2 σ b y (1), we hav e log κ ≤ σ . Assume log κ < σ . F rom (1) we conclude that log κ ≤ σ m for some m ∈ N . Therefore 2 log κ ≤ 2 σ m ≤ sup n ∈ N 2 σ n ≤ κ ≤ 2 log κ b y (1). Thus κ = 2 log κ is an exp onen tial cardinal, a con tradiction. This p ro v es th at σ = log κ . MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 7 T o pro v e the “if ” part, assume that cf ( σ ) = ω and log κ = σ . T hen there exists a sequence of cardinals { σ n : n ∈ N } suc h that σ = su p n ∈ N σ n and σ n < σ = log κ for eve ry n ∈ N . In particular, 2 σ n < κ for ev ery n ∈ N . Consequent ly , sup n ∈ N 2 σ n ≤ κ ≤ 2 log κ = 2 σ . That is, (1) holds. Th erefore, the sequence { σ n : n ∈ N } witnesses Min ( κ, σ ). Item (b) follo ws from item (a ).  Example 3.4. Let κ and σ b e cardinals. According to Example 3.2(a), Min ( κ, σ ) do es not imp ly cf ( σ ) = ω in case κ is exp onenti al. ( Ind eed, it suffices to tak e κ = 2 σ with cf ( σ ) > ω .) (a) Let us sho w that the assumption “ κ is n on -exp onen tial” in Lemma 3.3(a) is n ec- essary (to pro v e that Min ( κ, σ ) implies log κ = σ ) ev en in the case cf ( σ ) = ω . T o this end , use an ap p ropriate East on mod el [15] satisfying 2 ω ω +1 = ω ω +2 and 2 ω n = ω ω +2 for all n ∈ N . Let κ = ω ω +2 and σ = ω ω . Then 2 σ = κ as 2 ω ω +1 = 2 ω n = κ for ev ery n ∈ N . So Min ( κ, σ ) holds by Example 3.2(a). Moreo v er cf ( σ ) = ω and log κ = ω 0 < ω ω = σ . (b) Usin g the cardinals κ and σ from item (a) we ca n giv e no w the example an ticipated in Example 2.6. Let G = Z (2) σ . Then w ( G ) = σ , so cf ( w ( G )) = ω and y et log | G | = log 2 σ = log κ = ω 0 < σ = w ( G ). The n ext prop osition, summarizing th e ab o v e results, pro vides an alternativ e description of the in finite Sto y ano v cardinals that mak es no u se of the somewhat “external” condition (1). Prop osition 3.5. L et κ b e an infinite c ar dinal. (a) If κ is exp onential, then Min ( κ, σ ) holds for every c ar dinal σ with κ = 2 σ . (b) If κ is non-exp onential, then Min ( κ, σ ) is e quivalent to σ = log κ and cf (log κ ) = ω . Pr o of. Item (a) follo w s from Ex amp le 3.2(a), and item (b) follo ws from Lemma 3.3(a).  4. Card inal inv ariants rela ted to pseud o comp act groups Recall that a subset Y of a space X is said to b e G δ -dense in X pro vided th at Y ∩ B 6 = ∅ for eve ry non-empt y G δ -subset B of X . The follo w ing theorem describ es pseudo compact grou p s in terms of their completion. Theorem 4.1. [7, Theorem 4.1] A pr e c omp act gr oup G is pseudo c omp act if and only if G is G δ -dense in its c ompletion. Definition 4.2. (i) If X is a non-empt y set and σ is an infinite cardinal, then a set F ⊆ X σ is ω -dense in X σ , pro vided that for ev ery counta ble set A ⊆ σ and eac h function ϕ ∈ X A there e xists f ∈ F suc h that f ( α ) = ϕ ( α ) for all α ∈ A . (ii) If κ and σ ≥ ω are c ardinals, then Ps ( κ, σ ) abb reviates the sen tence “there exists an ω -dense set F ⊆ { 0 , 1 } σ with | F | = κ ”. (iii) F or an infinite ca rdinal σ let m ( σ ) denote the min imal c ardinal κ su c h that Ps ( κ, σ ) holds. Items (i) and (ii) of the ab o ve d efi nition are tak en from [2] except f or th e notation Ps ( κ, σ ) that app ears in [13, Definition 2.6]. Item (iii) is equiv alen t to the d efinition of the cardinal function m ( σ ) of Comfort and Rob ertson [4]. It is w orth noting that m ( σ ) = δ ( σ ) for ev ery infi nite cardinal σ , where δ ( − ) is the cardinal function d efined b y Cater, Er d¨ os and Galvin [2]. 8 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV The set- theoretical cond ition Ps ( κ, σ ) is ultimately related to the existence of pseudo- compact g roup top olo gies. Theorem 4.3. ([4]; see also [13, F act 2.12 and Theorem 3.3(i)]) L et κ and σ ≥ ω b e c ar dinals. Then Ps ( κ, σ ) ho lds if a nd only if ther e exists a gr oup G of c ar dinality κ which admits a pseudo c omp act gr oup top olo gy of weight σ . Moreo ver, the cond ition Ps ( κ, σ ) completely describ es free ab elian group s that admit pseudo compact group top ologies. Theorem 4.4. ([5], [13, Th eorem 5. 10]) If κ i s a c ar dinal, then F κ admits a pseudo c omp act gr oup top olo gy of weight σ if and only if Ps ( κ, σ ) holds. In the next lemma w e summarize some p rop erties of the cardinal function m ( − ) for future reference. Lemma 4.5. ([2]; see also [ 4, Theorem 2.7]) L et σ b e an infinite c ar dinal. Then: (a) m ( σ ) ≥ 2 ω and cf ( m ( σ )) > ω ; (b) log σ ≤ m ( σ ) ≤ (log σ ) ω ; (c) m ( λ ) ≤ m ( σ ) whenever λ is a c ar dinal with λ ≤ σ . Some useful pr op erties of the condition Ps ( λ, κ ) are collected in the next pr op osition. Items (a) and (b) are part of [13, Lemmas 2.7 and 2.8], and items (d) and (e) are particular cases of [13, Lemma 3 .4(i)]. Prop osition 4.6. (a) Ps ( c , ω ) holds, and mor e over, m ( ω ) = c ; also Ps ( c , ω 1 ) holds. (b) If Ps ( κ, σ ) holds for some c ar dinals κ and σ ≥ ω , then κ ≥ c , and Ps ( κ ′ , σ ) holds for every c ar dinal κ ′ such tha t κ ≤ κ ′ ≤ 2 σ . (c) F or c ar dinals κ and σ ≥ ω , Ps ( κ, σ ) holds if and only if m ( σ ) ≤ κ ≤ 2 σ . (d) Ps (2 σ , σ ) and Ps  2 σ , 2 2 σ  hold for every infinite c ar dinal σ . (e) If σ is a c ar dinal such that σ ω = σ , then Ps ( σ , 2 σ ) ho lds. Example 4.7. L et κ = i ω 1 (see the text p receding Example 2.9 for the defin ition of i ω 1 ). One can easily see that κ is n ot a Stoy ano v cardinal (this was first noted b y Sto yano v himself ). Therefore, the group F κ do es not admit any minimal gr oup top olo gy by Theorem 1.5(a). On the other hand , κ = κ ω and Prop osition 4.6(e) yield that Ps ( κ, 2 κ ) holds. Applying Th eorem 4.4 w e conclude that F κ admits a pseudo c omp act gr oup top olo gy of weight 2 κ . In particular, σ = 2 κ is not a strong limit. Example 4.7 sh ou ld b e compared w ith Theorem 2.8 wh ere w e show that if F κ admits a minimal group topology of w eigh t σ and σ is not a a strong limit, then F κ admits also a pseudo compact group top ology of w eight σ . Example 4.8. Let κ b e a non-exp onen tial cardinal with κ = κ ω (e.g., a strong limit cardinal of uncounta ble cofinalit y). Then, according to Pr op osition 4.6(e), Ps ( κ, 2 κ ) holds. Therefore F κ admits a pseudo c omp act gr oup top olo gy (of w eight 2 κ ) t hat is b oth c onne cte d and lo c al ly c onne cte d [13, Theorem 5.10]. By Th eorem 2.12, F κ do es not admit a c onne cte d minimal gr oup top olo gy as κ is non-exp onen tial. Lemma 4.9. If κ and σ ar e infinite c ar dinals such that σ is not a str ong limit c ar dinal, then Min ( κ, σ ) implies Ps ( κ, σ ) . Pr o of. Assume that Min ( κ, σ ) holds, and let { σ n : n ∈ N } b e a sequen ce of cardinals witnessing Min ( κ, σ ). Since σ is not a strong limit cardinal, there exists a cardinal µ < σ suc h that σ ≤ 2 µ . Since σ = sup n ∈ N σ n b y (1), µ ≤ σ n for some n ∈ N . Then σ ≤ 2 µ ≤ 2 σ n , and so log σ ≤ σ n . Applying Lemma 4.5 (b) and (1), we obtain m ( σ ) ≤ (log σ ) ω ≤ σ ω n ≤ 2 σ n ≤ κ ≤ 2 σ . MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 9 Hence Ps ( κ, σ ) h olds b y Prop osition 4.6(c).  Our n ext example demonstr ates that the restrictio n on the cardinal σ in Lemma 4.9 is necessary . Example 4.10. Let κ b e the S toy ano v cardin al from Exa mple 2.9. F rom calc ulations in that example one concludes that Min ( κ, κ ) holds. As was shown in Example 2.9, F κ do es not admit any pseudo co mpact group top ology . Th er efore, Ps ( κ, σ ) fails for ev ery cardinal σ (Theorem 4. 4). In the n ext lemma we sh o w that, if κ is a Sto y ano v card inal satisfying Ps ( κ, λ ) for some λ , th en Ps ( κ, σ ) holds also for the ca rdin al σ witnessing t hat κ is S toy ano v. Lemma 4.11. L e t κ and σ b e infinite c ar dinals satisfying Min ( κ, σ ) . If Ps ( κ, λ ) holds for some infinite c ar dinal λ , then Ps ( κ, σ ) holds as wel l. Pr o of. By Lemma 4.9, it suffices only to consider the case when σ is a strong limit cardinal. Let { σ n : n ∈ N } b e a s equence of cardinals witnessin g Min ( κ, σ ). If σ = σ n for some n ∈ N , then κ = 2 σ b y Example 3.2(b). Since Ps (2 σ , σ ) holds by Prop osition 4.6(d), we are done in this case. Su p p ose no w that σ > σ n for ev ery n ∈ N . Since Ps ( κ, λ ) holds, from Prop osition 4.6(c) w e get m ( λ ) ≤ κ ≤ 2 λ . I f λ < σ , then 2 λ < σ and so κ < σ . F rom (1) w e get κ < σ n for some n ∈ N , and then σ n < 2 σ n ≤ κ , a con tradiction. Hence σ ≤ λ . By Lemma 4.5(c) m ( σ ) ≤ m ( λ ) ≤ κ . Moreo ve r κ ≤ 2 σ b y (1). It now follo ws from Prop osition 4.6 (c) th at Ps ( κ, σ ) h olds .  Corollary 4.12. L et κ b e a non-zer o c ar dinal. If F κ admits a minimal gr oup top olo gy τ 1 and a pseudo c omp act gr oup top olo g y τ 2 , then F κ admits also a pseudo c omp act gr oup top olo gy τ 3 with w ( F κ , τ 1 ) = w ( F κ , τ 3 ) . Pr o of. F rom Theorem 1.8 w e get κ ≥ c . Define σ = w ( F κ , τ 1 ). C learly , σ is infin ite. Ap- plying Th eorem 2.1, we conclude t hat Min ( κ, σ ) holds. T heorem 4.4 yiel ds that Ps ( κ, λ ) holds, wh ere λ = w ( F κ , τ 2 ). Clearly , λ is infinite. Then Ps ( κ, σ ) holds by Lemma 4.11. Finally , applying Theorem 4.4 once again, w e obtain that F κ m ust admit a p seudo compact group top olo gy τ 3 suc h t hat w ( F κ , τ 3 ) = σ .  The pr o of of Corollary 4.12 r elies on Theorem 2.1, which is pr ov ed later in Section 7. Nev ertheless, this do es not create any problems, b eca use Corollary 4.12 is nev er u sed thereafter. 5. Bu ild ing G δ -dense V -indep endent sub s ets in products A variety of gr oups V is a class of abstract grou p s closed u nder sub groups, qu otien ts and pr o ducts. F or a v ariet y V a nd G ∈ V a subset X of G is V -indep endent if the subgroup h X i of G generated by X b elo ngs to V and for eac h m ap f : X → H ∈ V there exists a unique homomorp h ism f : h X i → H extending f . Moreo ve r, the V -r ank of G is r V ( G ) := sup {| X | : X is a V -indep endent subset of G } . In particular, if A is th e v ariety of all ab elian groups , then the A -rank is the us ual f ree rank r ( − ), and for the v ariety A p of all a b elian groups of exp onen t p (for a prime p ) the A p -rank is the usual p -rank r p ( − ). Our fi r st lemma is a generalizat ion of [1 3, Lemma 4.1] that is in fact equiv alen t to [13, Lemma 4.1] (as can be seen from its pro of b elow). Lemma 5.1. L et V b e a variety of gr oups and I an infinite set. F or every i ∈ I let H i b e a gr oup such that r V ( H i ) ≥ ω . Then r V ( Q i ∈ I H i ) ≥ 2 | I | . 10 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV Pr o of. Define N = N \ { 0 } . F or every n ∈ N , let F n b e th e free group in the v ariety V with n generators. Define H = Q n ∈ N F n , and note that r V ( H ) ≥ ω . Since I is infinite, there exists a bijection ξ : I × N → I . F or ( i, n ) ∈ I × N , fix a subgroup F in of H ξ ( i,n ) isomorphic to F n (this can b e d one b ecause r V ( H ξ ( i,n ) ) ≥ ω ). Then Q ( i,n ) ∈ I × N F in is a subgroup of the group Q ( i,n ) ∈ I × N H ξ ( i,n ) ∼ = Q i ∈ I H i , wh ere ∼ = denotes the isomorph ism b et we en groups. Clearly , Y ( i,n ) ∈ I × N F in ∼ = Y i ∈ I Y n ∈ N F in ∼ = Y i ∈ I Y n ∈ N F n ∼ = Y i ∈ I H ∼ = H I , so th ere exists a monomorphism f : H I → Q i ∈ I H i . No w r V Y i ∈ I H i ! ≥ r V  f  H I  = r V  H I  ≥ 2 | I | , where the the first inequ ality follo ws f rom [13, Corollary 2.5] and the last inequalit y has b een pro ve d in [13, Lemm a 4.1].  Lemma 5.2. Supp ose t hat I is an infinite set and H i is a sep ar able metric sp ac e for every i ∈ I . If Ps ( κ, | I | ) holds, then the pr o duct H = Q i ∈ I H i c ontains a G δ -dense subset of size at most κ . Pr o of. Let i ∈ I . Since H i is a separable metric s pace, | H i | ≤ c , and so w e can fix a surjection f i : R → H i . Let θ : R I → H b e th e map d efined by θ ( g ) = { f i ( g ( i )) } i ∈ I ∈ H f or ev ery g ∈ R I . Since Ps ( κ, | I | ) holds, [13, Lemma 2.9] a llo w s u s to conclude that R I con tains an ω -dense subset X of size κ . Define Y = θ ( X ). Then | Y | ≤ | X | = κ . It remains only to sho w that Y is G δ -dense in H . Indeed, let E b e a non -emp t y G δ -subset of H . Then there exist a coun table subset J of I and h ∈ Q j ∈ J H j suc h that { h } × Q i ∈ I \ J H i ⊆ E . F or ev ery j ∈ J select r j ∈ R suc h that f j ( r j ) = h ( j ). Since X is ω -dense in R I , there exists x ∈ X suc h that x ( j ) = r j for eve ry j ∈ J . No w θ ( x ) = { f i ( x ( i )) } i ∈ I = { f j ( x ( j )) } j ∈ J × { f i ( x ( i )) } i ∈ I \ J = { h ( j ) } j ∈ J × { f i ( x ( i )) } i ∈ I \ J ∈ { h } × Y i ∈ I \ J H i ⊆ E . Therefore, θ ( x ) ∈ Y ∩ E 6 = ∅ .  Lemma 5.3. L et κ ≥ ω 1 b e a c ar dinal and G and H b e top olo gic al gr oups in a variety V such that: (a) r V ( H ) ≥ κ , (b) H ω has a G δ -dense subset of size at m ost κ , (c) G has a G δ -dense subset of size at m ost κ . Then G × H ω 1 c ontains a G δ -dense V - indep endent subset of size κ . Pr o of. Since κ ≥ ω 1 , w e ha v e | κ × ω 1 | = κ , and so w e can use item (a ) to fix a faithfully indexed V -indep en d en t subset X = { x αβ : α ∈ κ, β ∈ ω 1 } of H . F or ev ery β ∈ ω 1 \ ω the top ological groups G × H ω and G × H β are isomorphic, so w e can use items (b) and (c) to fix { g αβ : α ∈ κ } ⊆ G and { y αβ : α ∈ κ } ⊆ H β suc h t hat Y β = { ( g αβ , y αβ ) : α ∈ κ } is a G δ -dense subset of G × H β . F or α ∈ κ and β ∈ ω 1 \ ω defi ne z αβ ∈ H ω 1 b y (2) z αβ ( γ ) =  y αβ ( γ ) , for γ ∈ β x αβ , for γ ∈ ω 1 \ β for γ ∈ ω 1 . MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 11 Finally , define Z = { ( g αβ , z αβ ) : α ∈ κ, β ∈ ω 1 \ ω } ⊆ G × H ω 1 . Claim 5.4. Z is G δ -dense in G × H ω 1 . Pr o of. Let E b e a non-empt y G δ -subset of G × H ω 1 . Then there exist β ∈ ω 1 \ ω and a non-empt y G δ -subset E ′ of G × H β suc h that (3) E ′ × H ω 1 \ β ⊆ E . Since Y β is G δ -dense in G × H β , th er e exists α ∈ κ suc h that ( g αβ , y αβ ) ∈ E ′ . F rom (2) it follo w s that z αβ ↾ β = y αβ . Com bining th is with (3), we conclude that ( g αβ , z αβ ) ∈ E . Th us ( g αβ , z αβ ) ∈ E ∩ Z 6 = ∅ .  Claim 5.5. Z is V -indep endent. Pr o of. Let F b e a non-empt y fi nite subset of κ × ( ω 1 \ ω ). Define (4) γ = max { β ∈ ω 1 \ ω : ∃ α ∈ κ ( α, β ) ∈ F } . F rom (2) and (4) it follo ws that z αβ ( γ ) = x αβ for a ll ( α, β ) ∈ F . Therefore, X F = { z αβ ( γ ) : ( α, β ) ∈ F } = { x αβ : ( α, β ) ∈ F } ⊆ X . Since X is a V -indep enden t su bset of H , so is X F [13, Lemma 2.3]. Let f : G × H ω 1 → H b e t he pro jection homomorphism defi n ed by f ( g , h ) = h ( γ ) for ( g , h ) ∈ G × H ω 1 . Define S F = { ( g αβ , z αβ ) : ( α, β ) ∈ F } . Since G ∈ V , H ∈ V , h S F i is a su bgroup of G × H ω 1 and V is a v ariet y , h S F i ∈ V . Since f ↾ S F : S F → H is an injection and f ( S F ) = X F is a V -indep endent subset of H , from [13, Lemma 2.4] w e obtain that S F is V -ind ep enden t. S ince F was ta k en arb itrary , fr om [13, Lemma 2.3] it follo w s that Z is V -ind ep endent.  F rom the last claim we conclude th at | Z | = | κ × ( ω 1 \ ω ) | = κ .  Lemma 5.6. Assume that κ is a c ar dinal, { H n : n ∈ N } i s a family of sep ar able metric gr oups in a variety V and { σ n : n ∈ N } is a se quenc e of c ar dinals such that: (i) r V ( H n ) ≥ ω for every n ∈ N , (ii) σ = sup { σ n : n ∈ N } ≥ ω 1 , (iii) Ps ( κ, σ ) holds. Then Q n ∈ N H σ n n has a G δ -dense V -indep endent subset of size κ . Pr o of. Define S = { n ∈ N : σ n ≥ ω 1 } , G = Y n ∈ N \ S H σ n n and H = Y n ∈ S H σ n n . F rom ite ms (i) and (ii ) of our le mma it follo w s that H ∼ = Y i ∈ I H ′ i , where | I | = σ and eac h H ′ i is a separable metric group (5) satisfying r V ( H ′ i ) ≥ ω , where ∼ = denotes th e isomorph ism b etw een top ological group s. Since | σ n × ω 1 | = σ n for ev er y n ∈ S , we ha ve H ω 1 ∼ = Y n ∈ S ( H σ n n ) ω 1 ∼ = Y n ∈ S H σ n × ω 1 n ∼ = Y n ∈ S H σ n n ∼ = H . In p articular, Y n ∈ N H σ n n = G × H ∼ = G × H ω 1 . 12 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV Therefore, the conclusion of our lemma w ou ld follo w from th at of Lemma 5.3 so long as we pro v e that G and H satisfy the assumptions of Lemma 5.3 . F r om (ii), (iii) and Prop osition 4.6(b) o ne conclud es that κ ≥ c ≥ ω 1 . Let us chec k that the assum p tion of item (a) of Lemma 5.3 h olds. F rom (5 ) and Lemma 5.1 we get r V ( H ) ≥ 2 σ . Since Ps ( κ, σ ) holds b y item (iii), w e ha ve 2 σ ≥ κ b y Prop osition 4.6(c). This sho ws that r V ( H ) ≥ κ . Let us c hec k that the assumption of item (b) of Lemma 5.3 holds. Recalling (5), we conclude t hat H ω ∼ = Y i ∈ I  H ′ i  ω , wh ere eac h  H ′ i  ω is a separable metric space . Since | I | = σ b y (5), and Ps ( κ, σ ) h olds b y item (iii), Lemm a 5.2 allo ws us to conclude that H ω has G δ -dense su bset of size at m ost κ . Let us chec k that the assump tion of item (c) of Lemma 5.3 holds. Since σ n ≤ ω for ev er y n ∈ N \ S , G is a s eparable metric group, and so | G | ≤ c . Since Ps ( κ, σ ) holds, c ≤ κ b y Pr op osition 4. 6(b), and so G itself is a G δ -dense subset of G of size a t most κ .  Corollary 5.7. L et P b e the set of prime numb ers and { σ p : p ∈ P } a se quenc e of c ar dinals such that σ = sup { σ p : p ∈ P } ≥ ω 1 . If κ is a c ar dinal such that Ps ( κ, σ ) holds, then the gr oup (6) K = Y p ∈ P Z σ p p c ontains a G δ -dense fr e e sub g r oup F such that | F | = κ . Pr o of. Since r ( Z p ) ≥ ω for ev ery p ∈ P , app lying Lemma 5.6 w ith V = A w e can find a G δ -dense A -ind ep endent sub s et X of K of size κ . Since A -ind ep end en ce coincides with the usual indep end ence for a b elian groups , th e subgroup F of K generated by X is free. Clearly , | F | = κ . Since X ⊆ F ⊆ K and X is G δ -dense in K , so is F .  As an a pplication, w e obtain the follo wing particular case of [13, Lemma 4.3]. Corollary 5.8. L et κ and σ ≥ ω 1 b e c ar dinals such that Ps ( κ, σ ) holds. Then for ev- ery c omp act metric non-torsion ab elian g r oup H the gr oup H σ c ontains a G δ -dense fr e e sub gr oup F such that | F | = κ . Pr o of. Since H is a compact non-torsion ab elian group , r ( H ) ≥ ω . Applying Lemma 5.6 with V = A , σ n = σ and H n = H for ev ery n ∈ N , we can find a G δ -dense ind ep endent subset X of K = H σ of size κ . Th en the subgroup F of K generate d by X is f r ee and satisfies | F | = κ . Sin ce X ⊆ F ⊆ K and X is G δ -dense in K , so is F .  6. Ess ential fre e subgroups of co mp act torsion-free abelian groups Lemma 6.1. L e t K b e a torsion-fr e e ab elian gr oup and let F b e a fr e e sub gr oup of K . Then ther e exists a fr e e sub gr oup F 0 of K c ontaining F as a dir e ct summand, such that: (a) F 0 non-trivial ly me ets every non-zer o sub gr oup of K , an d (b) | F 0 | = | K | . Pr o of. Let A := K/F and let π : K → A b e the canonical p ro jectio n. Let F 2 b e a fr ee subgroup of A with generators { g i } i ∈ I suc h that A/F 2 is torsion. S ince π is sur jectiv e, for eve ry i ∈ I there exists f i ∈ K , s u c h that π ( f i ) = g i . C onsider th e su bgroup F 1 of K generated by { f i : i ∈ I } . As π ( F 1 ) = F 2 is f ree, we conclude that F 1 ∩ F = { 0 } , so π ↾ F 1 : F 1 → F 2 is an isomorphism. Let u s see that the sub group F 0 = F + F 1 = F ⊕ F 1 has the required prop erties. Indeed, it is fr ee as F 1 ∩ F = { 0 } and b oth F , F 1 are fr ee. Moreo ver, K /F 0 ∼ = A/F 2 is torsion and F is a direct s u mmand of F 0 . As K/F 0 is torsion, MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 13 F 0 non-trivially meets ev ery non-zero subgroup of K , so (a) h olds tr ue. S ince K is torsion- free, (b ) easily follo ws from (a).  Lemma 6.2. L et K b e a c omp act torsion-fr e e ab elian gr oup and let F b e a fr e e sub gr oup of K . Then ther e exists a fr e e essential sub gr oup F 0 of K with | F 0 | = | K | , c ontaining F as a dir e ct summand. Pr o of. Apply Lemma 6.1.  Lemma 6.3. Supp ose Min ( κ, σ ) holds, and let { σ p : p ∈ P } b e the se quenc e of c ar dinals witnessing Min ( κ, σ ) . L et F b e a fr e e su b gr oup of the gr oup K as in (6 ) with | F | = κ . Then ther e exists a fr e e essential sub gr oup F ′ of K c ontaining F as a dir e ct summand such that | F ′ | = κ . Pr o of. Let (7) wtd( K ) = M p ∈ P Z σ p p and F ∗ = F ∩ wtd( K ) . Then F ∗ is a fr ee subgroup of wtd( K ), so applying Lemma 6.1 to the group w td( K ) and its s u bgroup F ∗ w e get a free s ubgroup F ∗ of wtd( K ) such that: (i) F ∗ ⊇ F ∗ and F ∗ = F ∗ ⊕ L fo r an appropr iate subgroup L of F ∗ ; (ii) F ∗ non-trivially meets ev ery non-zero sub group of wtd( K ); (iii) | F ∗ | = | wtd( K ) | ≤ κ = | F | . Ob viously , (ii ) yields that F ∗ is essenti al in wtd( K ). As w td( K ) is e ssenti al in K [12], w e conclude that F ∗ is essenti al in K as well. F rom (iii) w e conclude that F ′ = F + F ∗ is an essen tial su bgroup of K o f size κ con taining F . Finally , from (7) and (i) w e get F ′ = F + L , and sin ce L ⊆ wtd( K ), we ha v e F ∩ L = F ∩ wtd( K ) ∩ L = F ∗ ∩ L = { 0 } . Therefore, F ′ = F ⊕ L is f ree.  Lemma 6.4. L et κ and σ ≥ ω 1 b e c ar dinals such that b oth Min ( κ, σ ) and Ps ( κ, σ ) hold. Then F κ admits a zer o-dimensional minimal pseudo c omp act gr oup top olo g y of weight σ . Pr o of. Let { σ p : p ∈ P } b e a sequence of cardin als witnessing Min ( κ, σ ). In particular, σ = sup { σ p : p ∈ P } . Then the group K as in (6) is compact and zero- dimensional. Since σ ≥ ω 1 and Ps ( κ, σ ) h olds, by Corollary 5.7 th ere exists a G δ -dense f r ee sub group F of K with | F | = κ . Since Min ( κ, σ ) holds, according to Lemma 6.3 ther e exists a free essen tial subgroup F ′ of K c ont aining F with | F ′ | = κ . Ob viously F ′ is also G δ -dense. By Theorem 4.1 F ′ is pseu d o compact. On the other han d , b y t he essenti alit y of F ′ in K and Theorem 1.7, the subgrou p F ′ of K is also min im al. Being a su bgroup of the zero-dimensional group K , the group F ′ is zero-dimensional. Since F ′ is dense in K , from (6) and (1) we ha v e w ( F ′ ) = w ( K ) = sup { σ p : p ∈ P } = σ . Since F ′ ∼ = F κ , the subspace topology induced on F ′ from K will do the job.  Lemma 6.5. L et κ and σ ≥ ω 1 b e c ar dinals suc h that κ = 2 σ . Then F κ admits a c onne cte d minimal pseudo c omp act gr oup top olo g y of weight σ . Pr o of. The group K = b Q σ is compact and conn ected. S ince κ = 2 σ , Ps ( κ, σ ) holds b y Prop osition 4.6(d ). By Corollary 5.8 there exists a G δ -dense free sub grou p F of K with | F | = κ . According to Lemma 6.2 there exists a free essen tial subgroup F ′ of K con taining F with | F ′ | = | K | = κ . Ob viously F ′ is also G δ -dense. By Theorem 4.1 F ′ is p seudo compact. On the other hand , by the essen tialit y of F ′ in K and Theorem 1.7, the sub grou p F ′ of K is also minimal. Since G δ -dense subgroup s of compact connected 14 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV ab elian groups are connected [13, F act 2.10(ii)], w e conclude that F ′ is connected. Since F ′ is dense in K , we ha ve w ( F ′ ) = w ( K ) = σ . Clearly , F ′ ∼ = F κ as | F ′ | = | F | = 2 σ = κ . Therefore, the subspace top ol ogy i ndu ced on F ′ from K will do the job.  7. Proofs of the theore ms from Section 2 Lemma 7.1. L et G b e a minimal torsio n-fr e e ab elian gr oup and K its c ompletion. Then: (i) K is a c omp act torsion-fr e e ab elian gr oup; (ii) ther e exists a se qu e nc e of c ar dinals { σ p : p ∈ P ∪ { 0 }} such that (8) K = b Q σ 0 × Y p ∈ P Z σ p p . Pr o of. (i) By th e precompactness theorem of Pro danov and S to yano v ([1 2, Theorem 2.7.7]) , G is precompact, and so K is compact. Let u s sh o w that K is torsion-free. Let x ∈ K \ { 0 } . Assum e that the cyclic group Z = h x i generated by x is fi nite. Then Z is closed in K and non-trivial. S ince G is essential in K by Theorem 1.7, it follo w s that Z ∩ G 6 = { 0 } . Cho ose y ∈ Z ∩ G 6 = { 0 } . Since Z is fin ite, y must b e a torsion e lemen t, in con tr adiction with the f act that G is torsion-free. (ii) Since K is torsion-free b y item (i), the P ontry agin d ual of K is d ivisible. No w the conclusion of item (ii) of our lemma follo ws from [19, Theorem 25.8].  Pro of of Theorem 2.1. Let K b e the compact completion of G . Let σ = w ( K ) = w ( G ). Then clearly (9) | G | ≤ | K | = 2 σ . If σ = ω , then | G | ≤ | K | = 2 σ = c . Hence Min ( | G | , σ ) holds according to Example 3.1. Therefore, we assume σ > ω for the rest of the pro of. W e consider fi rst the case w hen G is torsion-fr ee. Although this part of the pro of is not used i n the second part co vering t he g eneral ca se, w e p refer to i nclude it b ecause this pro vides a self-con tained pr o of of Th eorems 2.10, 2.11 and 2.12 which concern only free (hence, torsion-free) groups. Let { σ p : p ∈ P ∪ { 0 }} b e the sequence from the conclusion of Lemma 7.1(ii). Clearly , our assu mption σ > ω imp lies that σ p > ω for some p ∈ P ∪ { 0 }} . Hence σ = su p { σ p : p ∈ P ∪ { 0 }} . S in ce G is b oth den se and essen tial in K , from [1, Theorems 3.12 and 3.14] w e get sup p ∈{ 0 }∪ P 2 σ p ≤ | G | . Therefore Min ( | G | , σ ) holds in view of (9 ). Since σ = w ( G ), w e are done. In the general case, we consider the connected comp onent c ( K ) of K and the totally disconnected quoti ent K/c ( K ). Then K/c ( K ) ∼ = Y p ∈ P K p , where eac h K p is a pro- p -group. Let σ p = w ( K p ) and σ 0 = w ( c ( K )). Our assump tion σ > ω implies that σ p > ω for some p ∈ P ∪ { 0 }} , so that σ = w ( G ) = w ( K ) = sup p ∈{ 0 }∪ P σ p . By [1, Theorems 3.12 and 3.14], one has | c ( K ) | · sup p ∈ P 2 σ p ≤ | G | . MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 15 Therefore, sup p ∈{ 0 }∪ P 2 σ p ≤ | G | ≤ | K | = 2 σ in view of (9). Thus Min ( | G | , σ ) holds. Since σ = w ( G ), w e are done.  Pro of of Theorem 2.2. Let G b e a minimal ab elia n group with w ( G ) ≥ κ . Define σ = w ( G ). T hen Min ( | G | , σ ) holds by Theorem 2.1. Let { σ n : n ∈ N } b e a sequ en ce of cardinals witnessing Min ( | G | , σ ). That is, (10) σ = sup n ∈ N σ n and sup n ∈ N 2 σ n ≤ | G | ≤ 2 σ . If cf ( σ ) > ω , then | G | = 2 σ ≥ 2 κ b y Example 3.2(c). Assum e that cf ( σ ) = ω . If σ n = σ for some n ∈ N , then | G | = 2 σ ≥ 2 κ b y Example 3.2(b). So we ma y add itionally assume that σ n < σ f or e ve ry n ∈ N . Since cf ( κ ) > ω = cf ( σ ), our h yp othesis σ ≥ κ giv es σ > κ . Then σ n ≥ κ for some n ∈ N , and so | G | ≥ 2 σ n ≥ 2 κ b y (10).  Pro of of Theorem 2.5. By Th eorem 2.1, Min ( | G | , w ( G )) holds. Since | G | is assumed to b e non-exp onentia l, the conclusion no w fol lo ws from Prop o sition 3.5(b).  Pro of of Theorem 2.8. S ince | F κ | = κ , from our assumption a nd Theorem 2.1 w e co n- clude that Min ( κ, σ ) holds. Lemma 4.9 yields that Ps ( κ, σ ) holds as w ell. Since σ is infinite and not a strong limit, it follo ws that σ ≥ ω 1 . No w Lemma 6.4 applies.  Pro of of Theorem 2.10. Th e implications (c) ⇒ (b) and (b) ⇒ (a ) are ob vious. (a) ⇒ (c) Assume th at τ 1 is a minimal top ology of w eigh t σ on F κ . Th en σ ≥ ω 1 as κ > c . According to Th eorem 2.1 Min ( κ, σ ) holds. Now assume that τ 2 is a minimal top ol ogy of w eigh t λ on F κ . According to Theorem 4.3 Ps ( κ, λ ) holds. Now Lemma 4.11 yields that also Ps ( κ, σ ) holds true. Finally , th e application of Lemma 6.4 finishes th e p ro of.  Remark 7.2. It is clea r from the abov e proof that the top olo gies from items ( b) a nd (c) of Theorem 2.10 can b e c hosen to ha v e the same w eigh t σ as the minimal top ology from item (a) of this t heorem. Pro of of Theorem 2.11. Th e implications (b) ⇒ (a) and (c) ⇒ (a) are obvious. (a) ⇒ (d) S upp ose that F c admits a minim al p s eudo compact group top ology . Since F c is free, F c do es not admit any compact group top ology , and so c = | F c | ≥ 2 ω 1 b y Corollary 2.7. The c on v erse inequalit y c ≤ 2 ω 1 is c lear. (d) ⇒ (b) F ollo ws from c = 2 ω 1 and Lemma 6.5. (d) ⇒ (c) F ollo ws from c = 2 ω 1 and Lemm a 6.4, as Min ( c , ω 1 ) h olds by Example 3. 2 (a), and Ps ( c , ω 1 ) holds b y Prop osition 4.6(a).  Pro of of Theorem 2.12. (a) ⇒ (b) is obvio us. (b) ⇒ (c) Assume that τ 1 is a connected min imal group top ology on F κ with w ( F κ , τ 1 ) = σ . Then the completion K of ( F κ , τ 1 ) satisfies the conclusion of Lemma 7.1(ii). Moreo v er, K is connecte d. Since the zero- dimensional group L = Y p ∈ P Z σ p p from (8) is a con tinuous image of the connected group K , w e must ha ve L = { 0 } . It follo ws that K = b Q σ 0 . Note that σ 0 = w ( K ) = w ( F κ , τ 1 ) = σ . That is, K = b Q σ . Since F κ is b oth dense and essentia l in K b y Th eorem 1.7, from [1, Th eorems 3.12 and 3.14] w e get 2 σ ≤ | F κ | ≤ | K | = 2 σ . Hence κ = 2 σ . (c) ⇒ (a) F ollo ws from κ = 2 σ and Lemma 6.5.  16 D. DIKRANJAN, A. GIORDANO BR UNO, AND D. SHAKHMA TOV Pro of of Theorem 2.13. Let G b e a lo cally connected m inimal ab elia n group and K its completion. Let U b e a non-empt y op en connected subset of G . Cho ose an op en subset V of K s uc h that V ∩ G = U . Since U is dense in V and U is connected, so is V . Therefore, K is lo cally connected. App lyin g Lemma 7.1(i), we conclude that K is compact and torsion-fr ee. F rom [13, Corollary 8.8] we get K = { 0 } . Hence G is trivial as well.  8. F ina l rem a rks and o pen que s tions The divisible ab elian groups that admit a minimal group top ology we re describ ed in [8]. Here we need only the part of this c haracterization for d ivisible ab elian groups of size ≥ c . Theorem 8.1. [8] A divisible ab elian gr oup of c ar dinality at le ast c a dmits some minima l gr oup top olo gy pr e cisely when it admits a c omp act gr oup top olo gy. The concept of p seudo compactness generalizes compactness f r om a different angle than that of min im ality . It is therefore qu ite surp rising that minimalit y and pseudo compactness c ombine d to gether “yield” compactness in the class of divisible ab elian groups. Th is should b e compared with Corolla ry 2.7, wh ere a simila r phenomenon (i.e., minimal and pseudo- compact t op ologizations imply compact topologization) occurs for all “ small” groups. The next theorem s ho ws that the coun terpart of the simultaneous minimal and p s eudo- compact top o logizatio n of divisible ab elian groups is m uc h easier than that of free ab elian groups. Theorem 8.2. A divisible ab e lian gr oup admits a minimal gr oup top olo gy and a pseudo- c omp act gr oup top olo gy if and only it admits a c omp act gr oup top olo gy. Pr o of. The necessit y is obvio us. T o pro ve the s u fficiency , supp ose that a divisible a b eli an group G admits b oth a minimal group topology and a pseudo compact group top ology . If G is fin ite, th en G admits a compact group top ology . If G is infinite, then | G | ≥ c by Theorem 1.8. No w the conclusion f ollo ws from Theorem 8.1.  Our n ext example demonstrates th at b oth the r estriction on the cardinalit y in T heorem 8.1 and the hyp othesis of the existence of a p s eudo compact group top ology in T heorem 8.2 are needed: Example 8.3. (a) Th e divisib le ab elian group Q / Z admits a minimal group top ology [10], b ut do es not admit a ps eu do compact grou p top ology (Theorem 1.8). (b) T h e divisible ab elian group Q ( c ) ⊕ ( Q / Z ) ( ω ) admits a (connected) pseudo compact group top ology [13], but do es n ot admit any minimal group top ology . The latter conclusion follo ws from Theorem 8.1 and the fact that this group does not admit an y compact group top ology [19]. Let us briefly discuss the p ossibilities to extend our resu lts for f ree ab elian groups to the case of torsion-fr ee ab elian group s. T heorem 8.2 shows that for divisible torsion-free ab elian groups the situation is in some s ense similar to that of free abelian groups describ ed in Theorem 2.10 : in b oth cases the existence of a pseudo co mpact group top ology and a minimal group top ology is equiv alen t to the existence of a minimal pseu d o compact (actu- ally , compact) group top olo gy . Nev ertheless, ther e is a substant ial difference, b ecause free ab elian groups a dmit no compact g roup top ology . Another imp ortant difference b etw een b oth c ases is th at Pr oblem 1.2 is still open f or torsion-free ab elian groups [9]: Problem 8.4. Char acterize the minimal torsio n-fr e e ab e lian gr oups. A quotien t of a minimal group need not b e m inimal ev en in the ab elian case. This justified the isolati on in [10] of the smalle r class of to tally minimal groups: MINIMAL PSEUDOC OMP ACT GROUP TOPOLOGIES ON FREE ABELIAN GROUPS 17 Definition 8.5. A top ological group G is calle d total ly minimal if ev ery Hausdorff quotient group of G is minimal. Equ iv alen tly , a Hausd orff top ological group G is totally m in imal if ev ery con tin uous group homomorph ism f : G → H of G onto a Hausdorff top ological group H is op en. It is clear that compact ⇒ totally minimal ⇒ minimal. Therefore, Th eorem 2.10 mak es it n atural to a sk the follo wing question: Question 8.6. L e t κ > c b e a c ar dinal. (a) When do es F κ admit a total ly minimal gr oup top olo g y? (b) When do es F κ admit a total ly minimal pseudo c omp act gr oup top olo gy? More sp ecificall y , one can ask: Question 8.7. L et κ > c b e a c ar dinal. Is the c ondition “ F κ admits a zer o-dimensional total ly minimal pseudo c omp act gr oup top olo g y” e quivalent to those of The or em 2.10? Since F c admits a total ly minimal group top ology [21] and a pseudo compact group top ology [1 3], the ob vious c ounter-part of Theorem 2.11 su ggests itself: Question 8.8. A ssu me the Lusin ’s Hyp othesis 2 ω 1 = c . (i) D o es F c admit a total ly minimal pseudo c omp act gr oup top olo gy? (ii) D o es F c admit a total ly minimal pseudo c omp act c onne cte d gr oup top olo gy? (iii) Do es F c admit a total ly minimal pseudo c omp act zer o-dimensional g r oup top olo gy? Let us men tion another class of ab elia n groups where b ot h problems (Problem 1.2 for minimal group top ologies [11] and its coun terpart for pseu do compact group top ologies [6, 13]) are completely resolv ed. These are th e torsion ab elian group s. Nev erth eless, w e do not kno w the answer of th e follo wing question: Question 8.9. L e t G b e a torsion ab elian gr oup that adm its a minimal gr oup top olo gy and a pseudo c omp act gr oup top olo gy. Do es G admit also a minimal pseudo c omp act gr oup top olo gy? W e finish with the q u estion ab out (non-ab elian) free groups. W e n ote that the top ology from Theorem 1.6 is even totally minimal. F ur thermore, a free group F admits a ps eu do- compact group top ology if and only if Ps ( | F | , σ ) h olds for some infinite cardinal σ [13]. This justifies our final Question 8.10. L et F b e a fr e e gr oup that admits a pseudo c omp act g r oup top olo g y. (i) D o es F have a minimal pseudo c omp act gr oup top olo gy? (ii) D o es F have a total ly minimal pseudo c omp act gr oup top olo gy? (iii) Do es F have a (total ly) minimal pseudo c omp act c onne cte d gr oup top olo gy? (iv) Do es F have a (total ly) minimal pseudo c omp act zer o-dimensional gr oup top olo gy? Referen ces [1] E. Bo schi and D. Dikranjan, Essential sub gr oups of top olo gic al gr oups , Comm. Algebra 28 (1996) no. 10, 2325–2339. [2] F. S. Cater, P . Erd¨ os and F. Galvin, On the density of λ -b ox pr o ducts , Gen. T op ol. App l. 9 (1978), 307–312 . [3] W. W. Comfort, T op olo gic al gr oups , in: K . Kunen and J. E. 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(Dikran Dikranjan) Unive rsit ` a di Udine , Di p ar time nto d i Ma tem a tica e Inf orma tica, via delle Scienze, 206 - 33100 Udi ne, It al y E-mail addr ess : dikran.dikranjan@ dimi.uniud.it (Anna Giordano Bruno) Universi t ` a di Udine, Dip ar timento di Ma tema tica e Info rma tica, via delle S cienze, 206 - 33100 Udine, It al y E-mail addr ess : anna.giordanobrun o@dimi.uniud.it (Dmitri Sh ak hmatov) Gradua te Schoo l of Scie n ce and Engi neering, Division of Ma thema tics, Physics and Ear th Scien ce s, Ehim e Unive rsity, Ma tsuy ama 790-8577, Jap an E-mail addr ess : dmitri@dpc.ehime- u.ac.jp