Compact-like abelian groups without non-trivial quasi-convex null sequences

Compact-lik e abelian groups without non-tri via l quasi-con v e x null sequences * D. Dikranjan 1 and G ´ abor Luk ´ acs 2 October 29, 2018 Abstract In this p aper , we stu dy precompac t abe lian groups G th at contain no seq uence { x n } ∞ n =0 such that { 0 } ∪ {± x n | n ∈ N } is infinite and quasi-co n v ex in G , and x n − → 0 . W e characteri ze group s with this property in the followin g classes of groups: (a) bound ed precompac t abeli an groups; (b) minimal abe lian groups; (c) totally minimal abelian group s; (d) ω -bou nded abelian groups. W e also prov ide examples of minimal abelian groups w i th this property , and sho w that there exi sts a minimal pseudocompa ct abelia n gr oup with the same property ; furthermore, under Martin’ s A x iom, the group may be chosen to be counta bly compac t minimal abelian. 1. Intr oduction This note is a sequel to [7], and its aim is to provide an answer to Problem I posed in that paper . One of the main sources of ins piration for the theory of topological groups is the theory of topological vector spaces, wh e re t he notion of con vexity pl a ys a prominent role. In this context, the reals R are replaced with the circle group T : = R / Z , and linear function a ls are repl a ced by character s , that is, continuo us homomorph isms to T . By making substantial use of characters, V ilenkin i ntroduced the notion of quasi-con v exity for abelian top ological groups as a counterpart of con ve xity in topological vector spaces (cf . [30]). Let π : R → T denote the canonical proj e ction. Since the restriction π | ( − 1 2 , 1 2 ] : ( − 1 2 , 1 2 ] → T is a bijection, we often identi f y in what follo ws, par ab us de language , a number a ∈ ( − 1 2 , 1 2 ] with its image (coset) π ( a ) = a + Z ∈ T . W e put T m : = π ([ − 1 4 m , 1 4 m ]) for all m ∈ N \{ 0 } . According to standard notation in this area, we use T + to denote T 1 . For an abeli a n topological group G , we denote by b G the P ontryagin dua l of G , that is, the group of all characters o f G end o wed with the compact-open topology . * 2010 Mathematics Subject Classification: Primary 22A05 54H11; Secondar y 22C05 54D30 1 The first au thor ack no wledges the financial aid received from “Pro getti di Eccellenza 2011/1 2” of Fondazio ne CARIP AR O. 2 The second author gratefully acknowledges the generou s financial support received from NSERC, which enabled him to do this research. 2 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences Definition 1.1. For E ⊆ G and A ⊆ b G , the polar of E and the pr epolar A are defined as E ⊲ = { χ ∈ b G | χ ( E ) ⊆ T + } and A ⊳ = { x ∈ G | ∀ χ ∈ A, χ ( x ) ∈ T + } . The set E is said to be quasi - con ve x if E = E ⊲⊳ . Obviously , E ⊆ E ⊲⊳ holds for every E ⊆ G . Thus, E is qu a si-con vex if and only if for e very x ∈ G \ E there e xists χ ∈ E ⊲ such that χ ( x ) / ∈ T + . The set Q G ( E ) : = E ⊲⊳ is the smallest quasi- con ve x set of G that contains E , and it is called t he quasi-con ve x hull of E . Definition 1 .2. A sequence { x n } ∞ n =0 ⊆ G is said to be quasi -con ve x if S = { 0 } ∪ {± x n | n ∈ N } is quasi-con ve x in G . W e say t hat { x n } ∞ n =0 is non-tri vial if t he set S is infinite, and i t is a n ull sequence if x n − → 0 . Examples 1.3. It tu rns out that most “common” groups contain a non-trivial qu asi-con ve x null sequence: (a) For e very prim e p , the group J p of p -adic integers admits a non-tri vial quasi-con ve x null sequence contained in the subgroup Z (cf. [6, 1.4], [8, Theorem D], and [7, Theorem B]). (b) For every p rime p , T admits a non-trivial qu asi-con ve x null sequence contained in the p -com- ponent of T (cf. [6, 1.1], [8, Theorem B], and [7, Theorem C]). (c) If { m k } ∞ k =1 is a sequence of inte gers such that m k ≥ 4 for e ver y k ∈ N , then ∞ Q k =0 Z m k admits a non-trivial quasi-con ve x null sequence contained in the subgroup ∞ L k =0 Z m k (cf. [7, 5.5]). In our paper [7], we characterized the locally compact abelian groups that admit no non-tri vial quasi-con ve x null sequences as follows. Theor em 1.4. ([7, Theorem A]) For e very locally compact abelian grou p G , the following st ate- ments are equiv alent: (i) G admits no non-trivial quasi-con vex null sequences; (ii) one of the subgroups G [2] = { g ∈ G | 2 g = 0 } or G [3] = { g ∈ G | 3 g = 0 } is open in G ; (iii) G contains an open compact subgroup of the form Z κ 2 or Z κ 3 for some cardinal κ . Furthermore, if G is compact, then these conditions are also equi valent to: (iv) G ∼ = Z κ 2 × F or G ∼ = Z κ 3 × F , where κ is some cardinal and F is a finite abelian group; (v) one of the subgroups 2 G and 3 G is finite. W e also asked wheth er it was possible to replace th e class o f locally compact abeli an groups with a d iffe rent class that c ontains all compact abelian groups (cf. [7, Theorem A]). In t his note, we present sever al classes, and chara cterizations of groups in these classes t hat admit no non-tri vial quasi-con ve x null sequences. T o that end, we recall some compactness-like properties. Definition 1.5. Let G be a (Hausdorff) topological group. (a) G is pr ecompact if it can be covered by finitely many translates of any neighborhood o f t he identity , or equi valently , if it is a dense subgroup of a compact group e G , its completion ; (b) G is minimal if there is no coarser (Hausdorff) group topolo gy (cf. [29] and [12]); (c) G is total ly minimal if e very (Hausdorff) quotient group G is m inimal (cf. [10]); D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 3 (d) G is pseudocompact if ever y real-va lued cont inuous fun ction on G is bounded (cf. [15, 3.1 0]); (e) G is countably compact if ev ery countable open co ver of G admits a finite subcov er ( cf. [15, 3.10]); (f) G is ω -bounded if e very countable subset of G is contained in a compact subgroup of G . By the celebrated Prodanov-Sto yanov T heorem, e ve ry minimal abeli an group is precompact (cf. [26] and [27]), and thus the relations hips amo ng the aforementioned p roperties can be de- scribed as follows. ω -bounded = ⇒ countabl y compact = ⇒ pseudocompact = ⇒ precompact totally minim al = ⇒ minimal abeli an = ⇒ p recompact For greater clar ity , all of these implications, e xcept for the v ery last one, hold without the assump- tion that the group in question is abelian. The classes of groups studi ed in this paper overlap wi th thos e in vestigated in [7] onl y to the smallest possible extent, because e very group that is both precompact and locally compact i s ac- tually compact. Thus , the present note is complementary to our work in [7]. Our first two results demonstrate the level of complexity of the problem of finding non -trivial qu asi-con ve x null se- quences when one lea ves the class of locally comp act abelian groups. Indeed, in the absence of local compactnes s, Theorem 1.4 may fail even in the presence of s trong compactness-li ke proper - ties. Recall that the exponent of an abelian group G is exp G : = inf { n > 0 | nG = 0 } . The g roup G is bounded if it has a finite exponent. Theor em A. Let p = 2 or p = 3 , and let κ be an infinite car dinal. (a) Ther e exists a min imal a belian gr ou p G of e xponent p 2 such that | pG | = κ a nd G admit s no non-trivial quasi-con ve x null sequences. (b) If κ ω = κ , then t her e ex ists a minima l pseudocompact abelia n gr oup of exponent p 2 such that | pG | = κ and G admits no non-tri vial quasi-con ve x null sequences. In part (b) of Theorem A, pG i s a pseudocompact group (being a cont inuous image of G ), and thus | pG | = κ must s atisfy certain constraints that the size of ev ery infinite p seudocompact homo- geneous space does: κ ≥ c , and κ cannot be a s trong lim it cardinal of countable cofinalit y (cf. [14 , 1.2, 1.3(a)]). W e note that both of these conditio ns foll ow from the hy pothesis κ ω = κ . Under the Generalized Conti nuum Hypot hesis (GCH), κ ω > κ for a cardinal κ ≥ c if and only if κ is a strong limit cardinal of countabl e cofinality . Therefore, under GCH, the hypothesi s κ ω = κ i s not only suffi cient but also necessary for the e xistence of a group G as in Theorem A(b). Theor em B. Under Martin’ s Axiom (or the Continuum Hypothesis), ther e e xists a countably com- pact minim al abelian gr oup of e xponent 4 such that 2 G is infinite and G admits no non-tri vial quasi-con ve x null sequences. The proofs of Theorems A and B are presented in § 3, and they are based on “lifting” kno wn examples of precompact abelian groups of exponent p that admit no non-trivial con ve rgent se- quences at all (cf. [16], [5, Theorem 3], [13, 8.1 & 9], and [18]). Although Theorem B may appear to suggest a negative answer to [7, Problem II], it is possible to establis h a criterion in the s pirit of Theorem 1.4 for precompact groups of a finite exponent. T o that end, recall that a subset A of a top ological space X is sequentially open if for e ver y con ver gent sequence { x n } ⊆ X such th at lim x n ∈ A , on e has x n ∈ A for all but finitely many n . 4 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences Theor em C. F or e very bounded pr ecompact abelian gr oup G , the following statements ar e equiv- alent: (i) G admits no non-trivia l quasi-con ve x null sequences; (ii) one o f the subgr oups G [2] = { g ∈ G | 2 g = 0 } or G [3] = { g ∈ G | 3 g = 0 } is sequ entially o pen in G . The proof of Theorem C is presented in § 4. Certainly , the entire problem of searching for non-trivial quasi-con vex null sequences is mean- ingless in a group that a dmits no non-trivial con ver gent sequences at all. While ω -bo unded groups are known to ha ve non-trivial c on ver gent sequences, our interest in minimal abelian groups is also motiv ated by a recent result of Shakhmatov , which guarantees th e existence of non-t rivial con ver - gent sequences in minim al abe lian groups (cf. [28, 1.3]). Theor em D. F or ev ery minimal abelian gr oup G , the following statements ar e equivalent: (i) G admits no non-trivia l quasi-con ve x null sequences; (ii) G ∼ = P × F , wher e P is a minimal bounded abelian p -gr oup ( p ≤ 3 ) admitt ing n o non-trivial quasi-con ve x null sequences, and F is a finite abelian gr oup; (iii) one of the subgr oups G [2] = { g ∈ G | 2 g = 0 } or G [3] = { g ∈ G | 3 g = 0 } is sequentially op en in G ; (iv) G contains a sequentially open compact subgr oup of the form Z κ 2 or Z κ 3 for some car dinal κ . The following result sh ows that much of Theorem 1.4 can be salvaged by imposing strong er compactness-like properties. Theor em E. The following statements ar e equivalent for every abelian gr oup G th at is ω -bounded or totally minimal: (i) G admits no non-trivia l quasi-con ve x null sequences; (ii) one of the subgr oups G [2] = { g ∈ G | 2 g = 0 } or G [3] = { g ∈ G | 3 g = 0 } is open in G ; (iii) one of the subgr oups 2 G an d 3 G is finite. Furthermor e, if G is total ly minimal, then these conditions ar e also equivalent to: (iv) G ∼ = Z κ 2 × F or G ∼ = Z κ 3 × F , wher e κ is some car dinal and F is a finite abelian gr oup. Theorems A and B sho w that in Theorem E , one cannot weak en “ ω -bounded” to “countably compact and minim al” (or to “pseudocompact and minimal”), and “totally minimal” to “minimal” in Theorem E. Therefore, Theorem D is the best on e can achieve in the class of mi nimal abelian groups. The proofs of Theorems D and E are presented in § 6. The totally mini mal case of The- orem E relies on intermediate steps in the proof of Theorem D. One of the main ingredients of Theorem D is the following result. Theor em F. Let { q n } ∞ n =0 be a sequence of positive inte ger s, and put b n = q 0 · · · q n for e very n ∈ N . If q n ≥ 8 f or every n ∈ N , then { 1 b n } ∞ n =0 is a quasi-con ve x sequence in T . Theorem F im plies that the s ubgroup of Q / Z generated by element s of prime o rder admits a non-tri vial quasi-con vex null sequence. W e note that t he condition q n ≥ 8 in Theorem F is un- necessarily restricti ve, and can be replaced with q n ≥ 5 ; howe ver , the proof of the latter is more complicated and longer , and Theorem F is suf ficient for establishing Theorem D. The proof of Theorem F is presented in § 5. D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 5 Finally , we t urn to posing s ome open problems. As we mentio ned earlier , under GCH, the hypothesis κ ω = κ is not only sufficient but als o necessary f or the e xistence of a group G as in The- orem A(b). Thus, in ZFC, it is not possible to prove Theorem A(b) without the assumptio n κ ω = κ . Pr oblem I. Let p = 2 or p = 3 , and let κ be a car dinal such that κ ≥ c and κ ω > κ . Is it consistent that ther e e xists a minimal pseudocompact abelian gr oup of e xponent p 2 such that | pG | = κ an d G admits no non-trivial quasi-con ve x null sequences? W e show in § 3 that if one of the subgroups G [2] or G [3] is sequential ly open in a (locally) precompact group G , then it contains no non-trivial quasi-con vex n ull sequences (Proposition 3.2). Theorems C , D, and E state that the con verse of this implication i s als o true in each o f the classes of bounded precompact, mi nimal, totally minimal, and ω -bounded abelian groups . Thus, it is natural to ask whether the implication re mains re versible in general, without a ssumi ng some compact-like properties. Pr oblem II. Let G be a (locally) pr ecompa ct gr oup, and suppose that G admits no non-trivial quasi-con ve x null sequences. Is one of the subgr oups G [2] or G [3 ] sequentially open in G ? 2. Preliminaries In this section, we provide a fe w well-known definit ions and results that we rely on i n the paper . W e start of f by re calling some terminology from duality theory . Let H be a subgroup of an abelian topological group G . The annihi lator of H in b G is the subgroup H ⊥ : = { χ ∈ b G | χ ( H ) = { 0 }} . The subgroup H is du ally closed i n G if H = T { k er χ | χ ∈ H ⊥ } . Since H ⊥ = H ⊲ for e very subg roup, H i s dually closed in G if and only if it is quasi-con ve x in G . The subgroup H i s dually embedded in G if e very continuou s character of H has an extension to a continu ous character of G , that is, the restriction homomo rphism b G → b H i s surjectiv e. Examples 2.1. (a) If H i s a n open subgroup of the abelian topo logical grou p G , t hen H is dually c losed and dually embedded in G (cf. [23, 3.3]). (b) If H is a clo sed subgroup of a locally compact abel ian group L , then H is d ually clo sed and dually embedded in L (cf. [24, Theorems 37 and 54]). (c) If H is a dense s ubgroup of an abelian topological group G , then H is dually embedded in G . (d) Every subgroup of a locally pre compact abelian group is dually embedded in it. Lemma 2.2. Let G be an abelian topological gr oup. (a) If H is a dually embedded subgr oup of G , then Q H ( S ) = Q G ( S ) ∩ H for e very subset S of H . (b) If H i s a du ally clos ed and dually embedded subgroup of G , then Q H ( S ) = Q G ( S ) for every subset S of H . Lemma 2.2 is si milar t o [7, 5.1], and i ts proo f, w hich relies on the foll owing general property of the quasi-con vex hull, is provided here only for the sake of completeness. Lemma 2.3. ([21, I.3(e)], [6, 2.7]) If f : G → H is a continuous homomorphism of abel ian topo- logical groups, and E ⊆ G , then f ( Q G ( E )) ⊆ Q H ( f ( E )) . 6 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences P R O O F O F L E M M A 2 . 2 . Let S ⊆ H be a subset. (a) Let ι : H → G denote the inclus ion map.By Lemma 2.3, Q H ( S ) ⊆ ι − 1 ( Q G ( S ) ) = Q G ( S ) ∩ H . T o sho w the reverse inclusion, let h ∈ Q G ( S ) ∩ H . If χ ∈ b H is such that χ ( S ) ⊆ T + , then χ = ψ | H for som e ψ ∈ b G (as H is dually embedd ed in G ), and one has ψ ( S ) = χ ( S ) ⊆ T + . Consequ ently , χ ( h ) = ψ ( h ) ∈ T + , because h ∈ Q G ( S ) . This shows that Q G ( S ) ∩ H ⊆ Q H ( S ) , as required. (b) Since H is dually closed, Q G ( S ) ⊆ Q G ( H ) = H . Consequently , the statement follows from part (a). Examples 2.1 combined with Lemma 2.2 yields the following consequences. Corollary 2.4. Let G be a ( locally ) p r ecompact abelian gr oup, and H a closed subgr ou p. Then H is dually closed and dual ly embedded in G , and if { x n } ⊆ H is a quasi-con ve x sequence, then { x n } is quasi-con vex in G . P R O O F . By Example 2.1(d), H is dually embedd ed in G . By Example 2.1(b), cl e G H is dually closed in the completion e G . Thus, by Lemma 2.2(a), Q G ( H ) = Q e G ( H ) ∩ G ⊆ Q e G (cl e G H ) ∩ G ⊆ (cl e G H ) ∩ G = H , because H is closed i n G . This shows that H is dually closed in G . The second statement follows now by Lemma 2.2(b). Corollary 2.5. Let G be a ( locally ) pr ecompact abelian gr oup, H a (not necessaril y closed) sub- gr oup, { x n } ⊆ G a quasi -con ve x sequence s uch that x n ∈ H for infi nitely many n ∈ N , and { x n k } the subsequence o f { x n } consisting of al l members that belong to H . Then { x n k } is q uasi-con ve x in H . P R O O F . By Example 2.1(d), H is dually embedded in G . Put S : = {± x n | n ∈ N } ∪ { 0 } and set S ′ : = {± x n k | k ∈ N } ∪ { 0 } . By Lemm a 2.2(a), Q H ( S ′ ) = Q G ( S ′ ) ∩ H ⊆ Q G ( S ) ∩ H = S ∩ H . It follows from the definition of the subsequence { x n k } that S ∩ H = S ′ , as desired. W e turn now to mini mality and total minimality . Theor em 2.6 . Let G be an abelian group with completion e G . Then: (a) ([29, Theorem 2],[25], [2, Proposit ions 1 and 2], [22, 3.31]) G is m inimal if and only if G is precompact and G ∩ H 6 = { 0 } for e very non-tri vial closed subgroup H of e G ; (b) ([10], [22, 3.31]) G is totally minimal if and only if G is precompact and G ∩ H is dense in H for e very closed subgroup of e G . Recall t hat the socle of soc( A ) of an abelian group A is the subgroup of torsion elements whose order is square-free (that is, n ot di visible by the square of a prime number), or equiv alently , the direct sum of the subgroups A [ p ] : = { x ∈ A | px = 0 } , where p ranges over all primes. W e put to r( A ) for the torsion subgroup of A . D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 7 Corollary 2.7. Let G be an abelian group with completion e G . (a) If G is minimal, then so c( e G ) ⊆ G . (b) ([11, 4.3.4]) If G is totally minimal, then tor( e G ) ⊆ G . Furthermore, if e G is a bounded compact abelian group, then the con ver se of (a) is also true. Definition 2 .8. ([9], [11, p. 141]) A compact abelian group is an e xotic tor us if it cont ains no subgroup that is topolog ically isomorph ic to the J p ( p -adics) for some prime p . The notion of exotic torus was introduced by Dikranjan and Prodanov in [9], who also provided, among other things, the following characterization for such groups. Theor em 2.9. ([9], [ 7, 2.6]) A c ompact abelian group K is an exotic torus if and only if i t contains a closed subgroup B such that (i) K/ B ∼ = T n for some n ∈ N , and (ii) B = Q p B p , where each B p is a compact bounded abelian p -group. Furthermore, if K is connected, then each B p is finite. Recall that a topological group i s pr o-finite if it is th e (projective) li mit of finite groups, or equiv alently , if it is compact and zero-dimens ional. For a prim e p , a topological g roup G is called a pr o- p -gr oup if it is th e (proj ectiv e) lim it of fini te p -groups, o r equiv alently , if it is p ro-finite and x p n − → e for every x ∈ G (or , in the abelian case, p n x − → 0 ). Theor em 2.10. ([1], [20, Corollary 8.8(ii)], [11, 4.1.3]) Let G be a n abelian pro-finite group. Then G = Q p G p , where each G p is a pro- p -group. Finally , we note for the sake of clarity that in our notation, N = { 0 , 1 , 2 , . . . } , that is, 0 ∈ N . 3. Counter examples Theor em A. Let p = 2 or p = 3 , and let κ be an infinit e c ar dinal. (a) Ther e exists a min imal a belian gr ou p G of e xponent p 2 such that | pG | = κ a nd G admit s no non-trivial quasi-con ve x null sequences. (b) If κ ω = κ , then t her e ex ists a minima l pseudocompact abelian gr oup of exponent p 2 such that | pG | = κ and G admits no non-tri vial quasi-con ve x null sequences. Theor em B. Under Martin’ s Axiom (or the Continuum Hypothesis), ther e exists a countably com- pact minim al abelian gr oup of e xponent 4 such that 2 G is infinite and G admits no non-tri vial quasi-con ve x null sequences. In this section, we prov e Theorems A and B. Lemma 3.1. ([7, 5.3]) If G is an a belian topologi cal group of exponent 2 or 3 , then G admits no non-trivial quasi-con vex null sequences. Pr oposition 3.2. Let G be a ( locally ) pr ecompact abelian gr oup such that G [2] o r G [3] is sequen- tially open in G . Then G admit s no non-trivial quasi-con vex null sequences. In particular , if 2 G or 3 G admit s no non-trivial null sequences, then G [2] or G [3] is sequentially open in G , and hence G admits no non-trivial quasi-con ve x null sequences. 8 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences P R O O F . Suppose that G [ p ] is sequ entially open i n G , where p = 2 o r p = 3 , and let { x n } ⊆ G be a quasi-con v ex null sequence. Then x n ∈ G [ p ] for all but finitely many n ∈ N . Let { x n k } denote the subsequence of { x n } consisting of all members that belong to G [ p ] . By C orollary 2.5, { x n k } is a quasi-con vex null sequ ence in G [ p ] . Therefore, by Lemma 3.1, { x n k } is trivial, and hence { x n } is trivial. In order to show the second st atement, we ob serve that pG is a continuou s homom orphic image of G/G [ p ] . Thus, if pG has no n on-trivial null sequences, t hen G /G [ p ] has no n on-trivial null sequences either , and therefore G [ p ] is sequentially open in G . Lemma 3.3. Let P be a topological pr operty t hat is an i n verse in variant of open perfect maps, p a pri me n umber , and D a non-tr ivial p r ecompact abelian gr oup of exponent p with pr operty P . Then ther e e xists a minimal abelian gr oup of e xponent p 2 with pr operty P s uch that pG ∼ = D . P R O O F . Since D has exponent p , so does its completion e D , and thus e D ∼ = Z λ p for some cardinal λ (cf. [11, 4 .2.2]). Put K : = Z λ p 2 , and let f : K → pK ∼ = e D denote the continuous hom omorphism de- fined by f ( x ) = px . Since K is compact, the map f is open and perfect. Consequently , G : = f − 1 ( D ) has property P , the exponent of G is p 2 (because pG = f ( G ) = D is non-trivial), and G is dense in K . In particular , K = e G , and thu s so c( e G ) = K [ p ] = ke r f ⊆ G . Therefore, by Corollary 2.7(a), G is minim al. P R O O F O F T H E O R E M A . (a) Let κ b e an i nfinite cardinal, and let D denote the di rect sum Z ( κ ) p equipped with t he Bohr-topology . By Lemma 3.3 (with P the t rivial property), there exists a mi n- imal abelian group G o f exponent p 2 such that pG ∼ = D . By a well-known theorem of Flor , the Bohr -topology o f a discrete abeli an group admits no non-trivial conv er gent sequences (cf. [16]). Hence, by Proposition 3.2, G admits no non-trivial quasi-con ve x null sequences. This completes the proof, because | pG | = | Z ( κ ) p | = κ . (b) Le t κ be an infinite cardinal such that κ ω = κ . By a theorem of Dijkstra and v an Mill (cf. [5, Theorem 3]; see also [17, 5.8]), the compact group Z κ p admits a subgroup D such that: (1) D contains no non-trivial con ver gent sequences; (2) D is dense in the G δ -topology of Z κ p ; and (3) | D | = κ . It follows from property (2) that D is pseudocompact (cf. [4, 1. 2]). Since pseudocom pactness is an in verse in va riant of open perfect maps (cf. [15, 3.10.H]), by Lemma 3.3, there exists a pseudo - compact minim al group G o f exponent p 2 such that pG ∼ = D . Hence, by Proposit ion 3.2, G admits no non-trivial quasi-con ve x n ull sequences. Thi s completes the proo f, because | pG | = | D | = κ by property (3). P R O O F O F T H E O R E M B . V an Douwen showe d that under MA, there e xists a n infinite countably compact abelian group D o f exponent 2 that admi ts no non-tri vial con ver gent sequences (cf. [13, 8.1]), and he also observed t hat under CH, a const ruction o f Hajnal and Juh ´ asz yi elds a group D with the same properties ( cf. [18] and [13, 9]) . Since countable compactness is an in verse i n v ariant of perfect maps (cf. [15, 3 .10.10]), by Lemma 3.3, there e xists a countably compact minimal group G of exponent 4 such that 2 G ∼ = D . Hence, by Proposition 3.2, G admits no non -trivial quasi-con vex null sequences. D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 9 4. Bounded pr ecompact groups without non-tri vial quasi-con vex null sequ ences Theor em C. F or ev ery bounded pr ecompact abelian gr oup G , the follo wing statements ar e equiv- alent: (i) G admits no non-trivia l quasi-con ve x null sequences; (ii) one o f the subgr oups G [2] = { g ∈ G | 2 g = 0 } or G [3] = { g ∈ G | 3 g = 0 } is sequ entially o pen in G . In this section , we prove Theorem C. Recall that a set { f 1 , . . . , f n } of non -zero elements in an abelian group G is independent if whenev er n P i =1 l i f i = 0 for s ome l i ∈ Z , then l i f i = 0 for every i , or equiv alently , if h f 1 , . . . , f n i = h f 1 i ⊕ · · · ⊕ h f n i . In what fol lows, o ( g ) denotes the order of an element g in a group. Lemma 4.1. Let { f 1 , . . . , f n } be an independent subset of a (locally) pr ecompact abelian gr oup G . If 4 ≤ o ( f i ) < ∞ for every 1 ≤ i ≤ n , then X : = { 0 } ∪ {± f 1 , . . . , ± f n } is quasi-con vex in G . P R O O F . Set m k : = o ( f k ) for 1 ≤ k ≤ n , m k = 4 for k > n , and P : = ∞ Q k =1 Z m k . For e ve ry k ≥ 1 , put e k : = (0 , . . . , 0 , 1 , 0 , . . . ) , with 1 at the k -t h coordinate a nd zero elsewhere . The authors showed in [7, 5.5] that S : = { 0 } ∪ {± e k | k ≥ 1 } is quasi-con ve x in P . Put F : = h f 1 , . . . , f n i = h f 1 i ⊕ · · · ⊕ h f n i . Clearly , F i s finite. Cons equently , th e homom orphism ϕ : F → P defined by ϕ ( f i ) = e i for 1 ≤ i ≤ n i s an embedding of topological groups. Therefore, by Lemma 2.3 , X = ϕ − 1 ( S ) is quasi-con v ex i n F . Hence, by Corollary 2.4 and Lemma 2 .2(b), X is quasi-con ve x in G . Pr oposition 4.2. Let E and F be finite ab elian gr oups, and su ppose that e xp E ≥ 4 . Then every dense subgr oup A ≤ F × E ω contains a n on-trivial null sequence th at is qua si-con ve x both in A and in F × E ω . P R O O F . For e very positive int eger n , let π n : F × E ω → F × E n denote the canonical projection of the first n + 1 coordinat es. Pick y ∈ E such that o ( y ) = exp E . Since A is dense in F × E ω , one has π n ( A ) = F × E n . Thus, for every n , we may pick x n ∈ A such that π n ( x n ) = (0 , . . . , 0 , y ) . W e claim that { x n } is a quasi-con vex null sequence in A and F × E ω . Step 1. W e show by induction on n that the set { π n ( x 1 ) , . . . , π n ( x n ) } is independent in F × E n . For n = 1 , the statement is trivial, because π 1 ( x 1 ) is n on-zero. Assume now that the statement hold s for n , and suppose that n +1 P i =1 l i π n +1 ( x i ) = 0 for l i ∈ Z . Then n +1 P i =1 l i π n ( x i ) = 0 , and thus n P i =1 l i π n ( x i ) = 0 , because π n ( x n +1 ) = 0 . By the inductive hyp othesis, it follo ws that l i π n ( x i ) = 0 for 1 ≤ i ≤ n . The ( i + 1) -th coordinate of π n ( x i ) is y , and so o ( y ) | l i for 1 ≤ i ≤ n . Therefore, l i x i = 0 for 1 ≤ i ≤ n , because o ( y ) = exp E . Hence, l n +1 π n +1 ( x n +1 ) = − n P i =1 l i π n +1 ( x i ) = 0 , as required. Step 2. Put S : = { 0 } ∪ {± x n | n ∈ N } . By Lemma 4.1, π n ( S ) = { 0 } ∪ {± π n ( x 1 ) , . . . , ± π n ( x n ) } is quasi-con ve x in F × E n , because o ( π n ( x i )) = o ( y ) = exp E ≥ 4 . Thus, by Lemma 2.3, Q F × E ω ( S ) ⊆ π − 1 n ( π n ( S ) ) = S + ke r π n 10 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences for e very n . Therefore, Q F × E ω ( S ) ⊆ ∞ \ n =1 ( S + k er π n ) = cl F × E ω S, (1) because { k er π n } ∞ n =1 is a base for the topology of F × E ω at ze ro. Since x k ∈ k er π n for e v ery k > n , it follo ws that { x n } is a null sequence, and S is closed in F × E ω . Hence , by (1), S is quasi-con v ex in F × E ω , as desired. By Corollary 2.5, this implies that { x n } is quasi-con vex in A as well . Corollary 4 .3. Let K be an infinite bounded compact metrizable ab elian g r oup t hat contains no open compact subgr oup of the form Z ω 2 and Z ω 3 . Then every dense subgr oup A of K contain s a non-trivial null sequence that is quasi-con ve x in both A and K . P R O O F . Since K is bounded compact abelian, it is topologi cally isomorphi c to a product of finite cyclic groups (cf. [11, 4.2.2]). The number of the fa ctors is countably infinite, because K i s me- trizable and i nfinite, and the n umber of non-iso morphic factors is finite, as K is bounded. Thus, K ∼ = F × Z ω m 1 × · · · × Z ω m l , where F is a fini te abelian group and m 1 , . . . , m l are d istinct integers. Consequently , for E : = Z m 1 × · · · × Z m l , one has K ∼ = F × E ω . Since K cont ains no open com pact subgroup of the form Z ω 2 and Z ω 3 , clearly E 6 = Z 2 and E 6 = Z 3 . Therefore, exp( E ) ≥ 4 , and hence the statement follows by Proposition 4.2. Corollary 4.4. Let A be a bounded p r ecompact metrizable abel ian gr oup. If the subgr oups A [2] and A [3] ar e n ot open in A , then A contains a non-trivial nul l sequence that is quasi-con ve x in both A and the completion e A of A . P R O O F . Put K : = e A . Since A is metrizable and bounded, so is K . One has A [ p ] = K [ p ] ∩ A , and thus K [2 ] and K [3] are not open in K . In particular , K i s infinite, and it c ontains no open c ompact subgroup of the form Z ω 2 and Z ω 3 . Hence, the statement follows from Corollary 4.3. Lemma 4.5. If A is a bounded pr ecompact abelian gr oup generated by a nul l sequ ence { w n } , then A is metriz able. P R O O F . Let m denote the exponent of A . Clearly , e very character χ ∈ b A i s completely determined by the values χ ( w n ) taken at the generators of A . Since { w n } is a nu ll sequence and χ is cont inuous, χ ( w n ) → 0 in T , and χ ( x n ) belongs to the cyclic subgroup of order m in T , because mA = 0 . Consequently , χ ( x n ) = 0 for all but finitely many indices n . Therefore, b A is countable. Hence, A is metrizable (cf. [19, 2.12]). P R O O F O F T H E O R E M C . (ii) ⇒ (i): This impl ication holds even without the assum ption that G is bounded, and has already been shown in Proposition 3.2. (i) ⇒ (ii): Suppose that neith er G [2] nor G [3] is sequ entially open in G . Then t here are se- quences { y n } and { z n } i n G that witness t hat G [2] and G [3] are not sequent ially op en. In other words, y n → y 0 ∈ G [2] , b ut y n / ∈ G [2] for infinitely many indices n , and z n → z 0 ∈ G [3] , b ut z n / ∈ G [3] for i nfinitely many indices n . By replacing { y n } with { y n − y 0 } and { z n } with { z n − z 0 } , we may assume that y n → 0 and z n → 0 . Let { w n } denote the alternating sequence y 1 , z 1 , y 2 , z 2 , . . . . Clearly , { w n } is a null sequence, and w n / ∈ G [2] for infinitely many indices n and w n / ∈ G [3] for i nfinitely m any indices n . Let A D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 11 denote the subgroup of G generated by { w n } . Then A is a bounded precompact abelian group, and by Lemma 4.5 , A i s metrizable. Although w n → 0 , one has w n / ∈ A [2] = G [2] ∩ A for infinitely m any indices n and w n / ∈ A [3] = G [3] ∩ A for infinitely m any indices n . Thus, the subgroups A [2] and A [3] are not open in A . Therefore, by Corollary 4.4, there is a non-trivial null sequence { x n } ⊆ A s uch that { x n } is quasi-con vex both in A and t he completion e A of A . Since the completion e A is a closed subgroup of the completion e G of G , by Corollary 2.4, { x n } is also quasi-con vex in e G . Hence, by Corollary 2.5, { x n } is quasi-con v ex in G , b ecause { x n } ⊆ A ⊆ G . 5. Seque nces of the form { 1 b n } ∞ n =0 in T Theor em F. Let { q n } ∞ n =0 be a sequence of positive inte ger s, and put b n = q 0 · · · q n for every n ∈ N . If q n ≥ 8 f or every n ∈ N , then { 1 b n } ∞ n =0 is a quasi-con ve x sequence in T . In this section, we prove T heorem F. Although in Theore m F itself we require q n ≥ 8 , a number of intermediate statem ents remain true under no condit ions at all or weaker conditions i mposed upon t he sequence { q n } . Consequentl y , we con sider { q n } ∞ n =0 (and thus { b n } ∞ n =0 ) a fixed sequence of positive int egers and set X : = { 0 } ∪ {± 1 b n | n ∈ N } thro ughout this section, but make no further assumption s about th eir properties; instead, we impose conditions on the { q n } in each statement as needed. The first step tow ard the proof of Theorem F is to establish a standard method for describing elements in Q T ( X ) . As we hav e seen i n [8] and [7], finding a con v enient way to represent elements of T is a useful t ool i n calculating quasi-con vex hu lls o f s equences. Let { d i } ∞ i =0 be an increasing sequence o f posit iv e integers such that d i | d i +1 for every i ∈ N . Then z ∈ T (which, as we st ated in the Introduction, is identified with ( − 1 2 , 1 2 ] ) can be expressed in the form z = ∞ P i =0 c i d i , with c i integers such that | c i | ≤ d i 2 d i − 1 . (W e consider d − 1 = 1 .) This representation, ho we ver , need not be unique: For example, if d 0 = 3 and d 1 = 6 , then 1 6 can be expressed with c 0 = 0 and c 1 = 1 , b ut also wit h c 0 = 1 and c 1 = − 1 . In order t o elim inate this anom aly , we say that t he representation of z is standa r d if the following c onditi ons are satisfied: (i) c i ∈ Z and | c i | ≤ d i 2 d i − 1 for all i ∈ N ; (ii)     z − k P i =0 c i d i     ≤ 1 2 d k for e very k ∈ N ; (iii) if     z − k P i =0 c i d i     = 1 2 d k for some k , then    c k d k    <     z − k − 1 P i =0 c i d i     . (In t he aforementioned example, c 0 = 0 and c 1 = 1 is a standard representation of 1 6 , but c 0 = 1 and c 1 = − 1 is not a standard one.) The ne xt theorem is in the vein of [8, 4.3] a nd [7, 3.2], and plays an important role in the proof of Theorem F. Theor em 5.1. If q k +1 ≥ 4 whene ver q k = 7 , t hen Q T ( X ) ⊆  ∞ P i =0 ε i b i | ε i ∈ {− 1 , 0 , 1 }  . Furth ermor e, every x ∈ Q T ( X ) admits a standar d r epr esentation ∞ P i =0 ε i b i with ε i ∈ {− 1 , 0 , 1 } . 12 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences The proof of Theorem 5.1 requires two preparatory steps. The first one is an analogue of [7, 2.1, 2.2, 2.4], for which we introduce our own rounding functions: For x ∈ R , we put ⌈ x ⌉ : = min { n ∈ Z | x < n } , [ x ] : = max { n ∈ Z | n ≤ x } , ⌊ x ⌋ : = max { n ∈ Z | n < x } . W e not e t hat these are not th e u sual definitions of the floor and ceiling funct ions (as we us e strict inequality in both). Lemma 5.2. Let z = ∞ P i =0 c i d i ∈ T be a standar d r epr esentation . (a) If mz ∈ T + for all m = 1 , . . . , ⌈ d 0 6 ⌉ , then c 0 ∈ {− 1 , 0 , 1 } . (b) If mz ∈ T + for all m = 1 , . . . , [ d 0 4 ] and d 0 6 = 7 , t hen c 0 ∈ {− 1 , 0 , 1 } . (c) If mz ∈ T + for all m = 1 , . . . , [ d 0 4 ] and for m = d 0 − 1 , t hen c 0 ∈ {− 1 , 0 , 1 } . P R O O F . (a) Put l = ⌈ d 0 6 ⌉ . Since mz ∈ T + for all m = 1 , . . . , l , one has z ∈ { 1 , . . . , l } ⊳ = T l , and thus | z | ≤ 1 4 l < 3 2 d 0 . Therefore,     c 0 d 0     ≤ | z | +     z − c 0 d 0     < 3 2 d 0 + 1 2 d 0 = 2 d 0 . Hence, | c 0 | < 2 , as desired. (b) For d 0 = 2 and d 0 = 3 , the conclusi on is tri vial. If d 0 6 = 6 , 7 , then one has ⌈ d 0 6 ⌉ ≤ [ d 0 4 ] , and the statement follows from part (a). Suppos e that d 0 = 6 . Th en it i s give n that z ∈ T + , and thu s | z | ≤ 1 4 . If | z − c 0 d 0 | < 1 2 d 0 , then     c 0 d 0     ≤ | z | +     z − c 0 d 0     < 1 4 + 1 2 d 0 = 2 d 0 , and t hus | c 0 | < 2 . If | z − c 0 d 0 | = 1 2 d 0 , t hen | c 0 d 0 | < | z | ≤ 1 4 , because the representation of z is standard. Hence, | c 0 | < 2 , as desired. (c) If d 0 6 = 7 , then the statement follows from (b), and so we may suppose that d 0 = 7 . Then it is giv en that z , 6 z ∈ T + , which means that z ∈ { 1 , 6 } ⊳ = T 6 ∪ ( − 1 6 + T 6 ) ∪ ( 1 6 + T 6 ) . Thus, | z | ≤ 5 24 . Therefore,     c 0 d 0     ≤ | z | +     z − c 0 d 0     ≤ 5 24 + 1 14 = 47 168 < 48 168 = 2 d 0 . Hence, | c 0 | < 2 . The second preparatory step t o precede the proof of Theorem 5.1 in v olves finding characters in X ⊲ . Let η 0 : T → T denote the identit y homomorphism, and for k ≥ 1 , set η k : = b k − 1 η 0 . Lemma 5.3. If q k ≥ 4 , then mη k ∈ X ⊲ for m = 1 , . . . , [ q k 4 ] . If in additi on q k +1 ≥ 4 , then mη k ∈ X ⊲ also for m = q k − 1 . D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 13 P R O O F . Fix n ∈ N . If n < k , then η k ( 1 b n ) = b k − 1 b n ≡ 1 0 , and thus mη k ( 1 b n ) ∈ T + . If n = k , then η k ( 1 b n ) = b k − 1 b k = 1 q k , and so mη k ( 1 b n ) = m q k ∈ T + for m = 1 , . . . , [ q k 4 ] and m = q k − 1 . If n > k , then η k ( 1 b n ) = b k − 1 b n = 1 q k q k +1 ··· q n and mη k ( 1 b n ) = m q k q k +1 ··· q n . Consequently , mη k ( 1 b n ) ∈ T + for m = 1 , . . . , [ q k 4 ] . If q k +1 ≥ 4 , then q k − 1 q k q k +1 ··· q n ≤ 1 q k +1 ≤ 1 4 , and hence ( q k − 1) η k ( 1 b n ) ∈ T + . W e are now ready to prove Theore m 5.1. P R O O F O F T H E O R E M 5 . 1 . Let x ∈ Q T ( X ) , and let x = ∞ P i =0 c i b i be a standard representation of x . Fix k ∈ N , and put d i : = b k + i b k − 1 = q k · · · q k + i for e very i ∈ N . (As usual, we consider b − 1 = 1 .) Then z : = η k ( x ) = b k − 1 x ≡ 1 ∞ P i =0 c k + i d i , and it is a stand ard representati on of z , because x = ∞ P i =0 c i b i is a stan- dard representation of x . (Indeed, | c k + i | ≤ b k + i 2 b k + i − 1 = d i 2 d i − 1 , while conditi ons (ii) and (iii ) follows by observing t hat     z − m P i =0 c k + i d i     = b k − 1     x − k + m P i =0 c i b i     for e very m ∈ N .) Furthermore, i f mη k ∈ X ⊲ , t hen mz = mη k ( x ) ∈ T + . If q k < 4 , then | c k | ≤ q k 2 < 2 , and there is nothing to prov e. So, we may assume that q k ≥ 4 . Thus, by Lem ma 5.3, mη k ∈ X ⊲ for m = 1 , . . . , [ q k 4 ] . Consequentl y , mz ∈ T + for m = 1 , . . . , [ q k 4 ] . Therefore, if q k 6 = 7 , then b y Lem ma 5.2( b), the first coef ficient o f z , that is c k , sat isfies c k ∈ {− 1 , 0 , 1 } . If q k = 7 , then by our assumption, q k +1 ≥ 4 , and by Lemma 5.3, ( q k − 1) η k ∈ X ⊲ . Consequently , by Lemma 5.2(c), the first coef ficient of z , that is c k , satisfies c k ∈ {− 1 , 0 , 1 } . The next lem ma is som e what similar to [7, 3.3], bo th in its content and its role in the proof of Theorem F. Lemma 5.4. Let k 1 , k 2 ∈ N be such that k 1 < k 2 . Then [ q k 1 4 ] η k 1 ± ⌊ q k 2 4 ⌋ η k 2 ∈ X ⊲ . P R O O F . Let n ∈ N . If n < k 2 , then η k 2 ( 1 b n ) = b k 2 − 1 b n ≡ 1 0 , and thus ([ q k 1 4 ] η k 1 ± ⌊ q k 2 4 ⌋ η k 2 )( 1 b n ) ≡ 1 [ q k 1 4 ] η k 1 ( 1 b n ) ∈ T + , because [ q k 1 4 ] η k 1 ∈ X ⊲ by Lemma 5.3. Suppose now that k 2 ≤ n . Th en | [ q k 1 4 ] η k 1 ( 1 b n ) | ≤ [ q k 1 4 ] q k 1 q k 2 ≤ 1 4 q k 2 , and |⌊ q k 2 4 ⌋ η k 2 ( 1 b n ) | ≤ ⌊ q k 2 4 ⌋ q k 2 ≤ 1 4 − 1 4 q k 2 . Therefore, | ([ q k 1 4 ] η k 1 ± ⌊ q k 2 4 ⌋ η k 2 )( 1 b n ) | ≤ | [ q k 1 4 ] η k 1 ( 1 b n ) | + |⌊ q k 2 4 ⌋ η k 2 ( 1 b n ) | ≤ 1 4 , and hence ([ q k 1 4 ] η k 1 ± ⌊ q k 2 4 ⌋ η k 2 )( 1 b n ) ∈ T + . This sho ws that [ q k 1 4 ] η k 1 ± ⌊ q k 2 4 ⌋ η k 2 ∈ X ⊲ . 14 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences W e introduce furt her notations to facilitate calculations i n T . Let z = ∞ P i =0 c i d i be a st andard rep- resentation of z ∈ T with respect to a sequence { d i } ∞ i =0 such that d i | d i +1 for e very i ∈ N . W e put Λ( z ) : = { i ∈ N | c i 6 = 0 } , q ( z ) : = min { d i +1 d i | i ∈ Λ( z ) } , and S ( x ) : = 1 q ( x ) − 1 . (As usual, we consider d − 1 = 1 .) Lemma 5.5. Let x = ∞ P i =0 ε i b i be a standar d r epre sentatio n of x ∈ T , with ε i ∈ {− 1 , 0 , 1 } . Then, for every k ∈ Λ( x ) , one has 1 q k (1 − S ( x )) ≤ | η k ( x ) | ≤ 1 q k (1 + S ( x )) . P R O O F . One has η k ( x ) = b k − 1 x ≡ 1 ∞ P i =0 ε k + i q k ··· q k + i , and thus one obtains (modulo 1 )     η k ( x ) − ε k q k     ≤ 1 q k ∞ X i =1 1 q k +1 · · · q k + i ≤ 1 q k ∞ X i =1 1 ( q ( x )) i = S ( x ) q k . Therefore, 1 q k (1 − S ( x )) ≤ | η k ( x ) | ≤ 1 q k (1 + S ( x )) , as required. Lemma 5.6. Suppose t hat q k +1 ≥ 4 wheneve r q k = 7 , and let x ∈ Q T ( X ) . If k 1 , k 2 ∈ Λ( x ) ar e such that k 1 < k 2 , then [ q k 1 4 ] q k 1 + ⌊ q k 2 4 ⌋ q k 2 ! (1 − S ( x )) ≤ 1 4 . P R O O F . Put χ : = ε k 1 [ q k 1 4 ] η k 1 + ε k 2 ⌊ q k 2 4 ⌋ η k 2 . By Lemma 5.4, χ ∈ X ⊲ , and so χ ( x ) ∈ T + . The condi- tions imposed upon { q n } ∞ n =0 guarantee that Theorem 5.1 is applicable, and thus x can be expressed in standard form as x = ∞ P i =0 ε i b i , with ε i ∈ {− 1 , 0 , 1 } . In what follows, we use Lemma 5.5 to estimate χ ( x ) from below . If q ( x ) = 2 or q ( x ) = 3 , then the statement is tri vial, and so we may assume that q ( x ) ≥ 4 . Then, by Lemma 5.5, | η k j ( x ) | ≤ 4 3 q k j ( j = 1 , 2 ), and thus ⌊ q k j 4 ⌋| η k j ( x ) | ≤ [ q k j 4 ] | η k j ( x ) | ≤ 4[ q k j 4 ] 3 q k j ≤ 1 3 . Therefore, | χ ( x ) | ≤ [ q k 1 4 ] | η k 1 ( x ) | + ⌊ q k 2 4 ⌋| η k 2 ( x ) | ≤ 2 3 . This impl ies that we can perform the remaining calculati ons in [ − 2 3 , 2 3 ] ⊆ R , and χ ( x ) ∈ T + if and only if − 1 4 ≤ χ ( x ) ≤ 1 4 . Since t he first term of ε k j η k j ( x ) is 1 q k j , one has 0 ≤ ε k j η k j ( x ) , because | ε k j η k j ( x ) − 1 q k j | ≤ 1 2 q k j . Consequently , by Lemma 5.5, 1 q k j (1 − S ( x )) ≤ ε k j η k j ( x ) , and hence [ q k 1 4 ] q k 1 + ⌊ q k 2 4 ⌋ q k 2 ! (1 − S ( x )) ≤ ( ε k 1 [ q k 1 4 ] η k 1 + ε k 2 ⌊ q k 2 4 ⌋ η k 2 )( x ) = χ ( x ) ≤ 1 4 , as desired. D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 15 W e are now ready to prove Theore m F. P R O O F O F T H E O R E M F . Let x ∈ Q T ( X ) , and assume th at x / ∈ X . Then | Λ( x ) | > 1 , and s o w e may pick k 1 , k 2 ∈ Λ( x ) s uch that k 1 < k 2 . As q n ≥ 8 for ev ery n ∈ N , one has 1 − S ( x ) ≥ 6 7 , and thus, by Lemma 5.6, [ q k 1 4 ] q k 1 + ⌊ q k 2 4 ⌋ q k 2 ≤ 7 24 . This inequality , howe ver , does not hold with any q k j ≥ 8 . Hence, x ∈ X , as desired. Example 5.7. Let { p n } n =0 be an enumeration of all p rimes greater than 8 , and put b n = p 0 · · · p n for e very n ∈ N . By Theorem F, { 1 b n } ∞ n =0 is quasi-con vex in T , and s ince each b n is square-f ree, { 1 b n } ∞ n =0 ⊆ so c( T ) . Using the next lem ma, one can lift Theorem F into R . Recall that in this note, π : R → T denotes the canonical projection. Lemma 5.8. ([8, 2.4]) Let Y ⊆ R be a compact subset. If there is α 6 = 0 such that αY ⊆ ( − 1 2 , 1 2 ) and π ( α Y ) is quasi-con ve x in T , then Y is quasi-con ve x in R . Corollary 5.9. Let { x n } ∞ n =0 ⊆ R be a null sequence in R su ch that q n : = x n − 1 x n ar e i nte ger s and q n ≥ 8 for every n ∈ N \{ 0 } . Then { x n } ∞ n =0 is quasi-con ve x in R . P R O O F . Put α = 1 8 x 0 , q 0 = 8 , and b n = q 0 · · · q n . Then α x n = 1 b n , and th us, by Theorem F, the se- quence { π ( αx n ) } ∞ n =0 is q uasi-con ve x in T . Since { α x n } ∞ n =0 ⊆ [ − 1 8 , 1 8 ] , by Lemma 5.8, the sequence { x n } ∞ n =0 is quasi-con vex in R , as required. 6. Compact-like abelian gr oups that admit no non-trivial quasi-con vex null sequences Theor em D. F or ev ery minimal abelia n gr oup G , the following statements ar e equivalent: (i) G admits no non-trivia l quasi-con ve x null sequences; (ii) G ∼ = P × F , wher e P is a minimal bounded abelian p -gr oup ( p ≤ 3 ) admitt ing n o non-trivial quasi-con ve x null sequences, and F is a finite abelian gr oup; (iii) one of the subgr oups G [2] = { g ∈ G | 2 g = 0 } or G [3] = { g ∈ G | 3 g = 0 } is sequentially op en in G ; (iv) G contains a sequentially open compact subgr oup of the form Z κ 2 or Z κ 3 for some car dinal κ . Theor em E. The following statements ar e equivalent for e very abelian gr oup G that is ω -bounded or totally minimal: (i) G admits no non-trivia l quasi-con ve x null sequences; (ii) one of the subgr oups G [2] = { g ∈ G | 2 g = 0 } or G [3] = { g ∈ G | 3 g = 0 } is open in G ; (iii) one of the subgr oups 2 G an d 3 G is finite. Furthermor e, if G is total ly minimal, then these conditions ar e also equivalent to: (iv) G ∼ = Z κ 2 × F or G ∼ = Z κ 3 × F , wher e κ is some car dinal and F is a finite abelian gr oup. 16 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences In this section, we present the proofs of Theorems D and E. Since the latter relies on the former one, we prove Theorem D first. Pr oposition 6.1. Let G b e a mi nimal abelian gr oup that ad mits no non-trivial qu asi-con ve x n ull sequences. T hen the completion e G of G contains no closed subgr oup H such that so c( H ) contains a non-trivial null sequence that is quasi-con ve x in H . P R O O F . Let { x n } ⊆ so c ( H ) be a non-trivial null sequence that is quasi-con vex in H . Since H is closed in e G , by Coroll ary 2.4, { x n } is quasi-con vex in e G . By C orollary 2.7(a), so c( e G ) ⊆ G , and thus { x n } ⊆ G . Therefore, by Corollary 2.5 , { x n } is a non-trivial quasi-con vex null sequence in G , contrary to our assumption . Pr oposition 6.2. Let G b e a mi nimal abelian gr oup that ad mits no non-trivial qu asi-con ve x n ull sequences. Then the completio n e G of G conta ins no subgr oups that a r e topologically iso morphic to: (a) J p for some prime p , (b) T , or (c) ∞ Q k =1 Z r k for squar e-fr ee numbers r k > 3 . P R O O F . (a) Assume that H is a subgroup of e G t hat i s topologically isom orphic to J p for some prime p . By Theorem 2.6(a ), t here is y ∈ G ∩ H such that y 6 = 0 . Since cl e G h y i ∼ = J p , by replacing H with cl e G h y i if necessary , we may ident ify h y i wi th Z in J p . Th us, by Example 1.3(a), H admits a non-trivial q uasi-con ve x sequence { x n } such that { x n } ⊆ G ∩ H . Since H is closed in e G , by Corollary 2.4, { x n } is quasi-con vex in e G . Therefore, by Corollary 2.5, { x n } is a non-trivial quasi- con ve x null sequence in G , contrary to our assumption. (b) Assume that H is a subgroup of e G t hat is topol ogically isomorphi c to T . Then H is closed in e G , and by Ex ample 5.7, H admit s a non-trivial quasi-conv ex nu ll sequence { x n } such that { x n } ⊆ so c ( H ) ∼ = so c( T ) . By Proposit ion 6.1, the statement follo ws. (c) Assume that H is a subgroup of e G t hat is topologically isomorphic to ∞ Q k =1 Z r k , where r k > 3 and r k is square-free for every k . Th en H is closed in e G , and by E xample 1.3(c), H admits a non- trivial quasi-con ve x null sequence { x n } such that { x n } ⊆ so c ( H ) ∼ = ∞ L k =1 Z r k . The statement follo ws now by Propositio n 6.1. Lemma 6.3. L et K be a compact abelian gr oup that contains no subgr oups that ar e topologically isomorphic to T , J p for some p , or ∞ Q k =1 Z r k for squar e-fr ee numbers r k > 3 . Then K = K p × F , wher e K p is a compact bounded abelian p -gr oup, p ≤ 3 , and F i s a finite abelian gr oup. P R O O F . Step 1. Suppose that K is a pro-finite group. Since K contains no subgroups that are topologically isom orphic to J p for some prime p , it i s an exotic torus. The group K has no con- nected quoti ents, because it i s pro-finite, and thus, by Theorem 2.9, K = Q p K p , where each K p is a compact bounded abelian p -group. C onsequently , each K p is topologi cally isomorphic to a prod- uct of finite cyclic p -groups (cf. [ 11, 4.2.2]), and K p is infinite if and only if it contains a subgroup D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 17 that i s t opologically isomorphic to Z ω p . By our assumption, K con tains no such subgroups for p > 3 . Hence, K p is finite for p > 3 . Put K ′ : = Q p> 3 K p . If K ′ is infinite, then th ere are infinitely m any p rimes p k > 3 such that K p k 6 = 0 . Consequently , K ′ (and thus K ) contains a subgroup that is topologically isomorphic to the product ∞ Q k =1 Z p k , contrary to our assumptio n. This sho ws that K ′ is finite. Finally , if both K 2 and K 3 are infinite, then K cont ains a subgroup t hat is topo logically iso- morphic to Z ω 2 × Z ω 3 ∼ = Z ω 6 , contrary to ou r assumption. Hence, one of K 2 and K 3 is finite, and either K = K 2 × F , where F : = K 3 × K ′ is finite, or K = K 3 × F , where F : = K 2 × K ′ is finite. Step 2. In the general case, we show that K is pro-finite. T o that end, let C be the connected component o f K . Since K is an exotic torus, so is C , and by Theorem 2.9, C contains a closed subgroup B s uch that B = Q p B p , where each B p is a finite p -group, and C /B ∼ = T n for some n ∈ N . In particular , B is a com pact pro-finite group t hat sati sfies the conditions of this lemma. Thus, by what we ha ve s hown so far , B ′ : = Q p> 3 B p is finite, and therefore B = B 2 × B 3 × B ′ is finite. Conse- quently , by Pontryagi n d uality , b B ∼ = b C /B ⊥ is fi nite (cf. [24, Theorem 54]), and B ⊥ ∼ = [ C /B = Z n (cf. [24, Theorem 37]). This implies that b C i s finitely generated. On the other hand, b C is t orsion free, beca use C is connected (cf. [24, Example 73]), which means that b C = Z n and C ∼ = T n . B y our assumption , ho we ver , K contains no subgroup that is topologically isomorphic to T . Hence, n = 0 and C = 0 . This sh ows that K = B is pro-finite, and the statement follows from Step 1. Proposition 6.2 combined with Lemma 6.3 yields the follo wing consequence. Corollary 6.4. Let G be a minimal abelian gr oup that admits no non-trivial quasi-con vex null sequences. Then the completi on e G of G i s a bounded compact abelian group, and e G = K p × F , wher e K p is a compact bounded abelian p -gr oup, p ≤ 3 , and F i s a finite abelian gr oup. P R O O F O F T H E O R E M D . (i) ⇒ (ii): Let G be a minimal abelian group that adm its no non-tri vial quasi-con ve x null sequences. By Corollary 6.4, e G = K p × F , where K p is a compact bounded abe- lian p -group ( p ≤ 3 ), and F is a fi nite abelian grou p. W ithout loss of generality , we may assume that F con tains no p -elements. Let e = p a m be the exponent o f e G , where m and p are coprime. Then p a G is dense in p a e G = F , and thus F = p a G ⊆ G . Therefore, for P : = G ∩ K p , one h as G = P × F , and P is a bounded abelian p -group. Since P is a clo sed subg roup of G , by Corollary 2.4, every quasi-con ve x sequence in P is also quasi-con vex in G , and so P admits no non-tri vial quasi-conv ex null sequences. In order t o show that P is m inimal, let H be a closed su bgroup o f e P ⊆ K p . Then, in particular , H i s a closed subgroup of e G . Consequently , by Theorem 2.6(a), P ∩ H = ( G ∩ K p ) ∩ H = G ∩ H 6 = { 0 } , because G is minim al. Hence, by Theorem 2.6(a), P is minim al. (It is well known that closed cen- tral subgroups of minimal groups are minimal, but in this case, a direct proof was also av ailable.) (ii) ⇒ (iii ): By Theore m 2.6(a), P is precompact. Thus, by Theorem C, P [2] or P [3] i s sequen- tially op en i n P . Since F is finite, P is open in G . Therefore, P [2] or P [3] is s equentially open in G . Hence, G [2] or G [3] is sequentially open in G . (iii) ⇒ (i v): Since G is mini mal, by Corollary 2.7(a), G [ p ] = e G [ p ] for ev ery prime p , and in particular G [ p ] is compact. Thu s, G [ p ] ∼ = Z κ p p for some cardinal κ p for ev ery prime p . 18 D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences (iv) ⇒ (i): If G contains a sequentially open compact subgroup of the f orm Z κ 2 or Z κ 3 , then G [2] or G [3] is sequentiall y open in G , and therefore the statement follo ws by Proposition 3.2. W e turn now to the case where G is ω -bound ed. Pr oposition 6 .5. Let G be an ω -bounded abelian group th at admits no non-trivial quasi-con ve x null sequences. Then: (a) G is bounded; (b) the subgr oup G p of p -elements is finite for p > 3 ; (c) at least one of G 2 and G 3 is finite; (d) 2 G 2 and 3 G 3 ar e finite. The proof of Proposition 6.5 is based on a well-known result of Comfort and Robertson. Theor em 6.6 . ([3, 7.4]) Every pseudocompact abelian torsio n group is bounded. P R O O F O F P R O P O S I T I O N 6 . 5 . (a) Let x ∈ G . Sin ce G is ω -bounded, h x i is contained in a com- pact subgrou p of G , and thus K : = cl G h x i is com pact. If x has an i nfinite order , then 2 K and 3 K are infinite, and so by Theorem 1.4, K admi ts a non-trivial quasi -con ve x null sequence { x n } . Since K is a closed subgroup of G , by Corollary 2.4, { x n } is a n on-trivial quasi-con vex null sequence in G , contrary to our assumpti on. Thi s sho ws t hat e very element in G has a finite order; in other words, G i s a torsion group. Sin ce G is ω -bounded, in particular , it is pseud ocompact. Hence, by Theorem 6.6, G is bounded. (b) Let p be a prime, and suppose that G p is infinite. Then G [ p ] contains a countably infinite subset S , which in t urn is contai ned in a compact s ubgroup K , because G is ω -bounded. Since K [ p ] is an infinit e com pact group of exponent p , it is topologically isomorphic to Z λ p for some infinit e cardinal λ (cf. [11, 4.2.2]). In particular , G contains a closed subgroup H that is topol ogically isomorphic to Z ω p . If p > 3 , t hen by Example 1.3(c ), H admits a non-tri vial quasi-con ve x null sequence, and since H is closed in G , this sequence will also be quasi-con v ex in G according to Corollary 2.4, contrary to our assumption. This shows that G p is finite for p > 3 . (c) Assume that both G 2 and G 3 are infinite. Then, by what we hav e seen so far , G cont ains a sub group H 2 that is topologically isomorphic to Z ω 2 , and a subgroup H 3 that is topologically isomorphic to Z ω 3 . Thus, H : = H 2 + H 3 is to pologically is omorphic to Z ω 6 . B y Example 1.3(c), H admits a non-trivial quasi-con vex null sequence, and sin ce H is closed in G , th is sequence wi ll also be quasi-con ve x in G according to Corollary 2.4, contrary to our assumpti on. This shows that at least one of G 2 and G 3 is finite. (d) Let p = 2 or p = 3 , and assume that pG p is infinite. Then, in particular , ( pG p )[ p ] is infinite, and so there is a countably infinit e subset S of G such th at pS is infinite and p 2 S = 0 . Since G i s ω -bounded, S is contained i n a com pact subgroup K of G . By replacing K wit h K [ p 2 ] if n eces- sary , we may assume that K has exponent p 2 , and so K is topologicall y isomorphi c t o Z λ 1 p × Z λ 2 p 2 for som e cardinals λ 1 and λ 2 . As pS ⊆ pK ∼ = Z λ 2 p is infinit e, λ 2 is infinit e, and thus G contains a subgroup H that is to pologically isomorphic to Z ω p 2 . By Example 1.3(c), H admits a non-trivial quasi-con ve x nu ll sequence, and since H is closed in G , this sequence wi ll also be quasi-con ve x in G according to Corollary 2.4, contrary to our assumption. This sho ws that 2 G 2 and 3 G 3 are finite. D. Dikra njan and G. Luk ´ acs / Compact-lik e abelian gr oups without quasi -con vex null sequ ences 19 W e are now ready to prove Theore m E. P R O O F O F T H E O R E M E . By Proposi tion 3.2, the implication (ii) ⇒ (i) h olds for e very precom- pact group, and obviously so does the equiv alence (ii) ⇔ (iii). The im plication (iv) ⇒ (iii) i s also clear . Thus, it suffices to prov e that (i) ⇒ (iii ), and if G is totally minim al, (i) ⇒ (iv). Suppose that G is ω -bounded and admits no no n-trivial quasi-con vex null s equences. Then, by Proposition 6.5(a), G is bounded, and so e G is boun ded. Thus , e G is a product of its subg roups of p -elements, and therefore G ∼ = G 2 × G 3 × G p 1 × · · · × G p l , where p 1 , . . . , p l > 3 are prime factors of the e xponent of G . By Proposition 6.5(b), each G p i is finite. By Propositi on 6.5(c), one of G 2 and G 3 is finite, and so G ∼ = G p × F , where p = 2 or p = 3 , and F i s a finite abelian group of order coprime to p . By Proposition 6.5(d), pG p is finite, and hence pG ∼ = pG p × F is fi nite, as desired. Suppose th at G is totally mini mal and admits no non-trivial quasi-con ve x null sequences. 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J ´ anos Bolyai , pages 493–50 8. Nor th-Hollan d, Amsterdam, 1985. [28] D. B . Shakhmatov . Con vergent sequences in minimal gro ups. T opo logy Appl. , (to appear). ArXiv: 0901 .0175 v1. [29] R. M. Stephenson, Jr . Min imal topolog ical g roups. Math. Ann. , 192:193– 195, 19 71. [30] N. Y . V ilenkin. The theory of characters of to pologica l Abelian group s with bound edness g iv en. Izvestiya Akad. Nauk SSSR. Ser . Mat. , 15:439 –462, 1 951. Department of Mathematics and Computer Science Halifax, No va Scotia Univ ersity of Udine Canada V ia delle Scienze, 208 – Loc. Rizzi, 33100 Udine Italy email: dikranja@dimi.uniud .it email: lukacs@topgr oups.ca