Some open problems in topological groups

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1 Some open problems in topological groups Javier Trigos-Arrieta California State University, Bakersfield Spring Topology and Dynamics Conference 2006 University of North Carolina at Greensboro Greensboro, North Carolina March 23, 2006 p. 1/91

2 Disclaimer This is p. 2/91

3 Disclaimer This is NOT a power point presentation. p. 3/91

4 Tychonoff, T,G All topological groups and spaces are Tychonoff. p. 4/91

5 Tychonoff, T,G All topological groups and spaces are Tychonoff. T := R/Z. As a model for T we take (( 1 2, 1 ] ), + mod 1 2 p. 5/91

6 Tychonoff, T,G All topological groups and spaces are Tychonoff. T := R/Z. As a model for T we take (( 1 2, 1 ] ), + mod 1 2 G =(Γ,τ) is an Abelian topological group with underlying group Γ and group topology τ. p. 6/91

7 Character group The character group of G is given by Ĝ := {f : G T : fτ-continuous homomorphism} Elements of Ĝ are called characters. p. 7/91

8 Character group The character group of G is given by Ĝ := {f : G T : fτ-continuous homomorphism} Elements of Ĝ are called characters. We assume G to be maximally almost periodic (MAP) i.e., g G \{0} f Ĝ [f(g) 0] p. 8/91

9 Bohr topology For G as above, we define τ w := weakest topology that makes the elements of Ĝ continuous G + := (Γ,τ w ) p. 9/91

10 Totally bounded groups A topological group is totally bounded, if it can be covered by finitely many translates of any of its open sets. p. 10/91

11 Totally bounded groups A topological group is totally bounded, if it can be covered by finitely many translates of any of its open sets. (Weil, 1937) G is totally bounded if and only if it is a dense subgroup of a compact group, which we denote by G. p. 11/91

12 Totally bounded groups A topological group is totally bounded, if it can be covered by finitely many translates of any of its open sets. (Weil, 1937) G is totally bounded if and only if it is a dense subgroup of a compact group, which we denote by G. (Comfort-Ross, 1964) G =(Γ,τ) is totally bounded if and only if τ = τ w. p. 12/91

13 Bohr compactification The Bohr compactification of G is defined by bg := G + p. 13/91

14 Bohr compactification The Bohr compactification of G is defined by bg := G + Id : G G + is continuous. Open if and only if τ = τ w, i.e., if τ is totally bounded. p. 14/91

15 Bohr compactification The Bohr compactification of G is defined by bg := G + Id : G G + is continuous. Open if and only if τ = τ w, i.e., if τ is totally bounded. As groups, Ĝ = bg. p. 15/91

16 Leptin, Glicksberg (Leptin, 1954) For G discrete (τ = P(Γ)) and K G: K finite K compact in G + p. 16/91

17 Leptin, Glicksberg (Leptin, 1954) For G discrete (τ = P(Γ)) and K G: K finite K compact in G + (Glicksberg, 1962) For G locally compact Kcompact in G Kcompact in G + p. 17/91

18 Leptin, Glicksberg (Leptin, 1954) For G discrete (τ = P(Γ)) and K G: K finite K compact in G + (Glicksberg, 1962) For G locally compact Kcompact in G Kcompact in G + Many authors have reproven Glicksberg s theorem. p. 18/91

19 Comfort, T-A, and Wu, 1993 For G locally compact, let N be a metrizable subgroup of bg (with N G + = {0}). IfK G, then Kcompact in G Kcompact in bg/n G G + bg bg/n p. 19/91

20 Comfort, T-A, and Wu, 1993 For G locally compact, let N be a metrizable subgroup of bg (with N G + = {0}). IfK G, then Kcompact in G Kcompact in bg/n G G + bg bg/n Condition may hold even for N non-metrizable. p. 20/91

21 Compact-open topology on Ĝ If K G is compact and ε>0, a typical neighborhood of the trivial character is (K, ε) :={f Ĝ : f(k) <ε k K} p. 21/91

22 Compact-open topology on Ĝ If K G is compact and ε>0, a typical neighborhood of the trivial character is (K, ε) :={f Ĝ : f(k) <ε k K} Then Ĝ is a topological group. p. 22/91

23 Compact-open topology on Ĝ If K G is compact and ε>0, a typical neighborhood of the trivial character is (K, ε) :={f Ĝ : f(k) <ε k K} Then Ĝ is a topological group. If G is discrete, then Ĝ is compact. p. 23/91

24 Compact-open topology on Ĝ If K G is compact and ε>0, a typical neighborhood of the trivial character is (K, ε) :={f Ĝ : f(k) <ε k K} Then Ĝ is a topological group. If G is discrete, then Ĝ is compact. If G is compact, then Ĝ is discrete. p. 24/91

25 Annihilator If H is a closed subgroup of G its Annihilator is defined by A(Ĝ, H) :={f Ĝ : f(h) =0 h H} p. 25/91

26 Annihilator If H is a closed subgroup of G its Annihilator is defined by A(Ĝ, H) :={f Ĝ : f(h) =0 h H} A(Ĝ, H) is a subgroup of Ĝ. p. 26/91

27 Annihilator If H is a closed subgroup of G its Annihilator is defined by A(Ĝ, H) :={f Ĝ : f(h) =0 h H} A(Ĝ, H) is a subgroup of Ĝ. As groups, Ĥ and Ĝ/A(Ĝ, H) are isomorphic. p. 27/91

28 Annihilator If H is a closed subgroup of G its Annihilator is defined by A(Ĝ, H) :={f Ĝ : f(h) =0 h H} A(Ĝ, H) is a subgroup of Ĝ. As groups, Ĥ and Ĝ/A(Ĝ, H) are isomorphic. Hence N compact and metrizable = N Ĝ/A( bg, N) is countable. p. 28/91

29 A( bg, N) for N metrizable Hence A( bg, N) is a subgroup of bg Ĝ of countable index. p. 29/91

30 A( bg, N) for N metrizable Hence A( bg, N) is a subgroup of bg Ĝ of countable index. If G is discrete, then Ĝ is compact. p. 30/91

31 A( bg, N) for N metrizable Hence A( bg, N) is a subgroup of bg Ĝ of countable index. If G is discrete, then Ĝ is compact. As groups, Ĝ/H and A(Ĝ, H) are isomorphic. p. 31/91

32 A( bg, N) for N metrizable Hence A( bg, N) is a subgroup of bg Ĝ of countable index. If G is discrete, then Ĝ is compact. As groups, Ĝ/H and A(Ĝ, H) are isomorphic. Therefore bg/n A( bg, N) is a (Haar) non-measurable subgroup of Ĝ. p. 32/91

33 Hence, in CTW [1993] For G discrete, let N be a metrizable subgroup of bg (with N G + = {0}). IfK G, then K finite K compact in bg/n G G + bg bg/n p. 33/91

34 Hence, in CTW [1993] For G discrete, let N be a metrizable subgroup of bg (with N G + = {0}). IfK G, then K finite K compact in bg/n G G + bg bg/n bg/n A( bg, N) is a (Haar) non-measurable subgroup of Ĝ. p. 34/91

35 Sequences If K is compact and countable, then a non-trivial sequence in K converges. p. 35/91

36 Sequences If K is compact and countable, then a non-trivial sequence in K converges. (CTW, 1993) If x n is a non trivial sequence of elements in (discrete) G and A := {f Ĝ : f(x n) 0} Then A is a Borel subgroup of measure 0. p. 36/91

37 A la Leptin (CTW, 1993) Let G be discrete and countable. If N is a subgroup of bg with non-measurable (in Ĝ) annihilator A( bg, N), then for K G K finite K compact in bg/n p. 37/91

38 A la Leptin (CTW, 1993) Let G be discrete and countable. If N is a subgroup of bg with non-measurable (in Ĝ) annihilator A( bg, N), then for K G K finite K compact in bg/n Again, bg/n = A( bg, N) bg Ĝ. p. 38/91

39 Kakutani (1943) Hom(Γ, T) := {f :Γ T : f group homomorphism} p. 39/91

40 Kakutani (1943) Hom(Γ, T) := {f :Γ T : f group homomorphism} Hom(Γ, T) has 2 2 Γ -many subgroups A ζ which separate points of Γ i.e., g Γ \{0} f A ζ [f(g) 0]. p. 40/91

41 Duality, CR (1964) If G =Γdiscrete, then Ĝ = Hom(Γ, T) is compact and a subgroup A separates the points of G if and only if A Ĝ is dense. p. 41/91

42 Duality, CR (1964) If G =Γdiscrete, then Ĝ = Hom(Γ, T) is compact and a subgroup A separates the points of G if and only if A Ĝ is dense. There exist a one-one relation between the above subgroups and the totally bounded topologies on Γ. A ζ τ ζ. p. 42/91

43 Duality, CR (1964) If G =Γdiscrete, then Ĝ = Hom(Γ, T) is compact and a subgroup A separates the points of G if and only if A Ĝ is dense. There exist a one-one relation between the above subgroups and the totally bounded topologies on Γ. A ζ τ ζ. The relation above is order preserving: A B τ A τ B. p. 43/91

44 Special case: Γ=Z If G = Z (discrete), then Ĝ = Hom(Γ, T) =T. This group has 2 c -many dense subgroups. Hence Z has 2 c -many totally bounded topologies. p. 44/91

45 Special case: Γ=Z If G = Z (discrete), then Ĝ = Hom(Γ, T) =T. This group has 2 c -many dense subgroups. Hence Z has 2 c -many totally bounded topologies. If A is a dense subgroup of T, then τ A denotes the weakest topology on Z such that the maps n na mod 1 are continuous, for each a A. p. 45/91

46 Diagram, N = A(bZ,A) Z A := (Z,τ A ) Z Z + Z A = Z A bz bz/n = Z A T T d bz/n = A p. 46/91

47 Raczkowski (1998) There exist 2 c -many dense non measurable subgroups A ζ of T of cardinality c, hence p. 47/91

48 Raczkowski (1998) There exist 2 c -many dense non measurable subgroups A ζ of T of cardinality c, hence there exist 2 c -many totally bounded topologies τ ζ of weight c on Z under which the only compact subsets of Z Aζ are the finite ones. p. 48/91

49 Raczkowski (1998) There exist 2 c -many dense non measurable subgroups A ζ of T of cardinality c, hence there exist 2 c -many totally bounded topologies τ ζ of weight c on Z under which the only compact subsets of Z Aζ are the finite ones. There also exist 2 c -many totally bounded topologies τ ζ of weight c on Z under which the sequence n! 0. p. 49/91

50 Raczkowski (1998) There exist 2 c -many dense non measurable subgroups A ζ of T of cardinality c, hence there exist 2 c -many totally bounded topologies τ ζ of weight c on Z under which the only compact subsets of Z Aζ are the finite ones. There also exist 2 c -many totally bounded topologies τ ζ of weight c on Z under which the sequence n! 0 (result eventually generalized by Barbieri, Dikranjan, Milan & Weber (2003)). p. 50/91

51 Raczkowski (1998) There exist 2 c -many dense non measurable subgroups A ζ of T of cardinality c, hence there exist 2 c -many totally bounded topologies τ ζ of weight c on Z under which the only compact subsets of Z Aζ are the finite ones. There also exist 2 c -many totally bounded topologies τ ζ of weight c on Z under which the sequence n! 0 (result eventually generalized by Barbieri, Dikranjan, Milan & Weber (2003)). Notice that if A ζ := (Z,τ ζ ), then ma ζ =0. p. 51/91

52 Raczkowski s question Question. Is there a subgroup A of T of measure 0, such that the only compact subsets of Z A are finite? p. 52/91

53 Answers to Raczkowski s question Question. Is there a subgroup A of T of measure 0, such that the only compact subsets of Z A are finite? (Barbieri, Dikranjan, Milan & Weber 2003). Yes under Martin s axiom. p. 53/91

54 Answers to Raczkowski s question Question. Is there a subgroup A of T of measure 0, such that the only compact subsets of Z A are finite? (Barbieri, Dikranjan, Milan & Weber 2003). Yes under Martin s axiom. (Hart & Kunen 2005). Yes. p. 54/91

55 Answers to Raczkowski s question Question. Is there a subgroup A of T of measure 0, such that the only compact subsets of Z A are finite? (Barbieri, Dikranjan, Milan & Weber 2003). Yes under Martin s axiom. (Hart & Kunen 2005). Yes. Both subgroups above have cardinality c. p. 55/91

56 Question 1. Is there a measurable subgroup A of T of cardinality less than c, such that the only compact subsets of Z A are finite? In particular... p. 56/91

57 Question 1 (ℵ 1 ) Is there a measurable subgroup A of T of cardinality ℵ 1, such that the only compact subsets of Z A are finite? p. 57/91

58 Question 1 (ℵ 1 ) Is there a measurable subgroup A of T of cardinality ℵ 1, such that the only compact subsets of Z A are finite? Remarks. We are assuming CH p. 58/91

59 Question 1 (ℵ 1 ) Is there a measurable subgroup A of T of cardinality ℵ 1, such that the only compact subsets of Z A are finite? Remarks. We are assuming CH There are models of CH in which there exist nonmeasurable subgroups of T of cardinality ℵ 1. p. 59/91

60 Question 1 (ℵ 1 ) Is there a measurable subgroup A of T of cardinality ℵ 1, such that the only compact subsets of Z A are finite? Remarks. We are assuming CH There are models of CH in which there exist nonmeasurable subgroups of T of cardinality ℵ 1. If the subgroup A of T is countable, then Z A is (totally bounded and) metrizable. Hence it has infinite compact sets. p. 60/91

61 Determined Subgroups If H is a dense subgroup of G, then every f Ĥ extends to some f Ĝ. f H T G T p. 61/91

62 Determined Subgroups If H is a dense subgroup of G, then every f Ĥ extends to some f Ĝ. f H T G f T p. 62/91

63 Determined Subgroups Also, if f Ĥ. Ĝ then its restriction to H belongs in H f H G f T T p. 63/91

64 Determined Subgroups Hence the map φ : f f H is an isomorphism from Ĝ onto Ĥ. p. 64/91

65 Determined Subgroups Hence the map φ : f f H is an isomorphism from Ĝ onto Ĥ. Since compact subsets of H are compact in G φ : Ĝ Ĥ is continuous in the compact-open topology. p. 65/91

66 Determined Subgroups Hence the map φ : f f H is an isomorphism from Ĝ onto Ĥ. Since compact subsets of H are compact in G φ : Ĝ Ĥ is continuous in the compact-open topology. Question. When is φ : Ĝ Ĥ a topological isomorphism? p. 66/91

67 Determined Subgroups Definition (CRT, 2001). If H is a dense subgroup of G, we say that H determines G if φ : Ĝ Ĥ is a topological isomorphism. p. 67/91

68 Determined Subgroups Definition (CRT, 2001). If H is a dense subgroup of G, we say that H determines G if φ : Ĝ Ĥ is a topological isomorphism. We say that G is determined if every dense subgroup of G determines G. p. 68/91

69 Determined Subgroups Definition (CRT, 2001). If H is a dense subgroup of G, we say that H determines G if φ : Ĝ Ĥ is a topological isomorphism. We say that G is determined if every dense subgroup of G determines G. Z + does not determine bz. p. 69/91

70 Determined Subgroups Definition (CRT, 2001). If H is a dense subgroup of G, we say that H determines G if φ : Ĝ Ĥ is a topological isomorphism. We say that G is determined if every dense subgroup of G determines G. Z + does not determine bz. For Ẑ+ = T, whereas bz = T d. p. 70/91

71 Few results (Chasco-Aussenhofer, 1998) Every metrizable group is determined. p. 71/91

72 Few results (Chasco-Aussenhofer, 1998) Every metrizable group is determined. (CRT, 2001) A compact group of weight c (or bigger) is not determined. p. 72/91

73 Few results (Chasco-Aussenhofer, 1998) Every metrizable group is determined. (CRT, 2001) A compact group of weight c (or bigger) is not determined. For example, T c is not determined. p. 73/91

74 Few results (Chasco-Aussenhofer, 1998) Every metrizable group is determined. (CRT, 2001) A compact group of weight c (or bigger) is not determined. For example, T c is not determined. Z c n is not determined. p. 74/91

75 Question 2. (CRT, 2001) Is a compact group of weight < c determined? In particular... p. 75/91

76 Question 2 (ℵ 1 ). (CRT, 2001) Is a compact group of weight ℵ 1 determined? p. 76/91

77 Question 2 (ℵ 1 ). (CRT, 2001) Is a compact group of weight ℵ 1 determined? We are assuming CH p. 77/91

78 Interplay between questions 1 and 2. Let A be a measurable subgroup of T of cardinality less than c, such that the only compact subsets of Z A are finite. p. 78/91

79 Interplay between questions 1 and 2. Let A be a measurable subgroup of T of cardinality less than c, such that the only compact subsets of Z A are finite. Set K = Z A. Then p. 79/91

80 Interplay between questions 1 and 2. Let A be a measurable subgroup of T of cardinality less than c, such that the only compact subsets of Z A are finite. Set K = Z A. Then K is a compact group of weight A < c, p. 80/91

81 Interplay between questions 1 and 2. Let A be a measurable subgroup of T of cardinality less than c, such that the only compact subsets of Z A are finite. Set K = Z A. Then K is a compact group of weight A < c, Z A is a dense subgroup of K, p. 81/91

82 Interplay between questions 1 and 2. Let A be a measurable subgroup of T of cardinality less than c, such that the only compact subsets of Z A are finite. Set K = Z A. Then K is a compact group of weight A < c, Z A is a dense subgroup of K, Ẑ A = A T, whereas p. 82/91

83 Interplay between questions 1 and 2. Let A be a measurable subgroup of T of cardinality less than c, such that the only compact subsets of Z A are finite. Set K = Z A. Then K is a compact group of weight A < c, Z A is a dense subgroup of K, Ẑ A = A T, whereas K = Ad. p. 83/91

84 Interplay between questions 1 and 2. Let A be a measurable subgroup of T of cardinality less than c, such that the only compact subsets of Z A are finite. Set K = Z A. Then K is a compact group of weight A < c, Z A is a dense subgroup of K, Ẑ A = A T, whereas K = Ad. Therefore K is a compact group of weight < c that is not determined. p. 84/91

85 Interplay between questions 1 and 2 (ℵ 1 ). If A is a measurable subgroup of T of cardinality ℵ 1, such that the only compact subsets of Z A are finite, then there is a compact group of weight ℵ 1 that is not determined. p. 85/91

86 Few more questions. Is T ℵ 1 non-determined? p. 86/91

87 Few more questions. Is T ℵ 1 non-determined? If F is a finite group, is F ℵ 1 non-determined? p. 87/91

88 Few more questions. Is T ℵ 1 non-determined? If F is a finite group, is F ℵ 1 non-determined? Last two questions are equivalent. p. 88/91

89 Few more questions. Is T ℵ 1 non-determined? If F is a finite group, is F ℵ 1 non-determined? Last two questions are equivalent. If G 1 and G 2 are determined topological groups, is G 1 G 2 determined? p. 89/91

90 Few more questions. Is T ℵ 1 non-determined? If F is a finite group, is F ℵ 1 non-determined? Last two questions are equivalent. If G 1 and G 2 are determined topological groups, is G 1 G 2 determined? In particular: If G is determined, is G G determined? p. 90/91

91 Respectfully dedicated To Ta-Sun Wu In memoriam Those fortunate enough to have known him personally will miss his smile and his friendly, self-effacing humility. The profession has been deprived of a significant figure of great creativity. WWC and SH p. 91/91

Some cases of preservation of the Pontryagin dual by taking dense subgroups

Some cases of preservation of the Pontryagin dual by taking dense subgroups To the memory of Mel Henriksen M. J. Chasco a, X. Domínguez b, F. J. Trigos-Arrieta c a Dept. de Física y Matemática Aplicada,