Convergent sequences in minimal groups

0.15 MB·English

by  Dmitri Shakhmatov

CONVERGENT SEQUENCES IN MINIMAL GROUPS
DMITRI SHAKHMATOV
Abstract. A Hausdorff topological group G is minimal if every continuous iso-
morphism f : G → H between G and a Hausdorff topological group H is open.
9 Clearly, every compact Hausdorff group is minimal. It is well known that every
0 infinite compact Hausdorff group contains a non-trivial convergent sequence. We
0 extendthis resultto minimalabelian groupsby provingthateveryinfinite minimal
2
abelian group contains a non-trivial convergent sequence. Furthermore, we show

n that “abelian” is essential and cannot be dropped. Indeed, for every uncountable
Ja regular cardinal κ we construct a Hausdorff group topology Tκ on the free group
F(κ) with κ many generators having the following properties:
1
(i) (F(κ),Tκ) is a minimal group;
] (ii) everysubsetofF(κ)ofsizelessthanκisTκ-discrete(andthusalsoTκ-closed);
N (iii) there are no non-trivial proper Tκ-closed normal subgroups of F(κ).
G In particular, all compact subsets of (F(κ),Tκ) are finite, and every Hausdorff
. quotient group of (F(κ),Tκ) is minimal (that is, (F(κ),Tκ) is totally minimal).
h
t
a
m
We denote by N the set the set of natural numbers.
[
Let X be a topological space. A convergent sequence in X is a sequence S = {xn :
1 n ∈ N} of points of X such that there exists a point x ∈ X (called the limit of S) so
v
that S \U is finite for every open subset U of X containing x. (We also say that S
5
7 converges to x.) A sequence S is non-trivial provided that the set S is infinite.
1
The identity element of a group G is denoted by 1. When G is abelian, the additive
0
. notation is used, and so 1 is replaced by the zero element 0 of G.
1
0
9 1. Results
0
: Our starting point in this manuscript is the following folklore fact.
v
i
X Fact 1.1. Every infinite compact Hausdorff group contains a non-trivial convergent
r sequence.
a
This result is a consequence of the theorem of Ivanovski˘ı [14] and Kuz’minov [15]
that every compact group is dyadic. We refer the reader to [20] for the proof of Fact
1.1 based on Michael’s selection theorem in the spirit of [25].
In Fact 1.1 compactness cannot be weakened to pseudocompactness or countable
compactness, even in the abelian case. Indeed, there exists an example (in ZFC) of a
pseudocompact abelian group without non-trivial convergent sequences [21]. Further-
more, there are numerous consistent examples of countably compact abelian groups
1991 Mathematics Subject Classification. Primary: 22A05; Secondary: 22C05, 54A10, 54A20,
54A25, 54D25, 54H11.
Key words and phrases. convergent sequence, minimal group, totally minimal group, compact
group, abelian group, free group.
The author was partially supported by the Grant-in-Aid for Scientific Research no. 19540092by
the Japan Society for the Promotion of Science (JSPS).
1
2 D.SHAKHMATOV
without non-trivial convergent sequences, see [12, 11, 16, 23, 3, 9, 24, 7]. However,
the existence of a countably compact group without non-trivial convergent sequences
in ZFC alone remains a major open problem in the area [8, Problem 22].
Anotherwell-known generalizationofcompactness intheclassoftopologicalgroups
is related to the fundamental fact that the topology of a Hausdorff compact space X
is a minimal element in the set of all Hausdorff topologies on the set X.
Definition 1.2. A Hausdorff group topology T on a group G is called minimal
provided that every Hausdorff group topology T ′ on G such that T ′ ⊆ T satisfies
T ′ = T . In such a case the pair (G,T ) is called a minimal group.
The notion of a minimal group was introduced independently by Choquet (see
Do¨ıtchinov [10]) and Stephenson [22]. We refer the reader to [6, 4] for additional
information on minimal groups.
While compactness cannot be replaced by pseudocompactness or countable com-
pactness in the statement of Fact 1.1, our first result demonstrates that compactness
can be weakened to minimality provided that the group in question is commutative.
Theorem 1.3. Every infinite minimal abelian group contains a non-trivialconvergent
sequence.
The particular version of Theorem 1.3 for countably compact groups has been
announced without proof on page 393 of [8] (see the text preceding [8, Problem 23]).
As usual, we say that a group G is non-trivial provided that |G| ≥ 2.
Our second result shows that the word “abelian” in Theorem 1.3 is essential and
cannot be omitted.
Theorem 1.4. For every uncountable regular cardinal κ there exists a Hausdorff
group topology T on the free group F(κ) with κ many generators having the following
κ
properties:
(i) (F(κ),T ) is a minimal group;
κ
(ii) every subset of F(κ) of size less than κ is T -discrete (and thus also T -
κ κ
closed);
(iii) there are no non-trivial proper T -closed normal subgroups of F(κ).
κ
In particular,
(a) all compact subsets of (F(κ),T ) are finite, and
κ
(b) every Hausdorff quotient group of (F(κ),T ) is minimal; that is, (F(κ),T )
κ κ
is totally minimal.
2. Proof of Theorem 1.3
Lemma 2.1. An infinite subgroup of a compact metric group has a non-trivial con-
vergent sequence.
Proof. Assume that G is an infinite subgroup of a compact metric group K. Then
G cannot be discrete, and thus the identity 1 of G is a non-isolated point of G.
Let {U : n ∈ N} be a decreasing local base at 1. By induction on n ∈ N choose
n
x ∈ U \{x ,...,x }. Then {x : n ∈ N} is a non-trivial sequence in G converging
n n 0 n−1 n
to 1. (cid:3)
CONVERGENT SEQUENCES IN MINIMAL GROUPS 3
Definition 2.2. [17,22] A subgroupG ofa topologicalgroupK is saidto beessential
in K provided that G∩H is a non-trivial subgroup of K for every non-trivial closed
normal subgroup H of K.
If K is abelian, then every subgroup H of K is normal, and so the word “normal”
can be omitted in the above definition.
The notion of an essential subgroup is a crucial ingredient of the so-called “mini-
mality criterion”, due to Prodanov and Stephenson [17, 22], describing dense minimal
subgroups of compact groups.
Fact 2.3. ([17, 22]; see also [5, 6]) A dense subgroup G of a Hausdorff compact group
K is minimal if and only if G is essential in K.
The straightforward proof of the following lemma is left to the reader.
Lemma 2.4. If G is an essential subgroup of an abelian topological group K, then
K[p] = {x ∈ K : px = 0} ⊆ G
for every prime number p.
Lemma 2.5. Let I be an infinite set, {K : i ∈ I} a family of non-trivial topological
i
groups and G an essential subgroup of the product K = K . Then G has a
i∈I i
non-trivial convergent sequence.
Q
Proof. We identify each K with the closed normal subgroup
i
{1}×···×{1}×K ×{1}×···×{1}
i
of K, where K occupies the ith place. For each i ∈ I, use essentiality of G in K to fix
i
g ∈ G∩K with g 6= 1. Since K ∩K = {1} whenever i,j ∈ I and i 6= j, it follows
i i i i j
that{g : i ∈ I}isaafaithfullyindexed familyofelements ofG. Choosinga faithfully
i
indexed subset {i : n ∈ N} of I, we obtain an infinite sequence {g : n ∈ N} of
n in
elements of G converging to 1. (cid:3)
Lemma 2.6. An essential subgroup of a non-trivial Hausdorff compact torsion-free
abelian group contains a non-trivial convergent sequence.
Proof. Assume that G is an essential subgroup of a non-trivial compact torsion-free
abelian group K. Since K is torsion-free, the Pontryagin dual of K is divisible,
and from [13, Theorem 25.8] we conclude that there exists a sequence of cardinals
{σ : p ∈ P∪{0}} such that
p
(1) K = Qσ0 × Zσp,
p
p∈P
Y
b
where Q denotes the Pontryagin dual of the discrete group Q of rational numbers, P
is the set of all prime numbers, and Z denotes the group of p-adic integers.
p
b
If the product (1) can be (re-)written as a product of infinitely many non-trivial
topological groups, then the conclusion of our lemma follows from Lemma 2.5. In
the remaining case K is metrizable being a finite product of compact metric groups.
Since K is non-trivial and G is essential in K, there exists g ∈ G ∩ K with g 6= 0.
Since K is torsion-free, g hasan infinite order, andso G is an infinite group. Applying
Lemma 2.1, we obtain a non-trivial convergent sequence in G. (cid:3)
4 D.SHAKHMATOV
Proof of Theorem 1.3. Assume that G is an infinite minimal abelian group. Then
its completion K is a compact Hausdorff abelian group [18] (see also [6]). Moreover,
G is essential in K by Fact 2.3. We consider two cases, depending on the size of the
torsion part
t(K) = {x ∈ K : nx = 0 for some n ∈ N\{0}}
of K.
Case 1. t(K) is uncountable. Then the p-rank r (K) of K must be uncountable
p
for some p ∈ P. In particular, K[p] is uncountable. Being a closed subgroup of the
compact group K, the group K[p] is compact. Hence K[p] contains a non-trivial
convergent sequence by Fact 1.1. Finally, K[p] ⊆ G by Lemma 2.4.
Case 2. t(K) is at most countable. Then U = K\(t(K)\{0}) is a G -subset of K
δ
containing 0. Therefore, there exists a closed G -subgroup N of K satisfying N ⊆ U
δ
(see, forexample, [13, Chapter II, Theorem 8.7]or[1]). In particular, N∩t(K) = {0}.
This means that N is torsion-free.
If N 6= {0}, then N is a non-trivial compact abelian group. Since G is essential in
K, G ∩ N is essential in N. Since N is torsion-free, Lemma 2.6 yields that G ∩ N
(and thus G as well) has a non-trivial convergent sequence.
If N = {0}, then {0} is a G -subset of K, and so K is metrizable. Applying Lemma
δ
2.1, we obtain a non-trivial convergent sequence in G. (cid:3)
3. Proof of Theorem 1.4
The construction in this section is inspired by an old construction of the author
[19].
Given a set X, the symbol S(X) denotes the symmetric group of X, i.e., the set of
all bijections of the set X with the composition of maps as the group operation. We
equip S(X) with the topology of pointwise convergence on X whose base is given by
the family
W (X) = {W(X,Z,ϕ) : Z is a finite subset of X and ϕ : Z → X is an injection},
where
(2) W(X,Z,ϕ) = {f ∈ S(X) : f ↾ = ϕ}.
Z
As usual, an ordinal α is considered to be the set consisting of all smaller ordinals;
that is, α = {β : β < α}. In what follows, F(α) denotes the free group with the
alphabet α. For special emphasis, we use ∗ to denote the group operation of F(α)
α
and e to denote the identity element of F(α).
α
Fix an uncountable regular cardinal κ. For γ ∈ κ+1 define
(3) T = {(α,β) ∈ (γ \ω)×γ : β < α}
γ
and
(4) X = {α}×F(α).
γ
α∈γ
[
For every γ ∈ κ\ω we have |T | = |γ|, so we can fix an injection j : T → γ.
γ+1 γ γ+1
Claim 1. The unique homomorphism J : F(T ) → F(γ) extending j is an injec-
γ γ+1 γ
tion.
CONVERGENT SEQUENCES IN MINIMAL GROUPS 5
For each γ ∈ κ \ ω, the family H of all bijections of X that move only finitely
γ γ
many elements of X , is dense in S(X ) and has size |X | = |γ|, so we can fix an
γ γ γ
enumeration
(5) H = {h : β ∈ γ}.
γ γβ
For (α,β) ∈ T define f ∈ S(X ) by
κ α,β κ
h (γ,g), for γ ∈ α
(6) f (γ,g) = αβ for (γ,g) ∈ X .
α,β (γ,g ∗ j (α,β)), for γ ∈ κ\α κ
( γ γ
Define
(7) Y = {f : (α,β) ∈ T } ⊆ S(X ),
κ α,β κ κ
and let G to be the subgroup of S(X ) generated by Y . Define the map θ : T → Y
κ κ κ κ κ
by
(8) θ(α,β) = f for (α,β) ∈ T ,
α,β κ
and let Θ : F(T ) → G be the unique homomorphism extending θ.
κ κ
Claim 2. Θ(g)(γ,e ) = (γ,J (g)) whenever γ ∈ κ\ω and g ∈ F(T ).
γ γ γ
Proof. The conclusion of our claim obviously holds for the identity element of F(T ),
γ
so we will assume that g is not the identity of F(T ). Then there exist n ∈ N,
γ
{(α ,β ) : k ≤ n} ⊆ T and {ε : k ≤ n} ⊆ {−1,1} such that
k k γ k
n
g = (α ,β )εk.
k k
k=0
Y
Together with (8) this yields
n n
(9) Θ(g) = Θ (α ,β )εk = θ(α ,β )εk
k k k k
!
k=0 k=0
Y Y
n
= (f )εk = fεn ◦fεn−1 ◦...◦fε0 .
αk,βk αn,βn αn−1,βn−1 α0,β0
k=0
Y
From (9) and (6) we get
Θ(g)(γ,e ) = fεn ◦fεn−1 ◦...◦fε0 (γ,e )
γ αn,βn αn−1,βn−1 α0,β0 γ
= fεn ◦fεn−1 ◦...◦fε1 (γ,j (α ,β )ε0)
αn,βn αn−1,βn−1 α1,β1 γ 0 0
...
n−1
= fεn γ, j (α ,β )εk
αn,βn γ k k !
k=0
Y
n
= γ, j (α ,β )εk = (γ,J (g)).
γ k k γ
k=0 !
Y
(cid:3)
Claim 3. Θ : F(T ) → G is an isomorphism.
κ κ
6 D.SHAKHMATOV
Proof. Since Y generates G and Θ(T ) = θ(T ) = Y by (7) and (8), it follows that
κ κ κ κ κ
Θ is a surjection.
To prove that Θ is an injection, assume that g ∈ F(T ) and Θ(g) is the identity
κ
map of S(X ). From (3) we have
κ
(10) F(T ) = F(T ),
κ α
α∈[κ\ω
and so there exists some γ ∈ κ\ ω with g ∈ F(T ). Since Θ(g) is the identity map
γ
of S(X ), from Claim 2 we get J (g) = e . Then g = 1 by Claim 1. Therefore,
κ γ γ
kerΘ = {1}, and so Θ is an injection. (cid:3)
Claim 4. Θ(F(T )) is discrete for every γ ∈ κ\ω.
γ
Proof. For each g ∈ F(T ) let ϕ : {(γ,e )} → X be the map defined by
γ g γ κ
(11) ϕ (γ,e ) = (γ,J (g)),
g γ γ
so that W(X ,{(γ,e )},ϕ ) ∈ W .
κ γ g
(i) For every g ∈ F(T ) one has
γ
Θ(g)(γ,e ) = (γ,J (g)) = ϕ (γ,e )
γ γ g γ
by Claim 2 and (11), which yields
Θ(g) ∈ W(X ,{(γ,e )},ϕ ).
κ γ g
(ii) Suppose that g ,g ∈ F(T ) and g 6= g . Then J (g ) 6= J (g ) by Claim 1,
0 1 γ 0 1 γ 0 γ 1
which together with (11) yields ϕ (γ,e ) 6= ϕ (γ,e ), and thus
g0 γ g1 γ
W(X ,{(γ,e )},ϕ )∩W(X ,{(γ,e )},ϕ ) = ∅.
κ γ g0 κ γ g1
Since W is a base for the topology of S(X ), from (i) and (ii) we conclude that the
κ
family
{W(X ,{(γ,e )},ϕ ) : g ∈ F(T )}
κ γ g γ
witnesses that the set Θ(F(T )) is discrete. (cid:3)
γ
Claim 5. If D ⊆ G and |D| < κ, then D is discrete.
κ
Proof. Since Θ is a bijection by Claim 3, |Θ−1(D)| = |D| < κ. By (3), F(T ) ⊆ F(T )
λ µ
whenever ω ≤ λ < µ < κ, so using (10) and regularity of κ we can find γ ∈ κ\ω such
that Θ−1(D) ⊆ F(T ) and so D ⊆ Θ(F(T )). Now Claim 4 applies. (cid:3)
γ γ
Claim 6. Y is dense in S(X ). In particular, G is dense in S(X ).
κ κ κ κ
Proof. Let Z be a finite subset of X and ϕ : Z → X be an injection. From (4) we
κ κ
get X = X , and X ⊆ X whenever ω ≤ λ < µ < κ. Since κ is uncountable
κ γ∈κ γ λ µ
and Z ∪ϕ(Z) is a finite subset of X , we have
S κ
(12) Z ∪ϕ(Z) ⊆ X
α
for some α ∈ κ\ω. The bijection ϕ between two finite subsets Z and ϕ(Z) of X can
α
be extended to a bijection of the whole X . Therefore, W(X ,Z,ϕ) is a non-empty
α α
open subset of S(X ). Since H is dense in S(X ), using (5) we can find β ∈ α such
α α α
that h ∈ W(X ,Z,ϕ). That is, h (γ,g) = ϕ(γ,g) for every (γ,g) ∈ Z. Combining
αβ α αβ
CONVERGENT SEQUENCES IN MINIMAL GROUPS 7
this with (12) and (6), we conclude that f (γ,g) = h (γ,g) = ϕ(γ,g) whenever
α,β αβ
(γ,g) ∈ Z. Together with (2) and (7) this yields
f ∈ W(X ,Z,ϕ)∩Y 6= ∅.
α,β κ κ
Since W (X ) is a base for the topology of S(X ), we conclude now that Y is dense
κ κ κ
in S(X ). (cid:3)
κ
Proof of Theorem 1.4. Since |T | = κ, the groups F(κ) and F(T ) are isomorphic.
κ κ
Combining this with Claim 3, we conclude that the groups F(κ) and G are isomor-
κ
phic. Let T be the topologyon F(κ) obtained by transferring the subgroup topology
κ
that G inherits from S(X ) via the isomorphism between F(κ) and G .
κ κ κ
Every dense subgroup of an infinite symmetric group S(X) is minimal and has no
proper non-trivial closed normal subgroups [2]. Combining this with Claim 6, we get
(i) and (iii). Claim 5 yields (ii). Finally, (a) follows from (ii), and (b) follows from
(iii). (cid:3)
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Graduate School of Science and Engineering, Division of Mathematics, Physics
and Earth Sciences, Ehime University, Matsuyama 790-8577, Japan
E-mail address: [email protected]