Download free topological groups with no small subgroups

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 46, Number 3, December 1974
FREE TOPOLOGICAL GROUPS
WITHNO SMALLSUBGROUPS
SIDNEY A. MORRIS AND H. B. THOMPSON
ABSTRACT.
topological
thus
group
answering
subgroups.
subgroups
exist
space
1. Introduction.
It is well
pact group with no small
subgroups.
subgroups
Kaplansky
by Morris
[15].
has no small
were three
that
topological
(3) To show
a necessary
group
on a
X admits
admits
a continuous
a quotient
subgroup
group
group
a continumetric
of a Lie group
of a locally
compact
group
is
com-
with no small
group with no small
of G, is G/H (necessarily)
question
steps
that
was answered
a
in the negative
in the argument:
topological
group
on any metric
space
We investigate
topological
group
is a quotient
of its free
group.
no small
be weakened?
any abelian
that
As a consequence
group with
here
groups
subgroups.
is a locally
a free abelian
group
metric
is that
a quotient
This
metric
However
if G is a topological
normal
argument
subgroups.
(2) To note
abelian
that
subgroups
subgroups?
There
which
of a
on any metric
topological
subgroups
known that
[9] asks:
(1) To show that
abelian)
group
The
group
It is shown
(free
group
subgroups,
any abelian
with no small
[4],
and H is a closed
group with no small
small
subgroups.
no small
of a group
or equivalently
a quotient
in the negative.
in profusion!
for a free
any topological
group
that
can have
topological
no small
X to have
Hence
is a quotient
shown
Consequently
with
condition
ous metric.
a Lie group,
a free
of a group
topological
has
subgroups
abelian
has no small
and sufficient
author
of Kaplansky
space
small
no small
that
on showing
with
first
a question
relied
is a quotient
The
with
there
exist
every
metric
abelian
abelian
metric
groups
with small
group is a quotient
subgroups.
group
of a
subgroups.
the questions:
We answer
Can "abelian"
both questions
be removed?
in the affirmative.
Can "metric"
Conditions
(2)
Received
by the editors March 9, 1973 and, in revised form, October 26, 1973.
AMS iMOS) subject classifications
(1970). Primary 22A05, 20E05; Secondary
22E99, 54E35.
Copyright © 1974, American Mathematical Society
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431
432
S. A. MORRIS AND H. B. THOMPSON
and (3) are readily
(1) requires
generalized.
a new approach
"abelian"
However
the necessary
as the original
generalization
proof of (1) relies
of
heavily
on the
assumption.
Our work yields:
Theorem.
A necessary
topological
group
subgroups
is that
Corollary.
(A(X))
F(X)
(A(X))
X admits
Corollary.
quotients
of topological
the question:
subgroups
Precisely
X to have
and paracompact.
group which
group
corollary
space
no small
Then
F(X)
if and only if X is metric.
Any topological
The second
for the free (free abelian)
metric.
compact
subgroups
of a topological
condition
on a topological
a continuous
Let X be locally
has no small
quotient
and sufficient
admits
with no small
shows
that a large
groups
class
with no small
what groups
a continuous
metric
is a
subgroups.
of topological
subgroups.
are quotients
groups
are
We leave unanswered
of groups
with
no small
?
2. Notation
be assumed
and preliminaries.
to be completely
with the notions
All topological
regular
of free topological
due to Markov [12].
free topological
(See [6].)
group
We will denote
spaces
and Hausdorff.
considered
We assume
group and free abelian
Given a topological
will
familiarity
topological
group
space
X, we denote
the
by e.
The set of all words
on X by F(X).
the identity
element
in F(X) of length < 72with respect
of a group
to X will be denoted
by F (X).
We note
that
F,(X) = XuX_1
for all 72 > 1.
Thus
We will need
an alternative
Theorem
closed
if X is compact,
the following
A.
Let X be a compact
FjiX) = (X U X~ l U <?)"
then each
Definition.
exists
groups
F (X) is compact.
structure
result
space.
if and only if A n F (X) is compact
As mentioned
These
basic
proof of a more general
in the introduction,
cal group G is a quotient
there
and
theorem
see
A topological
will be referred
group
(For
[ll].)
Then
a subset
it is easily
A of F(X)
is
shown
that every
topologi-
group F(G) [6, §8.23 (b)].
is said to have no small
of e which
[5].
for all n.
group of its free topological
a neighbourhood
of Graev
contains
to as NSS groups.
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no nontrivial
subgroups
subgroup.
if
FREE TOPOLOGICAL GROUPS
We will denote
topological
the set
space
will
j 1, 2, 3, • • * ! by N.
be denoted
433
The closure
of a set X in a
by cl(X).
3. Results.
Lemma 1. Any NSS group
Proof.
of e which
n £ N.
\e\.
G admits
Since
G is NSS, there
contains
no nontrivial
Clearly
each
'
V
We now define
exists
Define
neighbourhood
*
metric.
an open symmetric
subgroups.
is an open
77
symmetric
a continuous
neighbourhood
=\g-
U
V
gn £ V\, tot
of e such
"
neighbourhoods
V
that
(l
of e inductively
n
V
=
n
such
that
(i) (J C V and (ii) U2 , C U , tot n £ N. By Theorem 8.2 of [6], these sets
77—77
77+1—77
U77 can be used
\e\,
to define
a continuous
pseudo-metric
d on G.
r
Since
fl
71
Un =
d is a metric.
Remark.
topological
product
The converse
group which
group
II.
,G.
cal group isomorphic
tinuous
metric
of Lemma 1 is false.
admits
given
a continuous
Theorem
metric.
the Tychonoff
Define
topology,
to G, and / is countably
but it is not an NSS group
for free topological
For example,
Gn to be the
where
infinite.
[15].
G. is a topologi-
Then
However
let G be any
G admits
the converse
a con-
is true
groups.
1. The following
conditions
are equivalent:
(i) F(X) is NSS.
(ii)
F(X) admits
(iii)
X admits
(iv)
F(X)
Proof,
admits
from the proof
to show
that
It is sufficient
this,
(iii)
Now let
fication
[10].
cp: X —>ß(X).
that $|X
$(F(X))=
(given
metric
G(X), say.
(iii) =■=•
(iv) =» (iii) is trivial.
exists
when X is a metric
on the completely
d by Y. Then
and therefore
space
X is a dense
= cp. Clearly
the subspace
(ii) =» (iii) is trivial,
It
regular
the identity
extends
space
To see
X, and
mapping
z: X —»
to a continuous
and
/3(X) be its Stone-Cech
subspace
a continuous
of ß(X).
mono-
homomorphism
$ is an algebraic
in F(ß{X)))
<&: F(X) —» F(ß(X))
isomorphism
is NSS.
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compacti-
Let cp be the imbedding
Thus to prove F(X) is NSS it suffices
topology
space.
Thus if F(Y) is NSS, so is F(X).
(X, d) be a metric
There
1.
1 of [5].
the case
and one-to-one,
Then
metric.
=> (i).
to consider
morphism /: F(X) —>F(Y).
such
by Lemma
the set X with the metric
Y is continuous
invariant
of Theorem
let d be a continuous
denote
metric.
metric.
a two-sided
(i) =» (ii) is given
(iv) follows
remains
a continuous
a continuous
of F(X) onto
to show G(X)
434
S. A. MORRIS AND H. B. THOMPSON
Let
\p .'■j £ j\
restriction
topology
1].)
be a family
of continuous
to X defines the subspace
we may assume
Relabelling,
/ is countable.
we may assume
isa
continuous
pseudo-metric
the same
topology
1 of [5] yields:
F(ß(X));
whose
restriction
to a continuous
pseudo-metric
of p to X, call
of s1.
A1 is compact;
length
it s, can be extended
Define
(ii)
>n+i;
open sets
for sufficiently
have that
i and n,x
x £ G(X),
x 4 e, there
£ F .(ß{X))-
exists
i such
72such that
0 . Hence
1 '
any
exists
x"
77
large i and 72.] Put A = (j!° -,|kJ°°_j'4ï • By condition
A n Fn(ß(X))
= (J¿
*,<„ ^
Now by Theorem A, A is closed
no nontrivial
n Fjß(X)),
in F(ß(X)).
G(X) - A is an open neighbourhood
U contains
to a metric
fot n £ N, i £ N.
n G(X)= \e\, there exists
large
on
= ig: g £ F(ß{X)),
[Since if x £ G(X) there
77
n
p
e 4 A1 for all n £ N and i £ N; (iii)
Since f|"[0
x 4 0 , so for sufficiently
0
de-
of Theorem
OG(X) = {e!. Put
(iv) for any
72£ N and i £ N such that x"+l £ A1.
that x £ F.(/3(X)).
Then
to X is a metric
of the proof
K = kn+i- g é F.{ß{X)) - Oj,
(i) each
Theorem
integers.
examination
p1(g, e)< I/72S,for 72£ N. Then fl^LjO^
Then
IX, §4,
of the positive
r,
A careful
is an extension
word in A1 has
X has a metric
(See [2, Chapter
2
on ß(X)
as d.
(b) the restriction
on /3(X) whose
Since
v ,;1 1 +Pjp.
L
je]
(a) p can be extended
s1 on G(X); p
on X.
/ is a subset
9
fining
pseudo-metrics
topology
which
subgroups.
Hence
Further
77
(iii) we
is clearly
Thus by condition
of e in G(X).
l e A1
compact.
(ii), (7 =
by condition
G(X) is NSS and the proof
(iv),
is
complete.
Corollary
1. Let X be a locally
compact
paracompact
space.
Then
F(X)
is NSS if and only if X is metrizable.
Proof.
continuous
locally
It is readily
metric
metrizable
seen
that any locally
is locally
metrizable.
paracompact
space
compact
Since
space
by Theorem
is metrizable,
which
admits
a
2-68 of [7], any
Theorem
1 shows
that
F(X) is NSS if and only if X is metrizable.
Corollary 2. Let G be any locally compact group.
and only if G is metrizable.
The following
example
k -groups.
(A topological
where
X
each
is compact
shows
space
that
Corollary
X is said
and A is closed
Then F(G) is NSS if
2 cannot
to be a ¿^-space
be extended
if X = (J00
in X if and only if A O X
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to
.X
is
,
435
FREE TOPOLOGICAL GROUPS
compact
for all 72. As examples
locally
compact
groups
Example.
we have
all compact
spaces
and all connected
[13], [18].)
Let T denote
the circle
group
with the usual
compact
topology.
Then the free product [17], [16] T * T is a k -space group which is NSS [16].
Thus
T *T
admits
ever it is shown
Corollary
metric
a continuous
in [18] that
metric
3. // G is any topological
then
G is a quotient
We now have
tients
of NSS groups.
cisely
what topological
corollary
provides
Corollary
that
This,
is NSS), how-
group which
admits
a continuous
of F(G), which by Theorem 1 is NSS.
a large
class
however,
groups
some
F(T *T)
of an NSS group.
Proof. By §2, G is a quotient
Remark.
(and hence
T * T is not metrizable.
of topological
leaves
unanswered
are quotients
groups
are quo-
the question:
of NSS groups?
Pre-
The next
information.
4. Let G be a topological
group.
Then
the following
conditions
are equivalent.
(i) G is a quotient
(ii)
tinuous
G is a quotient
of an NSS group.
group
of a topological
group
space
of a topological
space
which
tinuous
G is a quotient
a con-
which
admits
a con-
metric.
(iv)
G is a quotient
of a free topological
(v) G is a quotient
invariant
of a topological
group
group
which
which
is NSS.
admits
a two-sided
metric.
Proof.
(i)=f=» (ii)by
Lemma
(iv), let G be a quotient
metric.
space
1.
(ii) =» (iii) is trivial.
of a topological
space
By Lemma 1 of [l], G is a quotient
We note that
Recently
group
that
admits
of F(X).
(iii)=*
a con-
By Theo-
we need (v) => (i).
(v) =» (iii) =» (iv) => (i), as required.
Joiner
G¡>-point, then each
[8] has
shown
that if a topological
of the topological
Our next corollary
point in F (X) unless
Corollary
To see
X which
rem 1, F(X) is NSS. (iv) => (v) by Theorem 1. Finally
points.
admits
metric.
(iii)
tinuous
group
shows
that
spaces
space
F (X) (C F(X))
if X is compact
then
X contains
contains
e cannot
be a Gj-
X is metrizable.
5. Let X be compact.
Then
the following
equivalent.
(i) Xz's metrizable;
(ii) F (X) is metrizable
for all n > 1 ;
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conditions
a
Gg-
are
S. A. MORRIS AND H. B. THOMPSON
(iii) e £ FAX) is a G^-set.
Proof.
zable
then
continuous
To see that
(i) =» (ii), we note that,
F(X)
a continuous
admits
metric.
Since
(ii) =» (iii) is trivial.
given
tinuous
Thus
F (X) is compact
Now assume
by f(x, y) = xy~
metric.
. Then
and e is a Gg-set,
by Theorem
F (X) admits
it is metrizable
(iii) is true.
f~
each
1, if X is metri-
Let /: X x X —» FAX)
(\e\) = A = \(x, x): x £ X\.
A is a G ..-set
a
for every 72.
in X x X.
Thus
Since
be
/ is con-
by [3, Problem
4,
p. 253], X is metrizable.
In conclusion
topological
either
we note
groups.
by our method
the above
results
Added
the second
no small
matical
all the above
in proof.
Further
subgroups
and will appear
in [15].
free topological
comments
is entitled
are true for free
of Theorem
indicated
hold for Graev
which
results
the analogue
or by the method
also
author,
that
In particular
on this
Remarks
note
that
[5].
are made in a paper,
on free
in the Journal
We also
groups
topic
abelian
1 can be proved
topological
by
groups
of the Australian
with
Mathe-
Society.
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MR
437
FREE TOPOLOGICAL GROUPS
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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NEW SOUTH WALES, KENSINGTON,
N. S. W. 2033, AUSTRALIA (Current address
of S. A. Morris)
SCHOOL OF MATHEMATICS, FLINDERS UNIVERSITY, BEDFORD PARK, S. A. 5042,
AUSTRALIA
Current
Queensland,
address
St. Lucia,
(H. B. Thompson):
Queensland,
Department
4067,
of Mathematics,
Australia
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University
of