NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 25 (1996), 59-72
F R E E T O P O L O G IC A L G -G R O U P S
M .G .
M e g r e l i s h v i l i ( L e v y )*
(Received August 1994)
Abstract. Necessary and sufficient conditions for the equivariant embeddability into:
topological G-groups, G-linear spaces, homogeneous spaces, are obtained.
ticular, free topological G-groups are investigated.
In par­
A “regionally-proximal type”
relation enables us to find compact homogeneous G-spaces G / H such that the free
topological G-group over G / H is trivial. W e construct also a G-group which is not
G-linearizable into a locally convex linear G-space. Using J. W est’s theorem about
extension of actions from Z-subspaces to an action on the Hilbert cube, we prove
that for every No-bounded group G and a G-compactificable space X there exists
an equivariant embedding of ( G ,X ) into ( H ( I T), I T) where r < w ( X ) ■w (G ).
In
particular, U spen skii’s theorem on the universality of H (I**°) is improved.
1. C on ven tion s and K n ow n Facts
In this paper, all topological spaces are Tychonoff (i.e., are completely regular
and Hausdorff) and all linear spaces are real locally convex. A topological trans­
form ation group (a ttg) or for short: a G-space is a system (G , X , a ) in which
G is a topological group, X is a topological space and a : G x X —> X is a
continuous action. As usual, we write gx instead of a (g , x ) . A g-transition is
the mapping a 9 : X —> X , a 9(x) = gx and an x-orbit mapping is the mapping
a x : G —>X , a x (g ) = gx. Denote by CompG and [Tych]G the classes of all compact
and all Tychonoff G-spaces respectively. A G-space X is called G-automorphic (or:
a G-group ) if X is a topological group and each a 9 is an automorphism. In the case
of a linear space X and linear a 9 — s, we obtain the definition of a linear G -space
[31].
Let H be a closed subgroup of G and G/H be the space of all left cosets endowed
with the quotient topology. The triple (G, G/H, at), where a i ( g , s H ) = gsH is
called homogeneous.
A ttg (G, X , a) will be called:
(a ) G-compactificable, or G-Tychonoff if X is a G subspace of a compact Gspace;
(b ) G-homogenizable, if there exists an equivariant embedding of ( G , X , a ) into
a homogeneous triple ( P ,P / H , a t ) (i.e., there exists a topological group
embedding </?: G
P and a topological embedding f : X <
—> P/H such
that at |v5(g)x/(X )= a).
( c ) G-automorphizable, if X is a G-subspace of an automorphic G-space.
(d) G-linearizable, if X is a G-subspace of a linear G-space.
Denote the corresponding classes by TychG, HTychG, ATychG and LTychG respec­
tively. Our aim will be the comparision of these classes.
1991 A M S Mathematics Subject Classification: Primary 22A 05, Secondary 54H15.
* Partially supported by the Israel Ministry of Sciences, Grant No. 3505
60
M.G. MEGRELISHVILI (LEVY)
L em m a 1.1. Let ( G , X , a ) be an automorphic triple, and let X X aG denote the
corresponding topological semidirect product. Then (G , X , a) is embedded into the
homogeneous triple ( X X aG, ( X X aG)/G, at), where G is identified with the sub­
group {e} x G.
P ro o f. Consider the mapping q: X —> ( X X aG)/G defined by the rule q(x) — xG.
According to [25, Prop. 6.17(a)], q is a homeomorphism. Moreover, it can be easily
checked that the restriction of at on G x X : = G x q [ X ) is a.
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Actually the same trick was used by Ta-Sun Wu in [33, p. 512].
L em m a 1 .2 . (J. de Vries [28])
Every homogeneous triple ( G, G/H, a g) is G -
Tychonoff.
P ro p o sitio n 1.3. For every topological group G the following holds:
LTychG C ATychG C HTychG C TychG C [Tych]G.
P ro o f. Non-trivial inclusions follow from Lemmas 1.1 and 1.2.
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If G is locally compact (G € L C ), then every G-space is G-linearizable (see for
example [26, 31]), and, hence all classes from Proposition 1.3 coincide.
E xam ple 1.4. [16] In general, TychG ^ [Tych]G. There exists a continuous
action a of a group G on J(No) (the so-called hedgehog space o f spininess Ho [11])
such that (G, J(y<0) ,a ) is not G-Tychonoff. Note that in this example G and J(No)
are both Polish (i.e., are separable complete-metrizable).
Our main Examples 3.7 and 4.7 will show that, in general, ATychG ^ HTychG
and LTychG / ATychG.
Furthermore, the filter of all neighborhoods (nbd’s) at a point x in a space X , is
denoted by Nx(X). If ^ is a compatible uniformity on a topological space X , then
for every e G [i and A C X denote by e(A) the set { y G X | { x , y ) G £ , x G A}.
Subsets A , B will be called e-near if e(A) fl e { B ) ^ 0.
The finest compatible uniformity we denote by /xmaxDue to [25] the left, right, upper and lower uniformities on a topological group
will be denote by C, H, C V 71, C A 1Z respectively. An action a o n a uniform space
(X , fi) is saturated if each ^-transition is uniformly continuous.
A system (G, (X , [i),a) is called:
(a ) bounded [29] (or: motion equicontinuous [8]) if for every e G / i there exists
U G N e ( G ), satisfying (x , g x ) G e for every ( g, x) G U x X .
(b) quasibounded [17] if for every e G n there exists a pair (S, U) G /i x N e(G)
such that (g x , gy) G £ whenever (x, y) G <5 and g G U.
For example, if 7Z(G/H) denotes the right uniformity (see [25, Th. 5.1]) on the
coset space G/H, then the action a t is 7Z(G/ #)-bounded [28].
We will say that ( X , n ) is a uniform G-space and write: (G, (X,/j,),a) G UnifG,
(or: [ X, fi ) G UnifG) if a is /i-quasibounded, /i-saturated and continuous. By [8]
every compact G-space is bounded with respect to its unique uniformity. Therefore,
CompG C UnifG. Recall some useful facts concerning the class Unif .
61
FREE TOPOLOGICAL G-GROUPS
Fact 1.5. ([15, Th. 2.2], [18, Th. 1.5]) Let a : G x X —» X be a continuous action.
The following statements are equivalent:
(i) (G ,X ,a> € T y ch G;
(ii) (G, ( X , p ) , a ) G UnifG for a certain compatible n\
(iii) (G, ( X , £ ) , a ) is quasibounded for a certain compatible £.
Fact 1.6. ([17, Th. 1.2]) Let X be a topological group and a : G x X —> X
be a continuous action of a topological group G on X by automorphisms. Then
(G, (X , p ) , a ) G UnifG for every p G { £ , 7£, £ V 7Z}.
Let G be a topological group and U G N e (G). Denote by U the set of all pairs
( x , y ) G G x G such that y € UxU. Then { U : U £ Ne ( G ) } is a base of the lower
uniformity on G (cf. [25, Def. 2.5]). Using this description, it is easy to check that
Fact 1.6 is true also for p, = £ A 1Z.
For every topological group G there are several natural actions:
(a ) { G , G , a i ) t a t ( g, x) = gx;
(b ) ( G , G , a r), a r { g, x) — x g ~ x\
( c ) (G ,G ,a # ), a # ( g , x ) = g x g ~ l \
(d ) (G x G ,G ,tt), n ( ( g i , g 2) , x ) = g ix g ^ 1.
Elementary verification shows that (G x G, (G,/i),7r)
G UnifG for any
H G {£ ,1 1 , £\ZH, £ A K } . This implies (by Fact 1.5) that for the actions a £ , a r , a #
there exists a common compact G-extension of G.
Denote by AUnifG and LUnifG the classes of all £ V 'R.-uniform G-subspaces in
automorphic and linear G-spaces respectively. Then, LUnifG C AUnifG C UnifG.
Indeed, AUnifG C UnifG directly follows from Fact 1.6 and the second inclusion is
trivial.
Fact 1.7. [17, Th. 1.10] Suppose that G is Baire and q : G x I -4 I is a d saturated- action on a metrizatye uniform space (X , d) such that for a certain dense
subset Y C X the orbit mapping a y : G —> X is continuous for every y e Y . Then:
(a) a is continuous;
(b) (G, (X , d ) , a ) G UnifG;
(c) X is G-Tychonoff.
C orollary 1.8. Let G be Baire and a : G x X —>X be an action on a metrizable
group by continuous automorphisms. Suppose that orbit mapping a y : G —►X
is continuous fo r every y G Y , where Y topologically generates X . Then a is
continuous.
The last result essentially improves Proposition 2 from [6, VIII, §2.1].
Fact 1.9. [18, Th. 3.1] Let (G, (X , p ), a) G UnifG. Denote by { X , p ) and G
the completions of ( X , p ) and (G, £ V 1Z). Then there exists a continuous action
a : G x X —> X which extends a and satisfies (G, ( X , p ) , a ) G UnifG.
62
M.G. MEGRELISHVILI (LEVY)
2. W e st’s T h eorem and Equivariant H om ogen ization
For each compact space X , the group H ( X ) of all self-homeomorphisms,
equipped with the compact-open topology, continuously acts on X by the rule
7r(h,x) = h(x). For a metrizable compact X , the group H ( X ) is Polish. If, in ad­
dition, 7r is transitive, then Effros’s Theorem [9] yields that the triple ( H ( X ) , X , 7r)
is homogeneous.
Lem m a 2.1. Let r be an infinite cardinal and I T be the Tychonoff cube. Then
the triple ( H ( I T), / r , n) is homogeneous.
P ro o f. The arguments preceding the Lemma and Theorem 4.1 from [5, p. 104]
show that
/ No, 7r) is homogeneous. By Exercise 3 of §5 in [25], the product
ttg ( ( i J ( /N° ) ) r , ( / N°)r , 7rr ), where 7rT is defined coordinate-wise, is also homoge­
neous. We can identify ( / N° )r with I T and ( # ( / N° ) ) r with a topological subgroup
of H ( I T). Now the following claim completes the proof.
Claim . Let tt: G x X —» I be a continuous transitive action and P be a topo­
logical subgroup of G such that ( P , X , n |P xX ) is homogeneous. Then ( G , X , n ) is
homogeneous too.
P r o o f o f Claim . Let x G X . By our assumption, the orbit mapping ( n |PxX J
:
P —*• X is open. For every nbd U G N e ( G ), the set ^7r |PxX J (U) belongs to
Nx ( X ) and is contained in 7rx {U). Then nX(U) G N x ( X ) . Therefore, the action
7r is micro-transitive in the sense of [1]. This implies the openness of ttx (see, for
example, the proof (a) =>• (b) of Lemma 1 in [1]).
R em ark 2.2. It should be mentioned that there exists (see [4]) a compact space
X , such that the natural action 7r: H ( X ) x X —> X is transitive but non-Effros.
Due to our terminology, this means that ( H ( X ) , X , a ) is not homogeneous. By
the claim, the same is true for every topological subgroup G of H ( X ) transitively
acting on X . Moreover, by Theorem 3.2 from [12], we can suppose that G carries
an arbitrary group topology that leaves a \GxX continuous.
Recall that a topological group G is called r-bounded [14] (we write: £U(G) < r)
if for every V € N e( G ) there exists a subset S with the cardinality l^l < r such
that S V = G. As is well-known, G is r-bounded iff it is a topological subgroup in
a product of groups with weight < r.
Fact 2.3. [17]
For every G-Tychonoff space X there exists a compact Gextension Y of weight w ( Y ) < w ( X ) ■£U(G).
A weaker version (for a locally compact G, and replacing t u( G ) by £(G) (= the
Lindelof degree of G)) is obtained in [30].
Every continuous action a : G x X —> X induces the continuous action G /ker
a x X —> X , where kera = { g € G | gx — x for every x G X } . We say that
a is topologically exact (for short: t-exact) [19], if a is algebraically exact (i.e.,
kem = {e }) and there is no strictly coarser group topology on G, that makes a
continuous.
FREE TOPOLOGICAL G-GROUPS
63
Lem m a 2.4. Let ( G , X , a ) G TychG. Then there exists a ttg (G ,Y ,ir) G CompG
and a G-embedding f : X
Y , such that w ( Y ) — w ( X ) •w ( G ) and 7r is t-exact.
P ro o f. Consider the action a f. G x G —> G and the natural action of G on the
disjoint sum X U G. Clearly, this action is t-exact. Since (G, G, a t) G TychG
(see [8] or Lemma 1.2). Then, X U G is G-Tychonoff as well. Finally, we apply
Fact 2.3.
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J. W e st’ s T h eorem . [32]
Any continuous action o f a topological group G on a
closed Z-subset o f the Hilbert cube JN° can be extended to a continuous action o f
G on I * 0.
T h eorem 2.5. Let ( G , X , a ) G TychG and w( G) ■w ( X ) < Ko. Then (G , X , a ) is
embedded into ( H( I* ° ), JN°, n).
P ro o f. By Lemma 2.4, without loss of generality, we may suppose that a is texact and X is a metrizable compact G-space. Embed X to 7N° as a Z-space, and
use West’s Theorem. Then there exists a continuous action (G, I**0, a*) such that
algebraically G is a subgroup of H ( I * ° ) and a* \G x X — a. Since a* is continuous,
then the given topology a on G contains the compact-open topology <rc defining
with respect to the action a*. The ^-exactness of a (and hence of a*) implies
a = ac.
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C orolla ry 2.6. (Uspenskii [27]) H ( I h°) is a universal separable metrizable topo­
logical group.
T h eorem 2.7. Let G be an NQ-bounded group.
Then fo r every G -Tychonoff
space X there exists an equivariant embedding o f ( G , X ) into (H ( I T) , I T) where
t < w ( X ) ■w(G).
P ro o f. It suffices to show that the given ttg (G, X , a) is embedded into
(see proof of Lemma 2.1). By Lemma 2.4 we can assume that
a is i-exact and X is compact. Using the equivariant (generalization of Mardesic’s)
approximation Theorem [17, Th. 2.19], we can represent X as a G-limit of an in­
verse G-system { ( G , X i , a i ) |i G M } (|M| < r), consisting of compact metrizable
G-spaces Xi. Denote by fa the i-th projection X —> Xi, and by ipi the canonical
homomorphism G —» G/kercn*. As in the proof of Theorem 2.5, the topology ai
of G /kera* contains the compact-open topology o f, defining with respect to the
action ai. Since w( Xi ) < No, then w (G / ker ai,cr^) < No- By Theorem 2.5, there
exists an equivariant embedding
( V v . f t i ) : ( ( G / k e r a 4, < ) > * < } ^
(■i s
M ).
Consider the diagonal products:
h= n
i^M
° /<) ■
- x -> ( / r»)t , ^ = n Wi ° V*) ■ g ->
ieM
Since X = lim{X^: i G M }, then h is a topological embedding.
Observe that
the topological subgroup ip(G) acts continuously on h (X ) . Since h at the same
time is equivariant, then the i-exactness of a implies that ^ is a topological group
embedding.
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64
M.G. MEGRELISHVILI (LEVY)
C orollary 2.8. For every Ho-bounded group G TychG = HTychG.
Q uestion 2.9. Is it true that TychG = HTychG for every G?
Probably, the most natural candidate to be a counterexample is the abovementioned (Remark 2.2) example from [4].
Concerning Theorem 2.7, let us note that a related result for an L C cr-compact
group G (but with additional important properties) is proved in [2 ].
3. Free T op olog ica l G—G rou ps
As usual, for a topological space X denote by F ( X ) , A ( X ) , L ( X ) the free topo­
logical group, the free Abelian topological group, and the free locally convex space
respectively.
Let (G, X , a) be a ttg. We will say that an automorphic triple (G, Fa ( X ) , a) with
a continuous G-mapping ia : X —> Fa ( X ) is the free topological G-group over X ,
if for every continuous G-mapping ip: X —>
►P to an automorphic G-space P there
exists a unique continuous G-homomorphism (p: Fa ( X ) —►P such that p j a = <p.
If p, is a uniformity on X , then considering uniform G-mappings and the upper
uniformities on topological groups, we obtain the definition of the uniform free
topological G-group over ( X , p ) . The corresponding universal morphism is denoted
by ia : [ X , p ) -> Fa ( X , p ) .
The ( uniform ) free locally convex G-space L a ( X ) (respectively : L a ( X , p ) ) can
be analogously defined.
An obvious equivariant generalization of the standard product procedure (see for
example, [13]), shows that the just defined free G-objects always exist. However,
it turns out that the embedding problem for ia is much more complicated. Clearly,
ia is a topological (uniform) embedding iff (G, X , a) G ATychG (respectively :
( G , ( X , p ) , a ) G AUnifG).
In this section we show that, in general, HTychG ^
ATychG. Our main tool will be a “regionally proximal-type” relation. We start
with the well known (at least for S = G) definition from topological dynamics.
Let (G, X , a) be a ttg, p be a uniformity on X and S C G. A pair (a, b) G X x X
is called regionally S-proximal [7] and is indicated: (a, b) G Q s, if for every e G p
and arbitrary nbd’s Oi G Na( X ) , 0 2 G N b ( X ) there exists g G S such that gO\
and g 0 2 are e-near. Otherwise, (a, b) is said to be regionally 5-distal. X is called
regionally S'-distal if Q s — A x : = {(x , x) |x G X } .
The following definition seems to be new.
D efinition 3.1. We say that a pair (a, b) G X x X is regionally S-pseudoproximal
and write : (a, b) G Qps (or: (a, b) G Qps ( X , p ) ) if there exists a finite set
{ a = Xq, xi , . . . , x n = bj with the following property:
(*5 ) for every e G p, and arbitrary nbd’s Oi G N Xi( X ) , i G { 0 , 1 , . . . ,n } there
exists g G S such that gOi and gO i+ 1 are e-near, for every i G { 0 , 1 , . . . ,
n — 1}.
A pair (a,b) will be called regionally *-pseudoproximal if (a, b) G Q y for every
V G N e(G). This defines a relation Q l = n { Q p
v \ V G N e ( G) j . If Q p = X x X
or Qp = A x , then we say that X is regionally *-pseudoproximal, or regionally
x=-pseudodistal respectively.
FREE TOPOLOGICAL G-GROUPS
65
Obviously, Qps and Q* are reflexive symmetric relations on X and always
Q s C QP
S,Q ! C Q ”s . I t s is symmetric and acts uniformly /z-equicontinuously
on X (i.e., if the family { a 9 | g G 5 } is uniformly ^-equicontinuous), then
Qp
s = Q s = A x - Thus, if G G L C and X is compact then Qp = A x - In general,
Q s ± Qp
s and Qp ± Qp
s.
E xam ple 3.2. Let G n = [h G H ( I ) | h(x{) = Xi,Xi =
G { 0 , 1 , . . . , n} }
be the topological subgroup of H ( I ) . Consider the ttg (G n, I , a ) and the canon­
ical uniformity on I. Then, for every natural n > 3, the elements 0 and 1 are
regionally Gn-distal. On the other hand, every pair (a, b) G I x I is regionally
*-pseudoproximal. Therefore, Q c n is a proper subset of Q c n f°r each n > 3.
E xam ple 3.3. Define the homeomorphism h : I —* I by the rule
3x2,
h(x) = < |
0< x < |
| \/3x — 1,
3x 2 — 4x + 2,
| < x < |
| < x < 1.
Consider the cyclic group G = { h nj nez and the natural action G x I —> I. Since
0, |, |, 1 are fixed then clearly (0,1) £ Q g • On the other hand, elementary com­
putations show that Q pg = 1 x 1. Note also that Qp = A x if G is discrete.
Lemma 3 .4. If f : ( X i,/ii)
—> ( X 2 ,H2) is a uniform G-mapping, then
( / x /)(Q 5(JY llMl)) c Qts ( X 2,^ 2) and ( f x /) ( Q ? ( X llW )) C Q l ( X 2,fi2). In
particular, if ( X, fi ) is regionally *-pseudodistal, then every uniform G-subspace
(Y,h\y) is regionally *-pseudodistal.
Theorem 3.5. Every G-group (G, (X , /i), a) is regionally *-pseudodistal fo r each
H G { c ,n ,c w n}.
Proof. First we consider the case /z = IZ. Assuming the contrary, take a pair
(a, b) G Qp of distinct elements. Since X is a Hausdorff topological group and a is
continuous, we can choose nbd’s Vo G Ne ( X ) , U G N e ( G ) such that
Vo n 9(Voa6-1) = 0
\ /ge U.
(1)
Since QZ C Q y - (0 , 6) € Q y .
Consider a finite set { a = x<). X i , . . . ,
x n = 6} satisfying Definition 3.1. Choose symmetric nbd’s Vi, V2 G Ne ( X ) with
the properties:
xqx ~ xV%
C V i X Q X V iG { 0 ,1 ,... ,n }
V?+1 c Vo-
(2)
(3)
Due to Definition 3.1, we pick for e : = { (x, y) S X x X | x y ~ 1 6 V2}
an element g G U such that g{V2Xi) and ^(V^^i+i) are VVnear with respect
to the right uniformity 11 on I . More precisely, there exist finite sequences
{po,P i, ••• ,P n -1}, {<71,92, - - - ,9n} in V2 such that g(piXi)(g(qi+1Xi+1))
every i G { 0 ,1 ,... ,n - 1}. Since a 9 is an automorphism,
g{pix ix i^ l qi^ 1) G V2
V i G { 0 ,1 ,... , n — 1}.
G V2 for
(4)
M.G. MEGRELISHVILI (LEVY)
66
Consider the element
2 = g i p o x o x ^ q i 1) g { p i x i x ^ 1q 21) •••g i p n - i x n - i x ^ q ^ 1).
Since V2 C Vi by (2), (4) and (3) imply z G V2 C V™ C Vq. Clearly,
2 = g ip oxox^ 1{ q ^ p ^ X i X z 1( q z 1p2) ■■•(g“ l 1pn_ i ) z n_ i£ ~ 1g~1)
and, q~xPi G V2 l V2 = V2 for each i G { 1 , . . . , n — 1}. Using (2) and the trivial
cancellations of the form x o x ~ 1x ix~+l = x o x ^ , (1 < i < n — 1) after n — 1 steps
we get
^
e giPoV^xox^q-1) c giVfxox-'q-1).
Using (2) (for i = n), we obtain
2 G g(V {l+1x o x ~ 1) = g {V ? +1ab~l ) C g{V0ab~l ).
Thus, z G Vo H 2 (Vba&_1), which contradicts (1). This proves the case fi = TZ.
For fi = C, use the G-unimorphism (X , C) —» (X , TZ),x —* x -1 and if /z = C V TZ
use Lemma 3.4 for the uniform G-mapping / = l x : (X , £ V TZ) —> (X , 7£).
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Theorem 3.6. Let ( X, fi ) be a *-pseudoproximal G-space. Then every uniform Gmapping (X , n) —> (Y, £) into a G-group Y is constant fo r each £ G {£ , TZ,CV TZ}.
In particular, the free uniform G-group Fa { X , p ) is cyclic discrete.
Proof. Combine Lemma 3.4 and Theorem 3.5.
I
Example 3.7. Let X — I n be the n-dimensional cube, or let X = § n be the
n-dimensional sphere (in both cases n G N). Then ( H ( X ) , X , a ) is a regionally *pseudoproximal ttg with respect to the unique uniformity on X . Then, by Theorem
3.5, the free topological G-group Fa ( X ) is cyclic discrete. Since (H( Sn), § n, a) is a
homogeneous triple, then we get that HTychG / ATychG and UnifG ^ AUnifG for
G: = H (§n). This example for X — I answers the question posed by the author
in [17, Problem 1.14].
Question 3.8. Under which conditions does the homogeneous triple (G ,G / H ,a e)
belong to ATychG?
This is so, for example, if H is a neutral [25] subgroup. Indeed, in such cases,
Theorem 5.8 and Proposition 7.7 from [25] imply that the action ae of G on G/H is
uniformly equicontinuous with respect to the quotient uniformity C/H. Therefore,
by [22, Th.1.2] (or by our Proposition 3.11) G/H is even G-linearizable.
Question 3.9. Under which conditions does the free uniform G-group Fa { X , y )
coincide with the free uniform group F ( X , f i ) over X ?
Fact 3.10. [17]
(For a stronger version see [18, Lemma 2.1]). Let an action
a : G x X —> X be quasibounded with respect to a uniformity /1 on X and let each
^-transition be continuous. Suppose that orbit mapping a y : G —* X is continuous
for each y £ Y , where Y is dense in X . Then a is continuous.
FREE TOPOLOGICAL G-GROUPS
67
Proposition 3.11. Let
be a saturated action on a uniform space
( X , y ) . Suppose that a certain V G N e (G) acts \i-uniformly equicontinuous. Then
Fa ( X , y ) = F(X,iu) and L a ( X , y ) — L ( X , y ) .
Proof. Let a : G x F ( X , y ) —> F ( X , y ) be the lifted action. Clearly, each gtransition a 9 is continuous. Since X algebraically generates F ( X , y ) , then the
continuity of orbit mappings a x : G —» X and of group operations in F ( X , y) imply
that for each w G F ( X , y ) the orbit mapping a w : G —►F ( X , y) is continuous. From
the constructive description of a neighborhood system of the identity in F ( X , y)
[20, 23] it follows that V acts C V 7£-uniformly equicontinuously on F ( X , y ) . In
particular, a is £ V 7^-quasibounded. By Fact 3.10, a is continuous. Obviously,
this implies that Fa (X, y) = F ( X , y). Essentially the same proof works for L ( X , y)
using [24].
I
Pestov [21] proved the continuity of the associated action a : G x F ^ ( X ) —>•Ff}(X)
for the uniformly equicontinuous action a : G x I - > I , where F * ( X ) denotes the
free uniform balanced (i.e., C = TV) group in a variety v. For an analogous “lifting”
Theorem for a modification of the free locally convex spaces, see [22, Th. 1.2].
Lemma 3.12. If G G L C and ( G , ( X , y ) , a ) G UnifG, then each compact subset
o f G acts y-uniform ly equicontinuous. In particular, this holds fo r y = y max.
Proof. The standard compactness arguments prove the first assertion. In order to
apply it to our second assertion, observe that (G , ( X , y max) , a) G UnifG (see [18,
Pr. 3.7]).
|
Proposition 3.13. I f G G L C and ( G , ( X , y ) , a ) G UnifG, then Fa ( X , y ) —
F ( X , y ) and L a ( X , y ) = L ( X , y ) .
L(X).
In particular, Fa ( X ) = F ( X ) and La ( X ) =
Proof. Apply Lemma 3.12 and Proposition 3.11.
|
Concerning the Proposition 3.13, it should be mentioned that the continuity of
the lifted actions on A ( X ) and L ( X ) (under the assumption G G LC) has been
established earlier by Eisenberg [10] and de Vries [31] respectively.
Example 3.14. We give an example where ia : X —> Fa { X ) is a topological em­
bedding and the group Fa ( X ) only algebraically coincides with F ( X ) . Let Q be
the topological group of all rational numbers. Consider the ttg (Q, (Q, ^max), &e)Then y mSiX is complete and saturated, but there is no continuous non-trivial ac­
tion of M on Q. By Fact 1.9, at is not ^max-quasibounded. Then Fact 1.6 im­
plies that i a : (Q, AWx) “ > ( - ^ ( Q ) , £ V 7Z) is not a uniform embedding. Thus,
Pa€(Q) 7^ P(Q)- On the other hand, by Proposition 3.11, Fae(Q, C) = F ( Q , C ) .
Since F(Q , C) is, in particular, algebraically free over Q, clearly Fa£(Q) is also
algebraically free over Q.
Question 3.15. Is this true that if ( G , X , a ) G ATychG, then Fa ( X ) is alge­
braically free over X I
M.G. MEGRELISHVILI (LEVY)
68
4. G-Linearizations
Definition 4.1. Let p be the uniformity generated by a system V = {pi)i^M of
pseudometrics on a space X . We say that an action a : G x X —> X is weakly
V-Lipschitz if for every (i, g) G M x G there exists (c, j, V ) G R x M x Ng( G ) such
that p i ( g x ,g y ) < c p j ( x , y ) whenever Pi ( x, y ) / 0 and g E V . If pj can be choosed
in such a way that the inequality holds for every (g, x, y) G V x X x X , then we say
that the action is V-Lipschitz. If a is continuous and "P-Lipschitz (resp.: weakly
P-Lipschitz) then we write (G, ( X , V ) , a ) G LipG (respectively: G LipG). If each
Pi is a metric then these concepts coincide.
Theorem 4.2. If (G, ( X , fi), a) G LUnifG then there exists a system V = { pi}ieM
o f pseudometrics generating /i such that ( G , ( X , V ) , a ) G LipG.
Proof. Let (X , /Lt) be a uniform G-subspace of a locally convex linear G-space E.
Consider a system {pi}i^M of seminorms on E which generates the usual structure
of E. We may assume that for every pair i, j G M there exists k G M such that
Pi + P j < Pk■ Fix g G G and i G M . From the continuity of the action G x E —» E
at (g, 0 # ), where 0# denotes the zero-element, it follows that for a certain <5 > 0,
V G Ng( G ) and j G M , the inequality Pj(a) < 6 implies Pi{ga) < 1 for each
(g, a) G V x E. We can suppose that pi < pj. Then Pi{x) ^ 0 implies that
Pj (x) ±
0. Since pj
f°r every x G X with the property Pi(x) ^ 0,
Pi ( g ( ^ y 27) ) ^ 1- Therefore, Pi{gx) < \pj(x) for every ( g, x) G V x E where
Pi(x) 7^ 0. Now, it is obvious that the system of pseudometrics {pi}i£M, where
Pi(x,y) = pi{x — y) for x , y G X , is the desired one.
I
Theorem 4.3. If ( G , ( X , V ) , a ) G LipG fo r a certain collection V generating p
then (G, ( X , p ) , a ) G LUnifG.
Proof. Without restriction of generality we can assume that X has a fixed point.
Apply the Arens-Eells [3] embedding (X , { pi }i e M) c~* {E, {pi}i^M)- The explicit
description of pi — s shows that the associated action a : G x E —> E is {pi}i^M~
Lipschitz. Since each orbit mapping G —> E is continuous, by Fact 3.10, a is
continuous.
I
We write (G, X , a) G BTychG if X is a G-subspace of a Banach G-space. Anal­
ogously, ( G , ( X , n ) , a ) G BUnifG means that { X , p ) is a uniform G-subspace in a
Banach G-space.
Theorem 4.4. For a metrizable uniformity p, the following conditions are equiv­
alent:
(i) (G ,(X ,/i),a ) G BUnifG;
(ii) (G, (X , d ) , a ) G LipG (or, equivalently: G Lip G) fo r a m etric d;
(iii) There exists a countable cover U{5'n} neN — G such that:
(a ) Each Sn acts p-uniformly equicontinuous.
(b ) A certain SnQ has an interior point.
FREE TOPOLOGICAL G-GROUPS
69
Proof, (i) =>• (ii) in fact, is the same as in Theorem 4.2. For (ii) =>- (iii), consider
Sn = {g G G I d(gx,gy) < n d ( x , y ) for every x , y e X } .
For (iii) =>■ (i) take a metric p < 1 on X which generates //, and define inductively
Aq = {e } and A n+1 = A n ■A n U Sn+\. Then {^4n} n(EN also satisfies (iii) and, in
addition, A n ■A m C A n+m. From this inclusion it easily follows that if we define
d {x, y) = sup {
neN I *
sup { p { g x , g y ) }
ge A n
then d is a compatible metric and d(gx,gy) < 2nd [ x, y) for each (g , x , y ) G A n x
X x X . Using the arguments from the proof of Theorem 4.3, by the Arens-Eells
construction, there exists an isometric G-embedding of (X , d) into a normed Gspace E. Since LUnifG C UnifG, Fact 1.9 completes the proof.
|
Theorem 4.5. Let G be an L C a-compact group, and X be a metrizable G-space.
Then (G , X , a ) G BTychG. Moreover, the following statements are equivalent:
(i) (G, ( X , f i ) , a ) G BUnifG;
(ii) Each g-transition is p-uniformly continuous and p, is metrizable.
Proof,
(i) =» (ii) is trivial. For (ii) =>• (i) use consequently Fact 1.7(b), Lemma
3.12 and apply Theorem 4.4. In order to check (G, (X,/j,),a) G BTychG, if suffices
to show that there exists a metrizable saturated uniformity fi. A stronger result
can be found in [18, Th. 3.8]. So our proof is completed.
|
Theorem 4.6. Let X be a compact metrizable G-space and G be a closed topo­
logical subgroup o f H ( X ) . Then the following statements are equivalent:
(i) G is L C and a-compact;
(ii) { G , X , a ) G BTychG.
Proof, (i) => (ii) directly follows from Theorem 4.5. For (ii) =>• (i) use the impli­
cation (i) =>• (iii) from Theorem 4.4 and apply the Ascoli-Arzela Theorem.
|
Theorems 4.4 and 4.5 have been proved also in [15]. We close the paper with
the following principal counterexample.
Example 4.7. There exists a G-group X which is not G-linearizable. This means
that, in general, ATychG ^ LTychG.
Construction. Let X be the measure algebra of all equivalence classes of
Lebesgue measurable subsets of the closed interval I. Under the operation A (the
symmetric difference) and the norm ||^4|| = meas(A), X is a separable completemetrizable topological Abelian group homeomorphic to the Hilbert space
(see
Theorem 7.1 and Corollary 7.2 in [5, VI. 7]). Let G be the group of all such homeomorphisms h: I —> I that h and h~x are Lipschitz and denote by a : G x I —> I
the action a ( h , x ) = h(x). Since each a h is a measurable function, the associated
action a * : G x X —►X is well defined. Moreover, each transition (a*) h : X —> X
is || • ||-uniformly continuous. Endow G with the topology T|| . || of || • ||-uniform
M.G. MEGRELISHVILI (LEVY)
70
convergence. Then a* is continuous. It turns out that the constructed G-group X
is not G-linearizable.
P ro o f. In order to show that X is the desired G-group, observe that the mapping
<p: I —>X , ip(t) = [0, t] is an isometric G-embedding. Therefore, it suffices to check
that (G, I, a) is not G-linearizable. We will prove in fact that I, as a G-space, is not
even G-mapped non-trivially into a linear G-space. Assume the contrary. Then,
due to Theorem 4.2, a is weakly "P-Lipschitz for a certain family V = {pi}i^M
of continuous pseudometrics on I containing a non-constant pi0. There exists a
sequence ( x n )n eN in I converging in the usual topology to a point xq such that
pio( x n, x o) ^ 0 for every n G N. Indeed, if this is not so, then for every x G I
one can choose 8X > 0 such that pio( y , x ) = 0 for every y G B ( x , 6 x ), where
B ( x , 6 x) = { y G I | \y — x\ < <5X}. There exists a finite cover { B ( x k , 8 Xk )}? = , o f / ,
such that B ( x k , 6 Xk) n B ( x k + i , S Xk+1) ± 0 for each A: G { 1 ,2 ,... , n —1}. Since pio is
a pseudometric, by the triangle axiom, we get pi0(x, y) — 0 for every x , y G I. This
contradiction proves the existence of the above-mentioned (xn)n6pj- Without any
loss of generality we may assume that this sequence is monotonic. For simplicity
suppose that (zn)neN is increasing and hence each x n lies in [0 , x q ) .
By Definition 4.1 there exists a triple ( c , p j , V ) from K x P x Ne ( G ) such that
V — V ~ l and pi0( x , y ) ^ 0 implies that
pio( h{ x ) , h{ y) ) < c p j { x , y )
V h G V.
(*)
Since pj is a continuous pseudometric, pj (xn, xo) tends to zero. Thus we can choose
a subsequence ( xnk)ke n such that x nk > Xk and
2c pj ( x nk,xo) < PiQ{ x k, x 0)
V k G N.
(**)
For each k G N there exists a homeomorphism hk'. I —+ I satisfying:
( 1) hk( xk) — x nk\
(2) hk(t) = t for every t G [0,Xfc_i] U [x0) 1];
(3) hk is linear on [xk-i,Xk] and [as*, aro]Clearly, each hk belongs to G. Moreover, (2) implies that
mea,s(hk(A)AA) < \xk-i — a^ol
V iG l.
Therefore, (hk)keN converges to the identity in G. Thus, hm G V for a certain
m e N. Then, by (**), (*), (1), (2) we get
Pio {Xmt Xq) ^ ‘I cpj (x nm , Xo) ^ 2Pi0 {hrn ix nm) ,h rn (s^o))
^Pio {.Xmi Xq).
Hence, pio( xrn, x o) > 2pio(xm, xo) > 0. This contradiction completes the proof. |
Acknowledgemement. I would like to express my gratitude to Yu.M. Smirnov,
J. de Vries and V.G. Pestov for valuable remarks. I am also grateful to the referee
for helpful comments.
FREE TOPOLOGICAL G-GROUPS
71
References
1. F.D. Ancel, An alternative proof and applications o f a theorem o f E.G. Effros,
Michigan Math. J. 34 (1987), 39-55.
2. S.A. Antonyan, J. de Vries, Tychonov’s theorem fo r G-spaces, Acta Math.
Hung. 50 (1987), 253-256.
3. R.F. Arens, J. Eells, On embedding uniform and topological spaces, Pacific J.
Math. 6 (1956), 397-403.
4. D.P. Bellamy, K.F. Porter, A homogeneous continuum that is non-Effros , Proc.
Amer. Math. Soc. 113 (1991), 593-598.
5. C. Bessaga, A. Pelczynski, Selected Topics in Infinite-Dimensional Topoloay,
P.W.N., Warszawa, 1975.
6 . N. Bourbaki, Elements de Mathematique: Integration, Ch. VIII, Hermann,
Paris, 1966.
7. I.U. Bronstein, Extensions o f Minimal Transformation Groups, Sijthoff & Noordhoff, Alpher aan den Rijn, 1979.
8 . R.B. Brook, A construction o f the qreatest ambit, Math. Systems Theory 4
(1970), 243-248.
9. E.G. Effros, Transformation groups and C*-algebras, Ann. of Math. 81 (2)
(1965), 38-55.
10. M. Eisenberg, Embedding a transformation group in an automorphism group,
Proc. Amer. Math. Soc. 23 (1969), 276-281.
11. R. Engelking, General Topology, P.W.N., Warszawa, 1977.
12. L.R. Ford, Homeomorphism groups and coset spaces, Trans. Amer. Math. Soc.
77 (1954), 490-497.
13. M.I. Graev, Theory o f topological groups I, (in Russian), Uspekhi Mat. Nauk
5 (1950), 2-56.
14. 1.1. Guran, On topological groups, close to being Lindelof, Soviet Math. Dokl.
23 (1981), 173-175.
15. M.G. Megrelishvili, Quasibounded uniform G-spaces, (in Russian), Manuscript
deposited at Gruz NIINTI (Tbilisi) on March 3, 1987, No. 331-G.
16. M.G. Megrelishvili, A Tychonoff G-space not admitting a compact Hausdorff
G -extension or G-linearization, Russian Math. Surveys 43:2 (1988), 177-178.
17. M.G. Megrelishvili, Compactification and factorization in the category of
G-spaces, in Categorical Topology and its Relation to Analysis, Algebra and
Combinatorics (J. Adamek and S. MacLane, eds.), World Scientific, Singapore,
1989, pp. 220-237.
18. M.G. Megrelishvili, Equivariant completions, Comment. Math. Univ. Carolinae 35:3 (1994), 539-547.
19. M.G. Megrelishvili, Group representations and construction o f minimal topo­
logical groups, Topology Appl. 62 (1995), 1-19.
20. V.G. Pestov, Neighborhoods o f unity in free topoloqical qroups, Moscow Univ.
Math. Bull. 40 (1985), 8-12.
M.G. MEGRELISHVILI (LEVY)
72
21. V.G. Pestov, To the theory o f free topological groups: free groups, exten­
sions, and compact coverability, (in Russian), Manuscript deposited at VINITI
(Moscow) on April 1, 1985, No. 2207-85.
22 . V.G. Pestov, On unconditionally closed sets and a conjecture o f A. A. Markov,
Siberian Math. J. 29 (1988), 260-266.
23. V.G. Pestov, Universal arrows to forgetful functors from categories o f topolog­
ical algebra, Bull. Austral. Math. Soc. 48 (1993), 209-249.
24. D.A. Raikov, Free locally convex spaces fo r uniform spaces, (in Russian), Mat.
Sb. (N.S.) 63 (1964), 582-590.
25. W. Roelcke, S. Dierolf, Uniform Structures in Topological Groups and their
Quotients, McGraw-Hill, New York, 1981.
26. Yu.M. Smirnov, On equivariant imbeddings o f G-spaces, Russian Math. Sur­
veys 31 (1976), 198-209.
27. V.V. Uspenskii, A universal topological group with a countable base, Functional
Anal. Appl. 20 (1986), 160-161.
28. J. de Vries, Can every Tychonoff G-space equivariantly be embedded in a com­
pact Hausdorff G-space?, Math. Centrum, Amsterdam, Afd. Zuivere Wisk.,
ZW 36 (1975).
29. J. de Vries, On the existence o f G-compactifications, Bull. Ac. Polon. Sci. Ser.
Math. 26 (1978), 275-280.
30. J. de Vries, Topics in the theory o f topological transformation groups, in Topo­
logical Structures II (Proc. Symp. Amsterdam, 1978), Math. Centre Tracts
No. 116 (P.C. Baayen and J. van Mill, eds.), Math. Centrum, Amsterdam,
1979.
31. J. de Vries, Linearization o f locally compact group actions, (in Russian), Trudy
Mat. Inst. Steklov 154 (1983), 53-70.
32. J.West, Extending certain transformation group actions in separable, infinite
dimensional Frechet spaces and the Hilbert cube, Bull. Amer. Math. Soc. 74
(1968), 1015-1019.
33. Ta-Sun Wu, Notes on coset transformation groups, in Topological Dynamics
(J. Auslander and W.H. Gottschalk, eds.), W.A. Benjamin, Inc., New York,
Amsterdam, 1968, pp. 507-512.
M .G . Megrelishvili (Levy)
Department of Mathematics and Computer Science
Bar-Ilan University
52900 Ram at-Gan
ISRAEL
megereli@bimacs.cs.biu.ac.il