A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a...

A Metamathematical Condition Equivalent to the Existence of a Complete Left InvariantMetric for a Polish GroupAuthor(s): Alex ThompsonSource: The Journal of Symbolic Logic, Vol. 71, No. 4 (Dec., 2006), pp. 1108-1124Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/27588505 .

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The Journal of Symbolic Logic Volume 71, Number 4, Dec. 2006

A METAMATHEMATICAL CONDITION EQUIVALENT TO THE EXISTENCE OF A COMPLETE LEFT INVARIANT METRIC FOR A

POLISH GROUP

ALEX THOMPSON

Abstract. Strengthening a theorem of Hjorth this paper gives a new characterization of which Polish

groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit

equivalence relation is not essentially countable.

?1. Introduction. The main theorem of this paper characterizes when a Polish

group has a complete left invariant metric compatible with its Polish topology:

Theorem 1.1. Suppose that G is a Polish group. Then the following are equivalent: G has a complete left invariant metric For any Polish space X which is a Polish G-space under some action

ju: G x X ? X, and any transitive M \= ZFC such that X,G,ju G M, //

(?,p,?) G M is such that P is a partially ordered set, p G P, ? is a F-name, and:

p lh g[H] g X

(p,p)\\-?[Hl]?%?[Hr]

then there is some x G XM such that:

p lh g[H] El x

where H denotes a F-generic over M, Hi x Hr denotes a P x F-generic over M,

andEQ denotes the orbit equivalence relation associated with ju.

Note that customarily G is used to denote a group when discussing groups, and G is used to denote a generic when discussing forcing. This could lead to some confusion so in this paper G will always denote a group, and H will always denote a generic.

In [4] Greg Hjorth proved the forward direction of Theorem 1.1:

Theorem 1.2 (Hjorth). Suppose that M \= ZF + DC, M is transitive, and X is a Polish G-space under the action ju. Suppose that X,G,ju G M. Suppose that there

Received May 20, 2005.

? 2006, Association for Symbolic Logic 0022-4812/06/7104-0003/$2.70

1108



COMPLETENESS OF INVARIANT METRICS FOR POLISH GROUPS 1109

is a complete left invariant metric compatible with the Polish topology for G. Then if (?,p,?) ? M is such that P is a partially ordered set, p G P, ? is a F-name, and:

P Ih ?[H] e X

(p,p)\\-?[Hl]?$?[Hr] then there is some x G XM such that:

p Ih ?[H] El x

So we are left only needing to prove the other direction of Theorem 1.1.

Definition. Let E be an equivalence relation on a Polish space X, and F an

equivalence relation on a Polish space Y. We say that E is Borel reducible to F, written E <b F, if there is a Borel function / : X ?> Y such that:

xEy ?=* f(x)Ff(y) for any x, y G X.

The notion of Borel reducibilty has been studied extensively for relatively simple, usually low in the projective hierarchy, equivalence relations on Polish spaces. The reader can find an overview of the theory in [5]. One natural subclass of equivalence relations which has been studied are the countable equivalence relations. An equiv alence relation is called countable if every equivalence class is countable. Though a

particular equivalence relation may not be countable, it may be essentially countable

up to Borel reducibility:

Definition. An equivalence relation E on a Polish space X is said to be essentially countable if there is some Borel equivalence relation F on a Polish space Y such that all of the equivalence classes of F are countable and E <B F.

For those who study Borel reducibility, Theorem 1.1 has an interesting corollary: Corollary 1.3. Suppose that G is a Polish group which does not have a com

plete left invariant metric. Then there is a Polish G-space X whose associated orbit

equivalence relation Eq is not essentially countable.

This is a thinly disguised version of the following: Theorem 1.4. Suppose that X is a Polish space with an analytic equivalence rela

tion E. Suppose that in some transitive M (= ZFC with X,E G M, there is (P, p, a) such that P is a partially ordered set, p G P, ? is a F-name, and:

(p9p)\r-?[Hl]E?[Hr] but there is no x G XM such that:

p Ih ?[H] E x

Then E is not essentially countable.

The proof of Corollary 1.3 given at the end of Section 2 may be viewed as a proof of Theorem 1.4.

Some readers will notice that in Theorems 1.1,1.2, and 1.4 we have been somewhat

sloppy by discussing a Polish G-space X in the ground model M, and another Polish

G-space, with P-name X, which exists in the generic extension, but we have not described how their actions are related. This is because we intend X to be the name



1110 ALEX THOMPSON

for the natural "version" of X in the generic extension. The action should be the natural extension of the action of G on X to an action of G[H] on X[H], where G is the name for the natural "version" of G in the generic extension.

One way to obtain X[H] (G[H]) is to fix a compatible complete metric d (df) for X (G) in M and take the Cauchy completion of X (G) in the generic extension. It is easy to see that this gives a Polish space (Polish group) in which X (G) sits and with the usual metric for Cauchy completions extending d.

Since XM x GM will be dense in X[H] x G[H] under d x d' we may extend the continuous action ju: X x G ?> Jtoa continuous action ju': X[H] x G[H] ?>

X[H]. This provides an extension of the equivalence relation Eq to an equivalence relation on X[H] by looking at the orbit equivalence relation of the action ju', whose P-name we denote by ?q. Since Eq is S

j there is C ? X x JV closed such that:

E% = {xeX\3y eJf(x,y) ? C} In fact it will be true that:

?g[H] = {xe X[H] \3yejr (x,y) e c} where C indicates the closure of C in X[H]. By Shoenfield absoluteness, if M is a transitive model of ZF+DC, then ?q[H] must be an equivalence relation such that

E%[H]nXM xXM = E%. From here on, when it is clear what model we are working in, we will simply

discuss X, G, or Eq without mentioning which specific "version" we intend. If it is

not clear, we will use XN, GN or Eq to indicate we are working with the "version" in N. Certainly when we say that "X is a Polish space in M" we are abbreviating the statement "X G M and (X is Polish)M".

The approach to proving Theorem 1.1 is to isolate the correct analogy with the

following theorem of Su Gao, which is proved in [3]:

Theorem 1.5 (Gao). Let N be a countable model in a countable language S?. The

following are equivalent:

1. There is no compatible complete left invariant metric for the Polish group Aut (N). 2. There is an S?m ̂ -elementary embedding j from N into N which is not onto and

is a limit of elements ?/Aut(N). 3. There is an uncountable model of(pN, the Scott sentence ofN.

We will develop our proof along the lines of (1) implies (2) and (2) implies (3) of the above. The analogy for (1) implies (2) is encapsulated in the following claim:

Claim 1.6. Suppose that G is a Polish group with no compatible complete left invariant metric. Fix d, a left invariant metric for G, and let X be the Cauchy completion of G in this metric. There is a natural Polish action of G on X by isometries such that there is:

p:X?>X

distance preserving and not onto which is a limit of isometries given by the action ofG.

Suppose we fix a countable model N and let <pN be its Scott sentence. Suppose that M is a uncountable model of <pN. Satisfaction for Sco^m is absolute between transitive models of ZF so that in any generic extension in which M becomes



completeness of invariant metrics for polish groups 1111

countable we find that M is isomorphic to N. Using the conclusion of Claim 1.6 we are able to prove the following claim analogous to (2) implies (3) of Theorem 1.5:

Claim 1.7. Suppose that G, (X,d), and p are as in the Claim 1.6. Then there is

(Xm ,df), a non separable complete metric space, such that in any generic extension in which |^y |

= Ho we have that:

(X,d)^(X i,df).

In V we will be able to find a countable sequence of Lipschitz functions, with

Lipschitz constant 1, from X i to [0,1] which code the nonseparability of Xm. Using that in any generic extension in which \coJ\

= Ko we have (X, d) = (X i ,dr) we will be able to give a Co\\{a>, co\ )-name for a corresponding sequence of Lipschitz functions on X. It will be easy to see then that under the natural action of G on

sequences of Lipschitz functions from X to [0,1] there cannot be a representative of the equivalence class (under orbit equivalence) for the interpretation of ? in V, but that any two different generic extensions generate equivalent interpretations o.

Given the above analogy the reader may find that an understanding of the proof of Theorem 1.5 in [3] is helpful for the reading of this paper.

?2. The Proof. Without loss of generality we can assume that any metric in the

following discussion is bounded by 1.

Greg Hjorth showed in [4] that if G has a complete left invariant metric then the 2nd condition of Theorem 1.1 holds, so that we only need to prove the other direction. We will show that if G does not have a complete left invariant metric then for some M and some (P, p,o) G M the 2nd condition fails.

Suppose G is a Polish group but does not possess a complete left invariant metric. G does possess a left invariant metric d compatible with its Polish topology so we

may consider the Cauchy completion X of (G, d). We may view X as equivalence classes of Cauchy sequences from G and obtain an induced metric by considering representatives (gv), (hi) for two equivalence classes:

d((gi),(hi))= limdig^hi). n?*oo

This makes X Polish. Note that d \ G = d. In the future we will refer to this extension of d also as d.

Consider the map from G x X to X:

(g>x) ^ (g-Xi) :=g x

where (x?) is a representative of the equivalence class corresponding to x. This map is an action of G on X.

Fact 2.1. The above gives a well defined continuous action of G on X.

Proof. By Theorem 9.14 in [7] in order to prove continuity it is enough to see that the action is separately continuous as a map from G x X to X. This is easy to

show, as is showing that the map is well defined. H

Fact 2.2. This action of G on X is "by isometries". That is for any fixed g G G the map x h-> g x is an isometry ofX.

Proof. Let g G G. For x, y G X with (xt) and (y?) representatives for x and y:

d(g-x,g-y) = Um d(g -Xi,g -yt)

= lim d{x?9yi) =

d(x,y). _.



1112 ALEX THOMPSON

At times we may have distinct G-spaces with nonempty intersection. In the case that there could be some confusion about which action is being applied we will use

gY - y to denote the G-action for Y by g G G on an element y G Y. We will never

have more than one action defined for a specific space. Let Iso(X) and DP(Z) denote the space of isometries and distance preserving

functions on X respectively. Using the above we may view G as contained in Iso(X) (and in turn DP(X)) via the map g i-> ng where we define ng by:

rcgM =g-x

for x G X. Let {qi}i>\ enumerate a dense subset of G (and hence of X). We may equip

DP(X) with the metric: oo

d (711,712) =^22~ld(ni(qi),7Z2(qi)). i=\

Any two distinct metrics obtained by applying the above definition to different countable dense subsets yield equivalent metrics. The topology given by any such

metric is just the topology where for any (qo,...,qn) X<co, any e > 0, and any g e DPtST):

# = {fe DP(X) I Vi < n d(f(qi),g(qi)) < e} is a basic open set.

Fact 2.3. d \ G andd are compatible metrics for G.

Proof. This follows easily from the definition of d, the nature of basic open neighborhoods mentioned above, the fact that the action of G on G is left multipli cation, and that multiplication on G is continuous. H

The following metric on Iso(X) gives it a Polish group topology in which its obvious action on X is continuous:

D(7iu7Z2) =

~[d (tzi, tz2) + d(n?\ 7i~1)].

Iso(Z) is complete in this metric.

Fact 2.4. D andd \ Iso(X) give compatible metrics on Iso(X). Proof. This is a well known fact for the spaces of continuous functions from Y

to Y and the space of homeomorphisms from Y to Y for any Polish spaces Y. H

Fact 2.5. G 2? X

Proof. Both d and d \ G are left invariant metrics for G compatible with its Polish topology. Therefore they are equivalent. So the Cauchy completion of G in d, which is X, and the Cauchy completion of G in d, which is G, are isomorphic. H

The connection between G and X is deeper than suggested by Fact 2.5:

Claim 2.6. Let G denote the closure of G in DP(X). Then G\G ^ 0

Proof. Let {g?) be a cauchy sequence in (G,d) which does not converge in G. Consider the map p : X ?> X:

p(x) = lim gi x.




It is clear that p G DP(X) and that (g?) converges to p in the topology of DP(X). Therefore p G G. Since p(e)

? lim,,-^^) G X\G then p ? G, since the action

of G on G is left multiplication. H

Claim 2.7. Any p G G\G is not onto.

Proof. Any subgroup of Polish group which is Polish in the inherited topology is closed in the larger group. Since the topologies of both Iso(X) and G are just those inherited from DP(X), then (G\G) n Iso(Z) =0. H

In order to show that (2) of Theorem 1.5 implies (3) Gao employed the following approach. Suppose that N and j satisfy case (2) of the theorem. Then j"N is a proper submodel of N and by hypothesis the models are 5?Wu(? elementarily equivalent. Rename these models as N = Jf\ and j"N

? Jô- By analogy with Jô and JV\ there is some J^ 2 ^\ with ̂i ? JV\ and JV\ -<%> J?2- Again by analogy we can obtain JV-$ 2 ^2 with J^ = J?2 and JV2 -<&a ^3. We can continue in this

way to obtain an ^ i, -elementary chain:

v -*'COl, CO * =^CO\,CO ?*' CWl , CO ll -*' CO] ,co

The theorem on elementary chains holds true for ̂ uco so automatically Jf =

\Jn? ̂n is elementarily equivalent to </F0. Hence we may apply Scott's Theorem to

get Jô = ̂. This allows us to continue this process through the countable ordinals

(Scott's Theorem only applies to countable models) so that we get a sequence

(/a)aeo)i suchthat:

1. If a < ? <co\ then Jfa C Jf? and </Fa = JV?. 2. If/? G co\ and ? is a limit ordinal then Jf?

= Uc*e/? -â

3. If a < ? < coi then jra -< ?^ Jf?.

Finally, if we let Jfm =

\Ja<COl â then JVm is uncountable, and since it is the limit

of an Secoua) -elementary chain, it models ipN. In order to obtain Xm we will proceed in a similar fashion. Fix p G G\G. Since

p is distance preserving then we may view p as giving an isomorphism between X and p"X. If we then identify X with p"X we can obtain X' 2 X with an extension of the metric on X such that there is some:

p':Xf^X

that, in the "natural" action of G by isometries on X', is a limit of actions by G. Since X7 is isomorphic to X we may repeat this process to obtain X" such that ICI'C X" each isomorphic to each other. We may continue this process so that we end up with:

icl'cfc-c x'(w) c .

such that for each i G N we have pt : X'? ^ X*i+l\ If we take X = \Jieco X'?

we may find that we have a metric space which is not complete, so that any hope of isomorphism with X, a complete space, is lost. However, should we wisely take X to be the Cauchy completion of \Jieco X'^l\ our reward will be to discover that

Xco = X. This allows us, like with the countable models, to build Xa = X for each a < co\, and eventually X i. Xm turns out to be non-separable and, if we carefully drag along the relationships between different levels of our construction, then in



1114 ALEX THOMPSON

any generic extension in which coj becomes countable we will be able to see that

It will be helpful to formalize a certain property.

Definition. Let (Xd1) and (Z, d") be Polish spaces, a a Polish action of G on 7, b a Polish action of G on Z, and functions n and </>. We will say that property P holds of (Y,d')9 (Z, d"), a, b, n, and 0, abbreviatedP(Y,d', Z, dn\ a,b,n, </>), if:

1. (?: (X,d)^ (Xd') 2. Y C Z; d" \Y = J7 3. n:Xxd')*i{Z,d") 4. For all g G G and y G Y:

a(g,y) =

gY -y =def (?(gx -cl>~1 (y))

and for ail g G G and z e Z:

b(g,z)=gz .z =dcf 7i(/>(gx (^-^-^z)).

5. The map n~l is a limit of isometries given by the action Z? of G on Z.

The reader should keep in mind the slogan for the above definition is "Z is

isomorphic to X and Y sits in Z in an analogous way to the image of some limit of isometries sitting in X". All other aspects of the definition are either made in order to make implicit assumptions explicit, or for later convenience.

For our purposes the objects d', d", a, b, and (sometimes) <t> will all be clear from context and therefore we will abbreviate P( Y, d', Z, d", a, b, n, </>) as P( Y, Z, n) or P( Y, Z, 71, (?>) (if (j) is not clear).

It is an obvious consequence of the definition that if P(Y,Z,n) then for any g eG,y G Y, and z G Z:

n(gY y) =gz-n(y),

n-l(gZ'z)=gY-n-l(z)i

For convenience for any set Y we will think of Y<co as not containing the empty sequence. Given a metric d' on a metric space Y we may extend the metric to Y<co

by defining for y, z G Y<co:

d,{ ,{\, iflh(jT)^lh(z), Ky' } \max{J/(7(0),z(0)),...,J,(;;(?-l),z(?-l))},

ifn = lh(y) = lh(z) and we can extend the action of G by having G act pointwise on vectors. In the same way we can extend any function from Y to S to a function from Y<C? to S<co

by applying the function pointwise. We will use such extensions without mention or distinction. We will frequently use the following claim:

Claim 2.8. Suppose that P(XZ,n) holds with the exception of condition 5. The

following are equivalent: Condition 5 of the definition ofP( Y,Z,n). For any y G Y< and any e > 0 there is g ? G such that d"(n(gY y), y) < e. For any ze Z<co andanye > Othereis g G G suchthat dn(n~x(z),gz-z) < e.




Proof. The equivalence of the first and third statements is an easy consequence of the fact that the sets of form:

{/ G DP(Z) | Vi < lh(z) d"(f(z(i)),g(z(i))) < e} for fixedzeZ,ge DP(Z) and e > 0 form a basis for DP(Z), and that P(XZ,n). Let us prove that the last condition implies the second. Suppose we are given arbitrary y G Y<co and e > 0. Let z=n(y). Let g G G be such that:

d"(n-\z),gz .z)<e.

Then, using our assumption on z:

d"(y,n(gY y)) =

d"{n-\z),gz n(y)) =

d"(n-\z),gz z) < e.

The other direction of the proof is similar. H

We are going to prove the following proposition: Proposition 2.9. There is (Xa \ a g co\) an increasing sequence of sets, a metric

d' on Ua<?)1 Xa, a map naj\ Xa ??

X? for each a < ? < co\,for each a < co\ an

action of G on Xa, and(j>: X ?> Xo such that:

1. For every a < ? < co\ we have P(Xa, X?,na}?, 7To,a ? </>)>

2. If y < a < ? < c?\ then nyj =

naj o

7ryja.

Finally let X i be the Cauchy completion of{Ja<COi Xa. X x is not separable. Fur

thermore, in any generic extension in which co\ is countable we will have that XWl is

isomorphic to X.

It is important that we explicitly mention that property P holds between any two

levels, so that when we reach a generic extension in which coj is countable we have the necessary information available to prove that Xm is isomorphic to X, and so that we will be able to apply Claim 2.13. We will first prove a series of claims. The first of these is to show that if we have

an isomorphic copy of X we can always extend this copy in a way that is analogous to the way that X extends p"X for some p G G\G. This is intuitively true, and the

following claim only seeks to show specifically how to do it. For the rest of this discussion fix p G G\G. Claim 2.10. Suppose that (X,d) = (Y,df). Then there is Y' D Y and a natural

extension ofd' (which we continue to calld') to Y',a map n: (Xd') = (Yf,df), and actions of G on Y' and Y such that P( X Y', n).

Proof. Let0: (X,d) ̂ (Y,df). Pick Y' 2 Y such that |r\7| - \X\p"X\ and let (p : X\p"X ?> Y'\ Y be a bijection witnessing this fact. Define <?>f : X ?> Yf by:

,l{ ,_Up-x(x), ifxep"x, ?KX} \tp(x), ifxex\p"x.

Then define d' for y, y' G Y' by:

d\y,y')=d{(j>'-\y)^'-\y')).

Define n = </>' o p~l o (j)'~ . Define the G-action at Y and Y' by:

gY'-y' =

4>'(gx-4>''\y')l

gY-y =

<l>{gx-r\y)).



1116 ALEX THOMPSON

These definitions force these actions to commute through n. Finally, in order to see that n~l is a limit of isometries given by G, note that:

</>' : (X,d,p",p, G action on X) ^ (Y',d', Y,n~x,G action on Y')

and p is a limit of isometries given by the action of G on X. So F(Y, Y', n). H

We need another claim to have effectively finished the successor stage of our

construction.

Claim 2.11. Suppose that P( Yq9 Y\,n\, </>) and P( Y\, Yi,7i2,n\ o 0). Then

V(Y^Y2,n207ix,(j)). Proof. It is clear by inspection that all conditions of the definition of P(Fo, Y2,

7i2 o7i\9<j>) are satisfied except possibly that rcf1 o n^1 is a limit of isometries given

by the action of G on Y2. Let y G Y< and e > 0. By P( Y0i Y\9n\9 (j>) let g G G be such that:

d'(n1(gY?-y),y)<?-.

n\ (g Y?

f) ? Y< so we may apply P( Y\, Y2,712, n\ o </>) to obtain h G G such that:

d'(7z2(hYi 7n(gFo /)),7Ti(gr? - y)) < ?-.

Using the triangle inequality, and commutativity of the G-action and n\, we can see that hg G G is as desired:

d\n2onl((hg)Y?.y),y)=d\n2onl(hY?-gY?.y),y) =

df(7i2(hYi-7il(gY?.y)),y)<e. H

Some readers will note that, although we have not done so explicitly, we have finished the successor case. The remaining claim builds the machinery for the limit case.

The limit case is basically done via a back and forth argument. Claim 2.12. Suppose that (Yn)neco is an increasing sequence of sets such that

(j>: {X,d) = (Yo,d')9 and for all n < m < co we have a map 7in>m such that

P(y?, Ym,%n^m,n?^n o

(j)), and for any i < j < k < co we have that n^ =

jtjj? o mj.

Let Y be the Cauchy completion of[jnec? Yn in df. Then there is a map noiCO such

that if we define nntCO = no, ? %? for each n > 1 then for all n we have that

P(X?, X , n?i(?, 7io,n ? <?>) (where we naturally extend d' to Y ). Proof. Fix {yi}ie Q Y0 dense. Fix {vi}ieco C

\Jneco Yn dense such that for i < co we have vi+\ G Yt. In order to obtain an isomorphism between (Yo,df) and

(Y , d') it is enough to obtain a sequence of maps {sn}neco such that:

1. sn: 7o= Yn.

2. sn is distance preserving. 3. For each / the sequence {sn(yi)}neco forms a Cauchy sequence. 4. For each i the sequence {s~l(vi)}n>i forms a Cauchy sequence.

Then we will define the map tco,? by:

7io,co(y) = lim sn(y)

for y G Yq.




It is easy to see that this will be a well defined distance preserving map. We must see that it is onto. Since the map is distance preserving it is enough to see that the image includes all of {vi}ieco. Fix some v?. Since {s~l(vi)}n>i forms a

Cauchy sequence let y G Yo be its limit. {sn(y)}ne forms a Cauchy sequence converging to noiCO(y). Let e > 0. Fix N such that for n > N we have both

d,(sn(y),7io>C?(y)) < e/2 and d'(s~l(vi),y) < e/2. Thenfor n > N we have:

df(vi,7ioiC?(y)) < d'(vi,sn(y)) + d,(sn(y),7z0, (y)) =

d'(s~l(vi),y) +d'(sn(y),7z0,co(y)) < e.

Hence 7z0, (y) = vt.

Each Si will take the form:

Si(y) =

7z0,i((gigi-i.. .go)7? y)

for y G 7o and some fixed gi,gi-\, ...,goeG.

Let so = idyo and go ? e. Suppose that for / < n we have defined st and gz. Define sn+\ andgn+i as follows. Using P(Y?, Yn+i,7tntn+i) pickgM+i G G suchthat for j < n + 1:

?'(^l?g^i '

^)),^)) < 2-(w+1),

^(^+ifeJ;i-^-),^)<2-(w+1).

Having chosen such a g?+i we define for all y G Yo:

S/i+lGO =def ^?,?+l(gw+i * ^GO)

= rc?,?+l(gw+i 7r0,?((g?gw-l

. - .go)Y? >>))

= 7T0,W+l((g?+lg?

- .go)7?

* J>)

First we should check that the necessary sequences are Cauchy. It is enough to see that for any n and j < n:

d'(sn+l(yj),s?(yj))<2-^\

d'(s^(vj),s-l(vj)<2-^\

The first follows immediately from noting from the definition of sn+\ and selection

ofg?+i:

d'(sn+x(yj),sn(yj)) =

df(nn,n+l(gYn+l sn(yj)),sn(yj)) < 2~{n+l\

For the second note that for j < n + 1:

?~+l(rc,i,?+l (gnll ' Vj))

= S~l((g~^)Yn ^+lU?,n+l(gJ;i

' Vj)))

= ^_1(^)

so that by applying s~^x to both sides of our second condition for selecting gn+\ we obtain:

d'(s?+M>S?l(vj)) =

d'(S?l^Vj)^S?l?nninAgnl\'Vj))) =

d'(vj,7zn>n+l(g%rvj))<2-^l\

We must show that P(Y?, Y ,7znt ) holds for every n. All conditions of the definition are true by inspection except for condition 5. Fix y G Y< and e > 0.



1118 ALEX THOMPSON

Yn ? Y so y G 7^??, therefore let n~^(y) = z. Let also 7i^\(z)

= ?T. Pick some N> n such that:

d'(sN(?),y) =

d'(sN(i?),n^ (U)) < -.

Note also that:

SN(u) =

7T?jAr((giV . .

-go)7" ^0,n(w)) =

nKiN{(gN -go)*" * *)

Now fix some ?gG such that:

d'(nn,N(hYn -y),y) < -

so that:

^r(^,iv((giV...go)rn -z),7Zn,N(hYn -y)) < .

By acting on both sides by (gx ... go)-1 we obtain:

d'(itntN(z)9nntN{(g?l .

.g^1^)7" /)) =

df(nn^(z),7in^((g~x . ..g~lh)Yn /)) =

df(y,nn^((g~l ...g~lh)Yn -y)) <e

so that go"1 ...g^lh gives the necessary element of G. H

Claims 2.10, 2.11, and 2.12 give us the tools to prove Proposition 2.9:

Proof. We proceed by transfinite induction. Let Xo be a set, disjoint from X, of the same cardinality as X. Pick a bijection cj> : X ?> Xo and equip Xo with a metric via:

d'{y,z) = d{<l>-\y),r\z)) for 7, z G Xo. This completes the base case.

Suppose that the construction is complete up to and including stage a, that is we have an increasing sequence of sets (Xy \ y < a), and for all y\ < y2 < o? maps 7iyu?2 : Xn

?? X72 such that:

1. For every y\ < 72 < ol we have P(Xyi, Xyi, nn?n, n^n o </>). 2. If y\ < 72 < 73 < ? then 7ryi;73

= 7cy2jy3

o 7ryi>y2.

By Claim 2.10 we may find Xa+\ D Xa and a map 7^+1 : Xa ?> Xa+\ such that P(Xa,Xa+i,7ca>a+i,7ro,a

o </>). Furthermore, since for each y < a we have

F(Xy,Xa,7iy>a,no,y o (j)), then by Claim 2.11 we have P(Xy,Xa+i,ny,a+i). This

completes the successor case.

Suppose now that the construction is complete up to (but not including) some limit ordinal ?, that is we have an increasing sequence of sets (Xa \ a < ?), and for all y\ < y2 < ? we have maps nyuy2 : Xn

?? Xy2 such that:

1. For every y\ < y2 < ? we have P(Xyi, Xy2, nn,n, 7r0,yi o c?). 2. If 71 < y2 < 73 < ? then nyu73

= ny2?n

o nyun.

Pick a strictly increasing sequence of ordinals 0 = o?o < a\ < which is cofinal in ?. Consider the increasing sequence of sets (Xan)neco. These sets, together with the maps nan^m for each n <m, meet the conditions of Claim 2.12. Therefore, if we take X? to be the Cauchy completion of \Jneco Xan

= \Ja<? Xa then we will have for

each n maps nan$ such that P(Xan, X?, nan$, n^an o 0). If y < ? then by picking n

large enough that an > y we have P(Xy, Xan, ny>0in, no,y oc?), and so by Claim 2.11 we




obtamF{Xy9X?97zant?0 7iy,an97iotyO(/)). If we then naturally define nyj

= nanjonyt0tn

(which does not depend on n) we achieve P(Xy, X?, nyj,no,y o 0) for any y < ?. Finally suppose that we define X v as the Cauchy completion of \Ja<cov Xa.

Suppose we are in a generic extension in which coj is countable. We may, in this

extension, choose as above a strictly increasing sequence of ordinals 0 = a0 < a\ < cofinal in coj. Considering the sequence (Xan)neco and corresponding maps

Kan,am for each n < m we see that Claim 2.12, which is true in any model of a

sufficiently large finite fragment of ZF + DC, applies. Thus the Cauchy completion ?f U?eo> Xan

= Uo<cyv ̂ *, which is exactly X v, is isomorphic to X.

To see that Xwv is not separable in ? note that for each a < coj the space Xa is closed in X v. If there was a countable dense subset of X v, we could obtain

a countable S dense in X^v which was contained in \Ja< v Xa. But, by the un

countable cofinality of co\, S would be contained in some Xa for a < coj. But then

X^v = S C Xa C X Y?a contradiction. H

We need one more claim in order to prove Theorem 1.1. We need to know that

any two distinct isomorphisms arising from Claim 2.12 cannot differ too badly, or

else we might end up with wildly different isomorphisms of X^v in distinct generic extensions.

Claim 2.13. Suppose that ? <co\ is a limit ordinal. Suppose that 0 = ao < a\ < an dO = yo < y\ <

- are two different sequences of ordinals cofinal in ?. We can

apply Claim 2.12 to either (Xan)n(Eco or (Xyn)ne obtaining isomorphisms with X?. Let n be the isomorphism arising from (Xan)neco and 9 the isomorphism arising from

(Xyn)neco. There is some g G G such that for y G Xo'.

n-l0(y)=gx?-y.

Proof. Clearly n~l ode Iso(X). So it is enough to show that n~l o 9 is a limit of isometries induced by the action of G, since G is closed in Iso(X).

Let y G X0<w and let e > 0 be fixed. We need g e G such that:

d'(gx?-y,n-l9(y))<e.

Let sn : Xo ?> Xan be the sequence of maps from Claim 2.12 converging to n, and gn the corresponding group elements. Let t? : Xo ?>

X7n be the sequence converging to 9 and hn the corresponding group elements. Let m be such that:

d'{tm{y),e{y))<\. Pick n such that Xan D X7m and:

d'{n-\tm{y)ls-\tm{y)))<\. This step is valid because once defined {s~l (q)} forms a Cauchy sequence converg

ing to n~l(q). Lastly pick a G G, u?ngY(Xym,XCin,7iym,oln), so that:

a\nym,aS?Xyn ' tm{y)),tm(y)) <

|.



1120 ALEX THOMPSON

The desired group element will be b =?ef g^1... g~lahm ...ho:

d>{bx?.y,n-'G{y))<d>{bx?.y,S;\tm{y)))

+ d'{s~x{tm{y)),n-\tm{y)))+d'{n-\tm{y)),n~\6{y)))

By choice the second term is less than e/3. For the third term note that

d'{n-\tm{y)),n-\d{y)))=d'{tm{y),e{y)) < f

So that:

d'{b*> y,n-'6{y)) < d'(bx? y,s;\tm{y))) + | +

|. For the remaining term note that:

(gol---gn1ahm...h0)Xo-y

= (g?1 g?V" ?fti.no.*,((ahm

... ho)x? y) =

(g?1 --g?1)** ^1annr^n(aXym n0,ym((hm h)Xa y)) =

^"1(^m,a?(aJr"" -tm{y))).

This gives that:

d'(bx? y,s-\tm{y))) =

d'{s-\nym,an{ax^ tm{y))),s~x\tm{y))

= d'(nym?an{ax^ tm(y))Jm(y)) <

|. Thus:

d'(bx?.y,n-'8(y)) <| +

| +

|=e. H

Now we are equipped to prove the new direction of the main theorem, Theo rem 1.1, which we state as Theorem 2.14:

Theorem 2.14. Suppose that M \= ZFC is transitive and G is a Polish group in M which does not have a complete left invariant metric. Then there is L G M, a Polish

G-space and there is Coll(co, co\)M-name ? such that:

iColWo^ lh ^ G ?'

?ColW*^' ^olWo") lh ?[Hi]?%?[Hr] but:

In the above, Coll(co, co\ )M refers to the canonical poset in M which collapses cof1. Proof. Let M be given and suppose G G M is a Polish group without a left

invariant metric, and let X be its Cauchy completion in some left invariant metric d. Let P = Coll(co,coi)M. Let (Xa \ a < cof), (naj \ a < ? < cof), and X u be

the result of applying Proposition 2.9 to G, and X. Without loss of generality let X = X0.

In M[H], for H a M-generic P-filter, there is a canonical strictly increasing sequence of ordinals 0 = c*o < c*i < cofinal in cof1. Using this canonical




sequence we may apply Claim 2.12 in M[H] to obtain a canonical P-name n

such that:

lPlh7T[i/]: (X,d) ^(X M,d').

Now consider a generic Hi x Hr over P x P. n[Hi] and n[Hr] were obtained via the methods of Claim 2.12 so that Claim 2.13 applies. Therefore there is some

g e GM[HlxHr] so that for aU x e jrMWxHr].

nlHtr'nlHrtx) =

gx? x.

Consider the space of infinite sequences of Lipschitz functions from Xo to [0,1] with Lipschitz constant 1 :

L = {((pi)ieN\ipi: X0 -> [0,1] suchthat \(Pi[x)-<pi(y)\<d{x,y) for all x,y GX0}.

There is a natural action of G on Lipschitz functions from Xo to [0,1] by:

(g ?>)(*) =

<f(g~l *)

Suppose that (p Lipschitz and x, y G Xo:

\g <p(x) -

g <p(y)\ =

\<f(g~l -x)-<p(g-1 -y)\ <d(g~l -x,g~x y) =d(x,y)

so this is indeed a well defined action by G. This extends to an action on L by:

g'(<Pi) =

(g-<Pi)

Since (X M)M is non-separable in M let (xa)a<(?M be a subset of (X^m^ in M, and let e > 0 in M be such that for a < ? < co\ :

d(xa,x?) > e.

Let (ra)a<C?M G M be a sequence of distinct elements of 2m such that for each

a, ra(0) = 1. Consider the following sequence of functions defined in both M and

M[H] on (X M)M and (X M)MW respectively:

cpi(x) =

mfa<C0Mlra{i)=ld(x, Be_(xa))

which may also be written as:

<pi(x)=d,(x, (J Bl(x0)). ra(i)=l

In M[H] the function:

(<Pi0 7i[H))eL as clearly each <pi is Lipschitz and n[H] is a metric space isomorphism. Let ? be the canonical name for (<pi o tc0com[H]).

Suppose that we have H? x Hr, a M-generic filter for P x P. As before there is

g G GM^/x^]suchthat:

niHtY'onlH^^g

in M[^/ x ifr]. Then for i G N:

(g"1 - (^ O ?[ff/]))(X)

= tpi O ?[ff/] O JC?Hj]-1 O 7T[i?r](x) = ^ O 7C[Hr](x)

so that:

(lp,lP)IH3g G(g.(7[i?/] =

(7[f?r]).



1122 ALEX THOMPSON

For the sake of contradiction suppose there was (^) G LM such that in some

generic extension M[H] there is g G GMm such that:

g ((/>>) =

(& o n[H]}.

Define for each a < cof1 :

Za = g"1 '7l[H]~l(xa).

Fix a countable basis {@j} G M for the topology of X. For each a < cof1 we may find some ja G co such that:

(9ja<ZBi{za).

Note that:

<Pi(?[H](gx y)) = (g 4>?(gx y) = <Pi(g~lX -gx-y) = 4>M. Furthermore:

d(y,za) =

d(y,(g-y ^n[H]-l(xa)) =

dr(n[H](gx y),xa).

In M we may define, for each j such that <j>'?@j =

{0}, a real qj G 2 by:

^(0 = 1 <= 4>!*j

= {0} and if 0^7 ^ {0} then define qj

= (000... ).

Now consider qja. Note that d(y, za) < e/4 if and only if d(n[H](gx y), xa) <

e/4. Also, if d(v9 xa) < e/4 then for ? ^ a we have rf(v, x^) > 3e/4 so that:

<pf(v) = 0 <^> ?>/(*<* )

= 0 <<=^ ra(i) = 1.

Thus, if y e0ja:

<t>M=0 <=> <Pi(7c[H](gx - y))

= 0 <=> ipi(xja) = 0.

Then by the definition of the qj sequence we have:

?/?(0 = ! ̂ ^ ?>i(*/J

= 0 <^=> ra(i) = 1.

Thus {qj }jeco 2 {^a}?<wM B?th these sets exist in M and the latter is uncountable, while the former is countable, which is a contradiction. H

In order to prove Theorem 1.3 we need a certain fact concerning countable Borel

equivalence relations. Let F2 denote the free group on two elements. Consider the

space:

2F2 = {/|/:F2-{0,1}}. We can define an action of F2 on 2?l via:

(g'f)(h)=f(g~1h) where g, h G F2 and / G 2?l. 2?l can be made Polish by identifying it with 2 and so this action gives an associated orbit equivalence relation Eoo which is Borel. It

was proved in [2] that E^ is "universal" for countable Borel equivalence relations in the following sense:

Theorem 2.15. Suppose that E is a countable Borel equivalence relation on a Polish

space X. Then E <b E^.

Now we are in a position to prove Corollary 1.3 which we restate as Corollary 2.16:




Corollary 2.16. Let G be a Polish group without a complete left invariant metric and suppose L as in Theorem 2.14. Then the orbit equivalence relation associated to

the action of G on L is not essentially countable.

Proof. Suppose not and let 9 : L ?> 2?l be a map such that:

xELGy <=> 9(x)EOQ9(y).

Let P = Coll(co, coj) and let Hi x Hr be V-generic for P x P. We may identify 9

with some Borel code 9. "9 codes a Borel reduction" is an absolute fact since it

requires only that we say:

Vx, y G X[3g eG (g x = y) <=>

3h G ?23w, z G 2?l(h w = z A 9(x) = w A 9(y)

= z)]

which is n^, and that 9 codes a function, which is also absolute. Now let ? be the P-name from the previous theorem. Consider a[H?], This is

a new element of X and it must have some corresponding point in 2?l since 9 is still a function, so consider 9(?[Hi]). F2 is definable subset of the hereditarily finite sets and therefore the makeup of F2 is absolute between transitive models of ZF. Therefore once one element of [9(?[Hi])]?2 is present the entire equivalence class must be present, since we may use F2 to compute the other elements. If we look at

V[H? x Hr] we can note that, because of the above, we must have that

[0(?[Hi])]?2 = [9(?[Hr])]w2

since ?[Hi]EQ?[Hr]. In fact this is forced by the top condition:

(1p, If) Ih [9(?[Hi])]?2 = [9(?[Hr])]?l.

We can then apply the following fact:

Fact 2.17. Let F be a partial order. Then if ? is a F-name such that for some

(p,q) GPxP:

(p,q)\h?[Hl] = ?[Hr]

then in fact there is A G V so that:

p\\-A = ?[H].

Proof. This can be proven by induction on the "rank over V". H

Applying 2.17 we obtain that [0(a[H?])]?2 G V. But "9~l([9(?[Hi])]?2) ? 0" is equivalent to:

3xeL3ye 2F2 [y G [0(a[H?])] A9(x)= y]

and therefore absolute. Hence there must be representative of [(o"[?//])]g present in V?a contradiction. H



1124 ALEX THOMPSON

REFERENCES

[1] H. Becker and A. S. Kechris, The descriptive set theory of Polish group actions, London Mathe

matical Society Lecture Notes Series, Cambridge University Press, Cambridge, 1996.

[2] R. Dougherty, S. Jackson, and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Transactions of the American Mathematical Society, vol. 341 (1994), pp. 193-225.

[3] S. Gao, On automorphism groups of countable structures, this Journal, vol. 63 (1998), no. 3,

pp. 891-896.

[4] G. Hjorth, Orbit cardinals: On the effective cardinalities arising as quotient spaces of the form

X/G where G acts on a Polish space X, Israel Journal of Mathematics, vol. Ill (1999), pp. 221-261.

[5] G. Hjorth and A. S. Kechris, Recent developments in the theory of Borel reducibility, Fundamenta

Mathematicae, vol. 170 (2001), pp. 21-52.

[6] T. Jech, Set theory, 3rd millenium ed., Springer Monographs in Mathematics, Springer, 2002.

[7] A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, Springer, 1995.

[8] Y. N. Moschovakis, Descriptive set theory, North-Holland Press, Amsterdam, 1980.

MATHEMATICS DEPARTMENT

UCLA

LOS ANGELES, CA 90095-1555, USA

E-mail: [email protected] URL: www.math.ucla.edu/~act2



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