A topological semigroup structure on the space of actions modulo weak equivalence

0.09 MB·English

by  Peter Burton

A topological semigroup structure on the space of actions modulo
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weak equivalence.
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Peter Burton
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a
J January 20, 2015

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Abstract

]
S Weintroduceatopologyonthespaceofactionsmoduloweakequivalencefinerthantheonepreviously
D studied in the literature. We show that the product of actions is a continuousoperation with respect to
this topology, so that the space of actions modulo weak equivalence becomes a topological semigroup.
.
h
t
a 1 Introduction.
m
[
Let Γ be a countable group and let (X,µ) be a standard probability space. All partitions considered in this

1 notewillbeassumedtobe measurable. Ifaisameasure-preservingactionofΓon(X,µ)andγ ∈Γwewrite
v γafortheelementofAut(X,µ)correspondingtoγundera. LetA(Γ,X,µ)bethespaceofmeasure-preserving
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actions of Γ on (X,µ). We have the following basic definition, due to Kechris.
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3 Definition 1. For actions a,b ∈ A(Γ,X,µ) we say that a is weakly contained in b if for every partition
4 (A )n of (X,µ), finite set F ⊆Γ and ǫ>0 there is a partition (B )n of (X,µ) such that
0 i i=1 i i=1
1. µ(γaAi∩Aj)−µ γbBi∩Bj <ǫ
0 (cid:12) (cid:0) (cid:1)(cid:12)
5 for all i,j ≤n and all γ ∈F. We w(cid:12)rite a≺b to mean that a is we(cid:12)akly contained in b. We say a is weakly
1 equivalent to b and write a∼b if we have both a≺b and b≺a. ∼ is an equivalence relation and we write
: [a] for the weak equivalence class of a.
v
i
X Formore informationonthe spaceofactions andthe relationofweakequivalence, werefer the readerto [3].
r LetA∼(Γ,X,µ)=A(Γ,X,µ)/∼bethesetofweakequivalenceclassesofactions. Freenessisinvariantunder
a weakequivalence,sothesetFR∼(Γ,X,µ)ofweakequivalenceclassesoffreeactionsisasubsetofA∼(Γ,X,µ).
Given [a],[b] ∈ A∼(Γ,X,µ) with representatives a and b consider the action a×b on X2,µ2 . We can
choose an isomorphism of X2,µ2 with (X,µ) and thereby regard a×b as an action on(cid:0)(X,µ).(cid:1)The weak
equivalence class of the re(cid:0)sulting (cid:1)action on (X,µ) does not depend on our choice of isomorphism, nor on
the choice of representatives. So we have a well-defined binary operation × on A∼(Γ,X,µ). This is clearly
associative and commutative. In Section 2 we introduce a new topology on A∼(Γ,X,µ) which is finer than
theonestudiedin[1],[2]and[4]. Wecallthisthefinetopology. Thegoalofthisnoteistoprovethefollowing
result.
Theorem 1. × is continuous with respect to the fine topology, so that in this topology (A∼(Γ,X,µ),×) is a
commutative topological semigroup.
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In [4], Tucker-Drob shows that for any free action a we have a×s ∼ a, where s is the Bernoulli shift
Γ Γ
on [0,1]Γ,λΓ with λ being Lebesgue measure. Thus if we restrict attention to the free actions there is
add(cid:0)itional alge(cid:1)braic structure.
Corollary 1. With the fine topology, (FR∼(Γ,X,µ),×) is a commutative topological monoid.
Acknowledgements.
WewouldliketothankAlexanderKechrisforintroducingustothistopicandposingthequestionofwhether
the product is continuous.
2 Definition of the fine topology.
Fix an enumeration Γ=(γ )∞ of Γ. Given a∈A(Γ,X,µ), t,k ∈N and a partition A=(A )k of X into
s s=1 i i=1
k pieces let MA(a) be the point in [0,1]t×k×k whose s,l,m coordinate is µ(γaA ∩A ). Endow [0,1]t×k×k
t,k s l m
with the metric given by the sum of the distances between coordinates and let d be the corresponding
H
Hausdorff metric on the space of compact subsets of [0,1]t×k×k. Let C (a) be the closure of the set
t,k
A
M (a):A is a partition of X into k pieces .
t,k
(cid:8) (cid:9)
We have a∼b if and only if Ct,k(a)=Ct,k(b) for all t,k. Define a metric df on A∼(Γ,X,µ) by
∞
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d ([a],[b])= supd (C (a),C (b)) .
f 2t (cid:18) H t,k t,k (cid:19)
Xt=1 k
This is clearly finer than the topology on A∼(Γ,X,µ) discussed in the references.
Definition 2. The topology induced by d is called the the fine topology.
f
We have [a ]→[a] in the fine topology if and only if for every finite set F ⊆Γ and ǫ>0 there is N so that
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when n≥N, for every k ∈N and every partition (A )k of (X,µ) there is a partition (B )k so that
l l=1 l l=1
k
|µ(γanA ∩A )−µ(γaB ∩B )|<ǫ
l m l m
l,Xm=1
for all γ ∈F and l,m≤k.
3 Proof of the theorem.
We begin by showing a simple arithmetic lemma.
Lemma 1. Suppose I and J are finite sets and (ai)i∈I,(bi)i∈I,(cj)j∈J,(dj)j∈J are sequences of elements of
[0,1] with a =1, d =1, |a −b |<δ and |c −d |<δ. Then |a c −b d |<2δ.
i j i i j j i j i j
Xi∈I Xj∈J Xi∈I Xj∈J (i,jX)∈I×J
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Proof. Fix i. We have
|a c −b d |≤ (|a c −a d |+|d a +d b |)
i j i j i j i j j i j i
Xj∈J Xj∈J
= (a |c −d |+d |a −b |)
i j j j i i
Xj∈J
≤δa +|a −b |.
i i i
Therefore
|a c −b d |≤ (a δ+|a −b |)≤2δ.
i j i j i i i
(i,jX)∈I×J Xi∈I
We now give the main argument.
Proof of Theorem 1. Suppose [a ] →[a] and [b ] →[b] in the fine topology. Fix ǫ >0 and t ∈N. Let N be
n n
large enough so that when n≥N we have
ǫ
max supd (C (a ),C (a)),supd (C (b ),C (b)) < . (1)
H t,k n t,k H t,k n t,k
(cid:18) (cid:19) 4
k k
Fix n≥N. Letk ∈N be arbitraryandconsiderapartitionA=(A )k ofX2 intok pieces. Find partitions
l l=1
D1 p and D2 q of X such that for eachl ≤k there are pairwise disjoint sets I ⊆p×q such that if we
i i=1 i i=1 l
w(cid:0)rit(cid:1)e D = (cid:0) (cid:1)D1×D2 then
l i j
(i,[j)∈Il
ǫ
µ2(D △A)< . (2)
l l 4k2
Write (γ )t =F. By (1) we can find a partition E1 p of X such that for all γ ∈F we have
s s=1 i i=1
(cid:0) (cid:1)
p
ǫ
µ γaD1∩D1 −µ γanE1∩E1 < (3)
i j i j 4
iX,j=1(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
and a partition E2 q of X such that for all γ ∈F we have
i i=1
(cid:0) (cid:1)
q
ǫ
µ γbD2∩D2 −µ γbnE2∩E2 < . (4)
i j i j 4
iX,j=1(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
Define a partition B =(B )k of X2 by setting B = E1×E2. For γ ∈F we now have
l l=1 l i j
(i,[j)∈Il
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k
µ2(γa×bD ∩D )−µ2(γan×bnB ∩B )
l m l m
l,Xm=1(cid:12) (cid:12)
(cid:12) (cid:12)
k
= (cid:12)µ2γa×b Di11 ×Dj21∩ Di12 ×Dj22
l,Xm=1(cid:12)(cid:12)(cid:12)  (i1,[j1)∈Il  (i2,j[2)∈Im 
(cid:12)
−µ2γan×bn Ei11 ×Ej21∩ Ei12 ×Ej22(cid:12)
 (i1,[j1)∈Il  (i2,j[2)∈Im (cid:12)(cid:12)(cid:12)
(cid:12)
k
= (cid:12)µ2 γaDi11 ×γbDj21∩ Di12 ×Dj22
l,Xm=1(cid:12)(cid:12)(cid:12) (i1,[j1)∈Il  (i2,j[2)∈Im 
(cid:12)
−µ2 γanEi11 ×γbnEj21∩ Ei12 ×Ej22(cid:12)
(i1,[j1)∈Il  (i2,j[2)∈Im (cid:12)(cid:12)(cid:12)
(cid:12)
k
= µ2 γaD1 ×γbD2 ∩ D1 ×D2 
(cid:12) i1 j1 i2 j2
l,Xm=1(cid:12)(cid:12)(cid:12)(cid:12) (i1∈,Ij[l1×,iI2m,j2)(cid:0) (cid:1) (cid:0) (cid:1)
−µ2 γanE1 ×E2 ∩ γbnE1 ×E2 
i1 j1 i2 j2 (cid:12)
(i1∈,Ij[l1×,iI2m,j2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)
k
= µ2 γaD1 ∩D1 × γbD2 ∩D2 
(cid:12) i1 i2 j1 j2
l,Xm=1(cid:12)(cid:12)(cid:12)(cid:12) (i1∈,Ij[l1×,iI2m,j2)(cid:0) (cid:1) (cid:0) (cid:1)
−µ2 γanE1 ∩E1 × γbnE2 ∩E2 
i1 i2 j1 j2 (cid:12)
(i1∈,Ij[l1×,iI2m,j2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)
k
≤ µ γaD1 ∩D1 µ γbD2 ∩D2 −µ γanE1 ∩E1 µ γbnE2 ∩E2
i1 i2 j1 j2 i1 i2 j1 j2
l,Xm=1(i1∈,IjXl1×,iI2m,j2)(cid:12)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)(cid:12)
≤ µ γaD1 ∩D1 µ γbD2 ∩D2 −µ γanE1 ∩E1 µ γbnE2 ∩E2
i1 i2 j1 j2 i1 i2 j1 j2
(∈i1p,×jX1q,×i2p,×j2q)(cid:12)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)(cid:12)
= µ γaD1 ∩D1 µ γbD2 ∩D2 −µ γanE1 ∩E1 µ γbnE2 ∩E2 . (5)
i1 i2 j1 j2 i1 i2 j1 j2
(i1∈,piX22,×j1q,2j2)(cid:12)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)(cid:12)
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ǫ ǫ
Now (3) and (4) let us apply Lemma 1 with I =p2,J =q2 and δ = to conclude that (5)≤ . Note that
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for any three subsets S ,S ,S of a probability space (Y,ν) we have
1 2 3
|ν(S ∩S )−ν(S ∩S )|=|ν(S ∩S ∩S )+ν((S \S )∩S )−ν(S ∩S ∩S )−ν((S \S )∩S )|
1 3 2 3 1 2 3 1 2 3 1 2 3 2 1 3
≤ν(S △S ),
1 2
hence for any l,m≤k and any action c∈A Γ,X2,µ2 we have
(cid:0) (cid:1)
µ2(γcA ∩A )−µ2(γcD ∩D )
l m l m
(cid:12)(cid:12)≤ µ2(γcAl∩Am)−µ2(γcDl∩Am(cid:12)(cid:12) ) + µ2(γcDl∩Am)−µ2(γcDl∩Dm)
(cid:12) (cid:12) ǫ(cid:12) (cid:12)
≤(cid:12)µ2(γcA△γcD )+µ2(A △D )≤(cid:12) (cid:12) , (cid:12)
l l m m 2k2
where the last inequality follows from (2). Hence for all γ ∈F,
k
µ2(γa×bA ∩A )−µ2(γan×bnB ∩B )
l m l m
l,Xm=1(cid:12) (cid:12)
(cid:12) (cid:12)
k
≤ µ2(γaA ∩A )−µ2(γaD ∩D ) + µ2(γa×bD ∩D )−µ2(γan×bnB ∩B )
l m l m l m l m
l,Xm=1(cid:0)(cid:12) (cid:12) (cid:12) (cid:12)(cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
k
ǫ
≤ + µ2(γa×bD ∩D )−µ2(γan×bnB ∩B )
2k2 l m l m
l,Xm=1(cid:16) (cid:12) (cid:12)(cid:17)
(cid:12) (cid:12)
ǫ
≤ +(5)≤ǫ.
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A B
Therefore M (a×b) is within ǫ of M (a ×b ) and we have shown that for all k, C (a×b) is contained
t,k t,k n n t,k
in the ball of radius ǫ around C (a ×b ). A symmetric argument shows that if n ≥ N then for all k,
t,k n n
C (a ×b ) is contained in the ball of radius ǫ around C (a×b) and thus the theorem is proved.
t,k n n t,k
References
[1] M. Abert and G. Elek. The space of actions, partition metric and combinatorial rigidity. preprint,
http://arxiv.org/abs/1108.2147,2011.
[2] P. Burton. Topology and convexity in the space of actions modulo weak equivalence. preprint,
http://arxiv.org/abs/1501.04079.
[3] A.S. Kechris. Global aspects of ergodic group actions, volume 160 of Mathematical Surveys and Mono-
graphs. American Mathematical Society, 2010.
[4] R.D. Tucker-Drob. Weak equivalence and non-classifiability of measure preserving actions. Ergodic
Theory and Dynamical Systems, to appear. http://arxiv.org/abs/1202.3101.
Department of Mathematics
California Institute of Technology
Pasadena,CA 91125
[email protected]
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