A topological semigroup structure on the space of actions modulo wea equivalence. Peter Burton January 8, 08 Abstract We introduce a topology on the space of actions modulo wea equivalence finer than the one previously studied in the literature. We show that the product of actions is a continuous operation with respect to this topology, so that the space of actions modulo wea equivalence becomes a topological semigroup. Introduction. Let Γ be a countable group and let X, be a standard probability space. All partitions considered in this note will be assumed to be measurable. If a is a measure-preserving action of Γ on X, and Γ we write a for the element of AutX, corresponding to under a. Let AΓ, X, be the space of measure-preserving actions of Γ on X,. We have the following basic definition, due to Kechris. Definition. For actions a, b AΓ, X, we say that a is wealy contained in b if for every partition A i n i of X,, finite set F Γ and ɛ > 0 there is a partition B i n i of X, such that a A i A j b B i B j < ɛ for all i, j n and all F. We write a b to mean that a is wealy contained in b. We say a is wealy 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 wea equivalence class of a. For more information on the space of actions and the relation of wea equivalence, we refer the reader to [3]. Let A Γ, X, AΓ, X, / be the set of wea equivalence classes of actions. Freeness is invariant under wea equivalence, so the set FR Γ, X, of wea equivalence classes of free actions is a subset of A Γ, X,. Given [a], [b] A Γ, X, with representatives a and b consider the action a b on X,. We can choose an isomorphism of X, with X, and thereby regard a b as an action on X,. The wea equivalence class of the resulting action on X, 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,. This is clearly associative and commutative. In Section we introduce ew topology on A Γ, X, which is finer than the one studied in [], [] and [4]. We call this the fine topology. The goal of this note is to prove the following result. Theorem. is continuous with respect to the fine topology, so that in this topology A Γ, X,, is a commutative topological semigroup.
In [?], Tucer-Drob shows that for any free action a we have a s Γ a, where s Γ is the Bernoulli shift on [0, ] Γ, λ Γ with λ being Lebesgue measure. Thus if we restrict attention to the free actions there is additional algebraic structure. Corollary. With the fine topology, FR Γ, X,, is a commutative topological monoid. Acnowledgements. We would lie to than Alexander Kechris for introducing us to this topic and posing the question of whether the product is continuous. Definition of the fine topology. Fix an enumeration Γ s s of Γ. Given a AΓ, X,, t, N and a partition A A i i of X into pieces let Mt,a A be the point in [0, ] t whose s, l, m coordinate is s a A l A m. Endow [0, ] t with the metric given by the sum of the distances between coordinates and let d H be the corresponding Hausdorff metric on the space of compact subsets of [0, ] t. Let C t, a be the closure of the set { M A t, a : A is a partition of X into pieces }. We have a b if and only if C t, a C t, b for all t,. Define a metric d f on A Γ, X, by d f [a], [b] t sup d H C t, a, C t, b. This is clearly finer than the topology on A Γ, X, discussed in the references. Definition. The topology induced by d f is called the the fine topology. t We have [ ] [a] in the fine topology if and only if for every finite set F Γ and ɛ > 0 there is N so that when n N, for every N and every partition A l l of X, there is a partition B l l so that for all F and l, m. l,m 3 Proof of the theorem. We begin by showing a simple arithmetic lemma. an A l A m a B l B m < ɛ Lemma. Suppose I and J are finite sets and a i i I, b i i I, c j, d j are sequences of elements of [0, ] with a i, d j, a i b i < δ and c j d j < δ. Then a i c j b i d j < δ. i I i I i,j I J
Proof. Fix i. We have a i c j b i d j i c j a i d j + d j a i d j b i a a i c j d j + d j a i b i δa i + a i b i. Therefore i,j I J a i c j b i d j i I a i δ + a i b i δ. We now give the main argument. Proof of Theorem. Suppose [ ] [a] and [ ] [b] in the fine topology. Fix ɛ > 0 and t N. Let N be large enough so that when n N we have max sup d H C t,, C t, a, sup d H C t,, C t, b < ɛ 4. Fix n N. Let N be arbitrary and consider a partition A A l l of X into pieces. Find partitions D p and i D q i i of X such that for each l there are pairwise disjoint sets I i l p q such that if we write D l Di Dj then i,j I l D l A l < ɛ 4. Write s t s F. By we can find a partition Ei p of X such that for all F we have i p i,j a Di Dj Ei Ej ɛ < 4 3 and a partition Ei q of X such that for all F we have i q i,j Define a partition B B l l of X by setting B l b Di Dj Ei Ej ɛ < 4. 4 i,j I l E i E j. For F we now have 3
l,m a b D l D m an bn B l B m l,m l,m l,m l,m a b Di Dj Di D j i,j I l i,j I m an bn Ei Ej Ei E j i,j I l i,j I m a Di b Dj Di D j i,j I l i,j I m an Ei bn Ej Ei E j i,j I l i,j I m l,m i,j,i,j i,j,i,j p q p q i,i,j,j p q i,j,i,j i,j,i,j a Di b Dj D i Dj i,j,i,j Ei Ej Ei Ej a Di Di b Dj Dj i,j,i,j Ei Ei Ej Ej a D i D i b D j D j E i E i E j E j a D i D i b D j D j E i E i E j E j a D i D i b D j D j E i E i E j E j. 5 4
Now 3 and 4 let us apply Lemma with I p, J q and δ ɛ 4 to conclude that 5 ɛ. Note that for any three subsets S, S, S 3 of a probability space Y, ν we have νs S 3 νs S 3 νs S S 3 + νs \ S S 3 νs S S 3 νs \ S S 3 νs S, hence for any l, m and any action c A Γ, X, we have c A l A m c D l D m c A l A m c D l A m + c D l A m c D l D m c A l c D l + A m D m ɛ, where the last inequality follows from. Hence for all F, l,m a b A l A m an bn B l B m l,m l,m ɛ + 5 ɛ. a A l A m a D l D m + a b D l D m an bn B l B m ɛ + a b D l D m an bn B l B m Therefore M A t,a b is within ɛ of M B t, and we have shown that for all, C t, a b is contained in the ball of radius ɛ around C t,. A symmetric argument shows that if n N then for all, C t, is contained in the ball of radius ɛ around C t, a b and thus the theorem is proved. References [] M. Abert and G. Ele. The space of actions, partition metric and combinatorial rigidity. preprint, http://arxiv.org/abs/08.47, 0. [] P. Burton. Topology and convexity in the space of actions modulo wea equivalence. preprint, http://arxiv.org/abs/50.04079. [3] Alexander S. Kechris. Global aspects of ergodic group actions, volume 60 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 00. [4] Robin D. Tucer-Drob. Wea equivalence and non-classifiability of measure preserving actions. Ergodic Theory Dynam. Systems, 35:93 336, 05. Department of Mathematics The University of Texas at Austin Austin, TX 787 pjburton@math.utexas.edu 5