aa r X i v : . [ m a t h . L O ] S e p NEW JUMP OPERATORS ON EQUIVALENCE RELATIONS
JOHN D. CLEMENS AND SAMUEL COSKEY
Abstract.
We introduce a new family of jump operators on Borel equivalencerelations; specifically, for each countable group Γ we introduce the Γ-jump. Westudy the elementary properties of the Γ-jumps and compare them with otherpreviously studied jump operators. One of our main results is to establishthat for many groups Γ, the Γ-jump is proper in the sense that for any Borelequivalence relation E the Γ-jump of E is strictly higher than E in the Borelreducibility hierarchy. On the other hand there are examples of groups Γfor which the Γ-jump is not proper. To establish properness, we produce ananalysis of Borel equivalence relations induced by continuous actions of theautomorphism group of what we denote the infinite Γ-tree, and relate these toiterates of the Γ-jump. We also produce several new examples of equivalencerelations that arise from applying the Γ-jump to classically studied equivalencerelations and derive generic ergodicity results related to these. We apply ourresults to show that the complexity of the isomorphism problem for countablescattered linear orders properly increases with the rank. § Introduction
The backdrop for our study is the Borel complexity theory of equivalence rela-tions. Recall that if
E, F are equivalence relations on standard Borel spaces
X, Y then E is Borel reducible to F , written E ≤ B F , if there exists a Borel function f : X → Y such that x E x ′ ⇐⇒ f ( x ) F f ( x ′ ).We say f is a homomorphism if it satisfies the left-to-right implication. We write E ∼ B F if both E ≤ B F and F ≤ B E , and we write E < B F if both E ≤ B F and E B F .The notion of Borel reducibility gives rise to a preorder structure on equivalencerelations. As with other complexity hierarchies, it is natural to study operationssuch as jumps. Definition 1.1.
We say that a mapping E J ( E ) on Borel equivalence relationsis a proper jump operator if it satisfies the following properties for Borel equivalencerelations E, F : ◦ Monotonicity: E ≤ B F implies J ( E ) ≤ B J ( F ); ◦ Properness:
E < B J ( E ) whenever E has at least two equivalence classes.Note that the terms jump or jump operator may be used for a monotone mappingwith E ≤ B J ( E ) in a context where strict properness is not relevant or has not beenestablished. We may also apply a jump operator to analytic equivalence relations;in this case we do not expect or require properness. Indeed, for all of the jump Mathematics Subject Classification.
Primary 03E15, Secondary 03C15, 06A05.
Key words and phrases.
Borel equivalence relations, jump operators, scattered linear orders. operators discussed below, one can find analytic equivalence relations which arefixed points for the mapping up to Borel bireducibility.One may also ask for some definability condition on a jump operator, but we donot need any particular conditions here. All the jump operators considered herewill be uniformly definable. For instance, for each jump operator discussed below,if R ⊆ Y × X is a Borel set so that each section R y is an equivalence relation on X , then the set ˜ R ⊆ Y × ˜ X given by ˜ R ( y, ˜ x , ˜ x ) iff ˜ x J ( E y ) ˜ x is also Borel,where J ( E ) is an equivalence relation on ˜ X .Several jump operators have been studied extensively, including the Friedman–Stanley jump [10] and the Louveau jump [20], which we discuss below. There isalso a jump operator on quasi-orders introduced by Rosendal [21].Here we introduce a new class of jump operators which are associated withcountable groups. Definition 1.2.
Let E be an equivalence relation on X , and let Γ be a countablegroup. The Γ -jump of E is the equivalence relation E [Γ] defined on X Γ by x E [Γ] y ⇐⇒ ( ∃ γ ∈ Γ) ( ∀ α ∈ Γ) x ( γ − α ) E y ( α ).We will use the term Bernoulli jump as a collective name for any member of thefamily of Γ-jumps. Indeed, note that if E = ∆(2) then E [Γ] is the orbit equivalencerelation induced by the classical Bernoulli shift action of Γ, and if E = ∆( X )for a Polish space X then E [Γ] is the orbit equivalence relation induced by the“generalized” Bernoulli action of Γ.We reserve the notation E Γ for the product of countably many copies of E with index set Γ. Thus E Γ is Borel isomorphic to E ω , and E [Γ] is an equivalencerelation of countable index over E Γ . Indeed, letting Γ act on X Γ by the left shift γ · x ( α ) = x ( γ − α ), we have that x, y are E [Γ] -equivalent iff there is γ ∈ Γ with γ · x E Γ y .It is clear that the Γ-jump operator is monotone for any Γ. We will be concernedwith whether and when the Γ-jump is proper. Before addressing this question, werecall the situation with the Friedman–Stanley and Louveau jumps. Definition 1.3.
Let E be a Borel equivalence relation on X . The Friedman–Stanleyjump of E is the equivalence relation E + defined on X ω by x E + y ⇐⇒ { [ x ( n )] E : n ∈ ω } = { [ y ( n )] E : n ∈ ω } . Theorem (Friedman–Stanley, [10]) . The mapping E E + is a proper jump op-erator. Definition 1.4.
Let E be a Borel equivalence relation on X and let F be a freefilter on ω . The Louveau jump of E with respect to F is the equivalence relation E F defined on X ω by x E F y ⇐⇒ { n ∈ ω : x ( n ) E y ( n ) } ∈ F . Theorem (Louveau, [20]) . For any free filter F , the mapping E E F is a properjump operator. The original proof of the Friedman–Stanley result used Friedman’s theorem onthe non-existence of Borel diagonalizers (recall a
Borel diagonalizer for E is ahomomorphism ϕ from E + to E so that ϕ ( x ) / ∈ { [ x n ] E : n ∈ ω } ). However, boththe Friedman–Stanley result and the Louveau result can be proved using the concept EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 3 of potential complexity of equivalence relations, which we briefly introduce. Firstwe will say that a
Borel class is a pointclass Γ consisting of Borel sets and closedunder continuous preimages. For example, Π ∼ α is a Borel class for any α < ω . Definition 1.5.
Let E be an equivalence relation on the Polish space ( X, τ ), andlet Γ be a Borel class. We say E is potentially Γ, written E ∈ pot(Γ), if there is atopology σ on X such that σ, τ have the same Borel sets and such that E is in Γwith respect to σ .We remark that E is potentially Γ if and only if E is essentially Γ, i.e., there issome E ′ in Γ with E ≤ B E ′ . Definition 1.6.
We say that a family F of Borel equivalence relations has cofinalpotential complexity if for for every Borel class Γ there is E ∈ F such that E / ∈ pot(Γ).Louveau established that if E is an equivalence relation with at least two classes,then the family of iterates of the Louveau jump of E with respect to a free filterhas cofinal potential complexity. Since being potentially Γ is equivalent to beingessentially Γ, a family of cofinal potential complexity cannot have a maximumelement with respect to ≤ B . Hence it follows that the Louveau jump with respectto a free filter is a proper jump operator.Meanwhile it is known that the family of Borel equivalence relations induced byactions of S ∞ has cofinal potential complexity. Moreover Hjorth–Kechris–Louveau[16] established that if E is an equivalence relation with at least two classes, thenany Borel equivalence relation induced by an action of S ∞ is Borel reducible tosome iterate of the Friedman–Stanley jump of E . Thus they arrived at a “potentialcomplexity” proof that the Friedman–Stanley jump is a proper jump operator.Returning to the Bernoulli jumps, we will establish the following. Theorem 1.
Let Γ be a countable group so that Z or Z <ωp for p prime is a quotientof a subgroup of Γ . Then the mapping E E [Γ] is a proper jump operator. In the proof, we will use a result of Solecki [24] which implies that if Γ is oneof the groups Z or Z <ωp for p prime, then the family of Borel equivalence relationsinduced by actions of the group Γ ω has cofinal potential complexity. Our mainwork will be to show that such equivalence relations are Borel reducible to iteratesof the Γ-jump. Before stating this result, we provide the following notation for theiterates of the Γ-jump. Definition 1.7.
For an equivalence relation E and a countable group Γ we definethe iterates J [Γ] α ( E ) of the Γ-jump recursively by: J [Γ]0 ( E ) = EJ [Γ] α +1 ( E ) = ( J [Γ] α ( E )) [Γ] J [Γ] λ ( E ) = M α<λ J [Γ] α ( E ) ! [Γ] for λ a limit . We use J [Γ] α to denote J [Γ] α (∆(2)) and Z α to denote J [ Z ] α (∆(2)).The tower of Z α is of particular interest and will figure in an application to theclassification of scattered linear orders. We note that ∆(2) may be replaced by EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 4 any nontrivial Polish space to produce equivalent iterates for α ≥
1. Also, in thedefinition of J [Γ] α ( E ), the third rule may be used for successor ordinals as well.Returning to the proof of Theorem 1, we will actually consider a more compli-cated group than Γ ω . For a countable group Γ, we will introduce the notion of aΓ -tree , which is a tree in which the children of each node carry the structure ofa subset of the group Γ. We will see that Γ-trees are closely tied to iterates ofthe Γ-jump in a way analogous to the way regular trees are tied to iterates of theFriedman–Stanley jump. Namely, the iterated Γ-jump J [Γ] α will be Borel bireduciblewith the isomorphism relation on well-founded Γ-trees of rank 1 + α .In particular, we will introduce the infinite Γ -tree T Γ , which may be naturallyidentified with Γ <ω . Its automorphism group, Aut( T Γ ), is a closed subgroup of S ∞ ,and the group Γ ω is a closed subgroup of Aut( T Γ ). We will establish the following. Theorem 2.
Let Γ be a countable group. Then for any Borel equivalence relation E such that E is induced by a Borel action of a closed subgroup of Aut( T Γ ) , thereexists α such that E is Borel reducible to J [Γ] α . Since this result applies to the group Γ ω , when it is combined with the resultof Solecki showing such groups have non-Borel orbit equivalence relations, it issufficient to prove Theorem 1.The Bernoulli jumps are not always proper jump operators. In particular, wewill establish the following. Theorem 3.
Let Γ be a countable group with no infinite sequence of strictly de-scending subgroups. Then the mapping E E [Γ] is not a proper jump operator. To establish this result, we directly calculate that for such Γ,we have that J [Γ] ω +1 is Borel bireducible with J [Γ] ω .Theorems 1 and 3 leave open the question of precisely when the Γ-jump is proper.Indeed, we shall see that there exist groups Γ which do not meet the hypothesis ofeither result.We will also study the structure of specific Borel equivalence relations with re-spect to the Bernoulli jumps and see that they provide new examples of equivalencerelations whose complexity lies between E ∞ and F . We first establish: Theorem 4. E [Γ]0 is generically E ω ∞ -ergodic. This result has subsequently been strengthened by Allison and Panagiotopoulos[2] to show that E [ Z ]0 is generically ergodic with respect to any orbit equivalencerelation of a TSI Polish group. Using this, we establish: Theorem 5.
We have the following: ◦ E ω < B E [ Z ]0 < B E [ Z ] ∞ < B F . ◦ E ω < B E ω ∞ < B E [ Z ] ∞ < B F . ◦ E [ Z ]0 and E ω ∞ are ≤ B -incomparable. Shani [22] has produced further non-reducibility results about Bernoulli jumpsof countable Borel equivalence relations. For instance, he has shown that E [ Z ]0 < B E [ Z ]0 , and that E [ Z ]0 and E [ Z <ω ]0 are ≤ B -incomparable. He has also produced an-other example of an equivalence relation strictly between E ω ∞ and F which is ≤ B -incomparable with E [ Z ]0 and E [ Z ] ∞ . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 5
One application of the theory of Bernoulli jumps is to the classification of count-able scattered linear orders. Recall that a linear order is scattered if it has nosubordering isomorphic to ( Q , < ). The class of scattered linear orders carries aderivative operation where one identifies points x, y such that the interval [ x, y ] isfinite. This derivative operation may furthermore be used to define a rank functionon the countable scattered orders with values in ω , as we will discuss in Section 9.We will establish the following. Theorem 6.
The isomorphism equivalence relation on countable scattered linearorders of rank α is Borel bireducible with J [ Z ] α . This result, together with the fact that the Z -jump is proper, implies that thecomplexity of the classification of countable scattered linear orders increases strictlywith the rank.This paper is organized as follows. In the next section we establish some ofthe basic properties of the Γ-jump. In Section 3 we compare the Γ-jump with theFriedman–Stanley jump. In Section 4 we introduce Γ-trees and relate iterates ofthe Γ-jump to isomorphism of well-founded Γ-trees. In Section 5 we study actionsof Aut( T Γ ) and prove Theorem 2. In Section 6 we investigate the properness of theΓ-jump and in particular prove Theorem 1 and Theorem 3. In Section 7 we revisitthe proof from Section 5 and show that in some cases, we can achieve better lowerbounds on the complexity of the iterated jumps. In Section 8 we investigate newequivalence relations arising from Bernoulli jumps, and compare them against well-known equivalence relations, establishing Theorems 4 and 5. In Section 9 we discussthe connection between the Z -jump and scattered linear orders, and in particularprove Theorem 6. Acknowledgement.
We would like to thank Shaun Allison, Ali Enayat, Aris-toteles Panagiotopoulos, and Assaf Shani for helpful discussions about the contentof this article. § Properties of the Γ -jump for a countable group ΓIn this section we explore some of the basic properties of the Γ-jump. We beginby recording the following three properties, whose proofs are immediate.
Proposition 2.1.
For any countable group Γ and equivalence relations E and F we have: (a) If E is Borel (resp. analytic), then E [Γ] is Borel (resp. analytic). (b) E ≤ B E [Γ] . (c) If E ≤ B F then E [Γ] ≤ B F [Γ] . The next result strengthens Proposition 2.1(b).
Proposition 2.2. If E is a Borel equivalence relation and Γ is an infinite group,then E ω ≤ B E [Γ] .Proof. If E has just finitely many classes, the left-hand side is smooth. Otherwisefor any a ∈ X , we have that E is Borel bireducible with E ↾ X r [ a ]. Hence it issufficient to define a reduction from ( E ↾ X r [ a ]) ω to E [Γ] . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 6
Let S ⊂ Γ be an infinite subset such that γ = 1 implies γS = S . (The set ofsuch S is of measure 1.) Define a function f : ( X r [ a ]) S → X Γ by f ( x )( α ) = ( x ( α ) if α ∈ Sa otherwise.Clearly if x E S x ′ then f ( x ) E Γ f ( x ′ ) and therefore f ( x ) E [Γ] f ( x ′ ). For the reverseimplication suppose f ( x ) E [Γ] f ( x ′ ), and let γ ∈ Γ be such that γ · f ( x ) E Γ f ( x ′ ).Then f ( x ′ )( γα ) E f ( x )( α ), so f ( x )( α ) = a ⇐⇒ f ( x ′ )( γα ) = a. This means precisely that γS = S . It follows that γ = 1, and therefore that x E S x ′ . (cid:3) The Γ-jump may be thought of as a kind of wreath product. Recall that if Λ , Γare groups then the full support wreath product Λ ≀ Γ is defined as follows. Let Λ Γ be the full support group product of Γ many copies of Λ, and let Γ act on Λ Γ bythe shift. Then Λ ≀ Γ is the semidirect product Γ ⋊ Λ Γ with respect to this action. Proposition 2.3. If E Λ is the orbit equivalence relation of some (possibly uncount-able) group Λ acting on X , and Γ is any countable group, then ( E Λ ) [Γ] is the orbitequivalence relation of Λ ≀ Γ acting on X Γ . This result implies that the Γ-jump preserves many properties of orbit equiva-lence relations. For the next statement, recall that a cli group is one which carriesa complete, left-invariant metric.
Corollary 2.4. If E is the orbit equivalence relation of a Polish group then E [Γ] is the orbit equivalence relation of a Polish group. The same statement holds if wereplace “Polish group” with any of the following: solvable group, cli group, or closedsubgroup of S ∞ . This follows for cli groups from the preservation of being a cli group under wreathproducts (Theorem 2.2.11 of [12]). Note, however, that being induced by a TSIgroup is not preserved (here a
TSI group is one which carries a two-sided invariantmetric). Indeed, it is shown in [2] that E [ Z ]0 can not be the orbit equivalence relationof a TSI group.By contrast, the Louveau jump does not satisfy any of these preservation prop-erties. For example, the Louveau jump of ∆( R ) with respect to the Fr´echet filteris E , and it is well-known E is not reducible to a Polish group action [13]. TheFriedman–Stanley jump does satisfy the above preservation property with respectto Polish groups and subgroups of S ∞ . However it does not satisfy the abovepreservation property with respect to solvable groups, TSI groups, or cli groups.For example, the Friedman–Stanley jump of ∆( R ) is F and it is well-known F isnot induced by a cli group action [17]. Proposition 2.5.
Let E be an equivalence relation on X . If Λ is a subgroup orquotient of Γ , then E [Λ] ≤ B E [Γ] .Proof. We adapt the argument from the case when E is an equality relation (seeProposition 7.3.4 of [12]). If Λ is a quotient of Γ, let π : Γ → Λ be a surjective
EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 7 homomorphism and define f : X Λ → X Γ by f ( x ) = x ◦ π . Then whenever λ = π ( γ )we have λx E Λ x ′ ⇐⇒ ( ∀ α ∈ Λ) x ( λα ) E x ′ ( α ) ⇐⇒ ( ∀ β ∈ Γ) x ( π ( γβ )) E x ′ ( π ( β )) ⇐⇒ ( ∀ β ∈ Γ) f ( x ( γβ )) E f ( x ′ )( β ) ⇐⇒ γ · f ( x ) E Γ f ( x ′ )It follows that f is a reduction from E [Λ] to E [Γ] .Next assume that Λ ≤ Γ. Fix an element a ∈ X , and let f : X Λ → X Γ be definedby f ( x )( α ) = ( x ( α ) α ∈ Λ a otherwise . Clearly if λ · x E Λ x ′ then λ · f ( x ) E Γ f ( x ′ ) as well, which means x E [Λ] x ′ implies f ( x ) E [Γ] f ( x ′ ). For the reverse implication, suppose f ( x ) E [Γ] f ( x ′ ), and let γ ∈ Γbe such that γ · f ( x ) E Γ f ( x ′ ). If γ = λ ∈ Λ, it is clear that λ · x E Λ f ( x ′ ) and so x E [Λ] x ′ . On the other hand if γ / ∈ Λ, then for all λ ∈ Λ, we have x ′ ( λ ) = f ( x ′ )( λ ) E γ · f ( x )( λ ) = f ( x )( γ − λ ) = a. An identical calculation with x, x ′ exchanged and γ, γ − exchanged shows the samefor x . Thus in this case x E λ x ′ and hence we have x E [Λ] x ′ , as desired. (cid:3) Next we can relate jumps for finite powers of a group Γ to iterates of the jumpfor Γ.
Lemma 2.6.
For any countable group Γ , we have E [Γ k ] ≤ B J [Γ] k (cid:16) E Γ k (cid:17) . In partic-ular, E [Γ k ] ≤ B J [Γ] k +1 ( E ) .Proof. Beginning with the first statement, we show the case of k = 2, with larger k being similar. Given x ∈ X Γ × Γ let f x be the function from Γ × Γ to X Γ × Γ sothat f x ( α, β ) is a code for all of the x ( γ, δ ), viewed from the base point ( α, β ),i.e., f x ( α, β )( γ, δ ) = x ( α − γ, β − δ ).We now define a reduction ϕ from E [Γ ] to(( E Γ ) [Γ] ) [Γ] by setting ϕ ( x )( α )( β ) = f x ( α, β )If x is equivalent to y , witnessed by ( α, β ), then ϕ ( x ) is equivalent to ϕ ( y ) withthe witness α for the outer jump and β for each coordinate of the inner jump.Conversely if ϕ ( x ) is equivalent to ϕ ( y ), then there exists ( α, β ) such that f x (1 , f ( y )( α, β ). It follows that ( α, β ) witnesses that x is equivalent to y . The second statement follows from Proposition 2.2. (cid:3) We can also absorb countable powers in certain Γ-jumps.
Proposition 2.7. If E is a Borel equivalence relation and Γ and ∆ are infinitegroups, then ( E ω ) [Γ] ≤ B E [Γ × ∆] .Proof. Arguing as in Proposition 2.2, we may assume E has infinitely many classes,and find a reduction from (( E ↾ X r [ a ]) ω ) [Γ] to E [Γ × ∆] for some a ∈ X . Let∆ = { δ n : n ∈ ω } . For x ∈ (( X r [ a ]) ω ) Γ , define f ( x ) by f ( x )(( α, δ n )) = ( x ( α )( n −
1) if n > a if n = 0 . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 8 If x ( E ω ) [Γ] x ′ then f ( x ) E [Γ × ∆] f ( x ′ ). Conversely, suppose f ( x ) E [Γ × ∆] f ( x ′ ) andlet ( γ, δ ) be such that ( γ, δ ) · f ( x ) E Γ × ∆ f ( x ′ ). Then we must have δ = 1 ∆ , so that γ · x ( E ω ) Γ x ′ and hence x ( E ω ) [Γ] x ′ . (cid:3) In particular, if Γ has a subgroup isomorphic to Γ × ∆ for some infinite group∆, then ( E ω ) [Γ] ≤ B E [Γ] .Another property of equivalence relations is that of being pinned , which is definedusing forcing. We briefly recall the definition; we refer the reader to [17] or [26] forproperties of pinned equivalence relations. Definition 2.8.
Let E be an analytic equivalence relation on X . A virtual E -class is a pair h P , τ i where P is a poset and τ is a P -name so that (cid:13) P × P τ ℓ E τ r , where τ ℓ and τ r are the interpretations of τ using the left and right generics, respectively.A virtual E -class h P , τ i is pinned if there is some x ∈ X from the ground model sothat (cid:13) τ E ˇ x . We say that E is pinned if every virtual E -class is pinned. Theorem 2.9. If E is pinned then E [Γ] is pinned.Proof. Suppose E is pinned and let h P , τ i be a virtual E [Γ] -class. Then (cid:13) P × P τ ℓ E [Γ] τ r . Hence (cid:13) ( ∃ γ )( ∀ λ ) τ ℓ ( λ ) E τ r ( γλ ). We can therefore find a condition ( p, q ) anda γ ∈ Γ such that ( p, q ) (cid:13) ∀ λ ) τ ℓ ( λ ) E τ r (ˇ γλ )Then temporarily considering the forcing P and the three factor terms τ , τ , τ ,we have: ( p, q, p ) (cid:13) ( ∀ λ ) τ ( λ ) E τ (ˇ γλ ) E τ ( λ )Since E is transitive this implies that( p, p ) (cid:13) ( ∀ λ ) τ ℓ ( λ ) E τ r ( λ )Now since E is pinned we can find x ∈ X Γ such that for all λ we have p (cid:13) τ ( λ ) E ˇ x ( λ ). It follows that p (cid:13) τ E [Γ] ˇ x . Finally we claim this is in fact unconditionallyforced. Indeed if q is any other condition then ( p, q ) (cid:13) τ ℓ E [Γ] ˇ x ∧ τ ℓ E [Γ] τ r . Againusing the transitivity of E , we conclude that q (cid:13) τ E [Γ] ˇ x too. (cid:3) By contrast, the Friedman–Stanley jump does not preserve pinned-ness, as therelation ∆( R ) + ∼ B F is not pinned (see 17.1.3 of [17]). Kanovei also shows in [17]that pinned-ness is preserved under Fubini products, so that the Louveau jumpwith the respect to the Fr´echet filter does preserve pinned-ness.We feel that the remarks following Corollary 2.4 and Theorem 2.9 justify thefollowing: Proclamation 2.10.
The Γ -jump is a kindler, gentler jump operator than theLouveau jump or the Friedman–Stanley jump. We may view the Friedman–Stanley jump and the Γ-jumps as special cases of amore general construction. Let E be an equivalence relation on X , and let ( G, A )be a permutation group of a countable set A . We define the jump E [ G,A ] on X N by x E [ G,A ] x ′ ⇐⇒ ( ∃ g ∈ G )( ∀ a ∈ A ) x ( g ( a )) E x ′ ( a )The permutation group S ∞ = (Aut( N ) , N ) corresponds to the Friedman–Stanleyjump. In general the ( G, A )-jump of a Borel equivalence relation need not be Borel;for example if E has at least two equivalence classes, then the (Aut( Q ) , Q ))-jumpof E is Borel complete. One may ask which (uncountable) group actions on N EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 9 give rise to various types of equivalence relations, e.g., Borel complete relations,Borel equivalence relations, or the Friedman–Stanley jump. In [2], the authorsinvestigate this generalization, studying a P -jump operator for a Polish group P ofpermutations of N .We may also ask how Γ-jumps for different groups Γ compare to one another. Question 1.
Given a fixed E , how many distinct complexities can arise as E [Γ] ?For countable groups Γ and Γ , when is E [Γ ] ≤ B E [Γ ] ?Here, Shani [22] has characterized strong ergodicity between Γ-jumps of count-able Borel equivalence relations in terms of group-theoretic properties, showing inparticular: Theorem 2.11 (Corollary 1.4 of [22]) . Let E be a generically ergodic countableBorel equivalence relation. Then: ◦ E [ Z ] < B E [ Z ] < B E [ Z ] < B · · · < B E [ Z <ω ] < B E [ F ] . ◦ E [ Z ] and E [ Z <ω ] are ≤ B -incomparable. This contrasts sharply with the case of group actions, where actions of any twoinfinite countable abelian groups produce hyperfinite equivalence relations. It alsoshows that there is no “least” Γ-jump, although the F -jump is the most compli-cated. We do not know whether incomparability can be extended to iterated jumps,such as through some notion of rigidity. Note, for instance, that by Lemma 2.6 wehave E [ Z ] ≤ B J [ Z ]3 ( E ). Question 2.
Are there countable groups Γ and Γ so that J [Γ ] α ( E ) and J [Γ ] β ( E )are ≤ B -incomparable for all α, β < ω ?We also introduce two restrictions of Bernoulli jumps. Definition 2.12.
Let E be an equivalence relation on X and Γ a countable group.The free part of X Γ and the pairwise-inequivalent part of X Γ are given by: X Γfree = { ¯ x ∈ X Γ : ∀ γ ( γ = 1 Γ → γ · ¯ x E Γ ¯ x ) } X Γp.i. = { ¯ x ∈ X Γ : ∀ γ, δ ( γ = δ → ¯ x ( γ ) E ¯ x ( δ )) } . We let E [Γ]free = E Γ ↾ X Γfree and E [Γ]p.i. = E Γ ↾ X Γp.i. .We immediately have E [Γ]p.i. ≤ B E [Γ]free ≤ B E [Γ] . We will see below that in certaincases all three are bireducible, but this is not true in general. Indeed, this failsalready for E = ∆(2), as ∆(2) [ F ] ∼ B E ∞ whereas ∆(2) [ F ]free ∼ B E ∞ T . Question 3.
For which E and Γ do we have E [Γ] ≤ B E [Γ]free or E [Γ]free ≤ B E [Γ]p.i. ? § Comparing Γ -jumps to Friedman–Stanley jumps We begin by recalling that the Friedman–Stanley tower F α for α < ω is definedanalogously to the iterated Γ-jump. EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 10
Definition 3.1.
The equivalence relation F α for α < ω is defined recursively by F = ∆( ω ) F α +1 = F + α F λ = M α<λ F α ! + for λ a limit . Equivalently, F α is the isomorphism relation on countable well-founded trees on ω of height at most 1 + α .We compare the Z -jump to the Friedman–Stanley jump. Definition 3.2.
We say that E is weakly absorbing if E has perfectly many classesand ( E <ω ) + ≤ B E + .Note that if E has perfectly many classes and E × E ≤ B E , then E is weaklyabsorbing, so in particular this holds for E , E ∞ , and F α for all α ≥ Lemma 3.3.
For any E with at least two classes, E [ Z ] is weakly absorbing.Proof. When E has at least two classes, E [ Z ] is above E and hence has perfectlymany classes. We define a reduction from (cid:16)(cid:0) E [ Z ] (cid:1) <ω (cid:17) + to (cid:0) E [ Z ] (cid:1) + as follows. Fix E -inequivalent points z and z . Given { ( x m, , . . . , x m,k m ) : m ∈ ω } , let it map tothe set { y mi ,...,i km : m ∈ ω, i , . . . , i k m ∈ Z } , where y mi ,...,i km = . . . ,z , z , x m, , x m, , x m, i , x m, i , . . . , x m,k m i km , x m,k m i km , z , z ,x m, , x m, , x m, i , x m, i , . . . , x m,k m i km , x m,k m i km , z , z , . . . , i.e., y mi ,...,i km consists of a Z -sequence of blocks, indexed by k ∈ Z , of the form x m, k , x m, k , x m, k + i , x m, i , . . . , x m,k m k + i km , x m,k m k + i km , separated by the pair z , z . If( x m, , . . . , x m,k m ) (cid:16) E [ Z ] (cid:17) <ω (˜ x ˜ m, , . . . , ˜ x ˜ m, ˜ k ˜ m )then k m = ˜ k ˜ m and there are j , . . . , j k m so that x m, j E ˜ x ˜ m, j + j ∧ x m, j E ˜ x ˜ m, j + j ∧ . . . ∧ x m,k m j E ˜ x ˜ m,k m j + j km for all j , and hence y mi ,...,i km E [ Z ] ˜ y ˜ mi + j − j ,...,i km + j km − j . Thus the function definedis a homomorphism. Conversely, because of the repetition in the blocks, we canrecover the (cid:0) E [ Z ] (cid:1) <ω -class of ( x m, , . . . , x m,k m ) from any y mi ,...,i km , so that it is alsoa cohomomorphism and hence a reduction. (cid:3) Lemma 3.4. If E has perfectly many classes then E [ Z ] free ≤ B ( E <ω ) [ Z ] p.i. .Proof. Given a non-periodic Z -sequence x of E -representatives, we construct a newsequence y is follows. For each n ∈ Z we first define a real p n by p n ( k, l ) = 1 iff x n + k E x n + l . We then let d n be the least d > x n E x n + d and p n = p n + d ,if such exists, and d n = 0 otherwise. Finally we let y n = ( p n , x n , . . . , x n + d n ). Thuseach coordinate y n has several pieces, and we say that two such coordinates y n and y n ′ are equivalent if they are equivalent with respect to ∆( R ) × E <ω .We claim that the entries y n are pairwise inequivalent. If this is not the case,we can find n < n such that y n is equivalent to y n . Let d = d n = d n . Note EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 11 that d >
0. Then the blocks x n , . . . , x n + d and x n , . . . , x n + d are pointwise E -equivalent. Next, since we have x n + i E x n + i for i ≤ d , and since p n + d = p n , wecan conclude x n + d + i E x n + d + i . Using this reasoning inductively, we can concludethat x n + kd + i E x n + kd + i for all k >
0. The fact that p n + d = p n can be usedright-to-left to obtain the same for k negative as well. In other words, x is periodicwith period n − n . This contradicts our assumption, and completes the claim.Thus the map x y is a reduction from the free part of E [ Z ] to the pairwiseinequivalent part of (∆( R ) × E <ω ) [ Z ] . Since E has perfectly many classes we have∆( R ) × E <ω ≤ B E <ω , so E [ Z ]free ≤ B ( E <ω ) [ Z ]p.i. . (cid:3) Theorem 3.5. If E is weakly absorbing then E [ Z ] ≤ B E + .Proof. Since E has perfectly many classes, we may fix a, b ∈ X with a E b sothat E ≤ B E ↾ X r [ { a, b } ]. From the previous lemma we then have that thereis a reduction g from E [ Z ]free to (( E ↾ X r [ { a, b } ]) <ω ) [ Z ]p.i. . Let P = S k ≥ P k bethe set of periodic elements, where P k consists of x ∈ X Z such that for all n wehave x n E x n + k . Let k ( x ) be the least k ≥ x ∈ P k if x ∈ P , and k ( x ) = 0 if x / ∈ P . We now define a reduction f from E [ Z ] to ( E <ω ) + as follows;since E is weakly absorbing this will be sufficient. If k ( x ) >
0, we let f map x ∈ P k ( x ) to { ( a, x i , x i +1 , . . . , x i + k ( x ) − ) : 0 ≤ i < k ( x ) } . If k ( x ) = 0, let f map x to {h g ( x ) n , b, g ( x ) n +1 i : n ∈ Z } , where h· , b, ·i is concatenation and provides areduction from (( E ↾ X r [ { a, b } ]) <ω ) to E <ω .Suppose first that x E [ Z ] x ′ . If x ∈ P then x ′ ∈ P and k ( x ) = k ( x ′ ), so that { [( a, x i , x i +1 , . . . , x i + k ( x ) − )] E <ω : 0 ≤ i < k ( x ) } = { [( a, x i , x i +1 , . . . , x i + k ( x ′ ) − )] E <ω : 0 ≤ i < k ( x ′ ) } . If x, x ′ / ∈ P , then there is m ∈ Z with g ( x ) n E <ω g ( x ′ ) m + n for all n . Then { [ h g ( x ) n , b, g ( x ) n +1 i ] E <ω : n ∈ Z } = { [ h g ( x ′ ) m + n , b, g ( x ′ ) m + n +1 i ] E <ω : n ∈ Z } = { [ h g ( x ′ ) n , b, g ( x ′ ) n +1 i ] E <ω : n ∈ Z } , i.e., f ( x ) ( E <ω ) + f ( x ′ ).Suppose conversely that f ( x ) ( E <ω ) + f ( x ′ ). We must have either both x, x ′ ∈ P or both x, x ′ / ∈ P . If x, x ′ ∈ P then we must have k ( x ) = k ( x ′ ), and thereare i and i ′ so that x i + j x ′ i ′ + j for 0 ≤ j < k ( x ), and hence x i + j x ′ i ′ + j for all j , so x E [ Z ] x ′ . Suppose then x, x ′ / ∈ P . Then { [ h g ( x ) n , b, g ( x ) n +1 i ] E <ω : n ∈ Z } = { [ h g ( x ′ ) n , b, g ( x ′ ) n +1 i ] E <ω : n ∈ Z } , so for each n there is an m ( n ) with g ( x ) n E <ω g ( x ′ ) m ( n ) and g ( x ) n +1 E <ω g ( x ′ ) m ( n )+1 . Because the g ( x ′ ) m ’s arepairwise inequivalent, we conclude that m ( n + 1) = m ( n ) + 1 for all n , so there is m with m ( n ) = n + m for all n . Thus g ( x ) ( E <ω ) [ Z ] g ( x ′ ) so x E [ Z ] x ′ . (cid:3) It turns out that E [Γ] is not reducible to E + in general. In particular the followingis shown in [3]: Theorem 3.6 (Shani) . ( E ω ) [ Z ] is not potentially Π ∼ . Hence, neither ( E ω ) [ Z ] nor ( E ) [ Z ] is reducible to F . Noting that ( E ) + ∼ B F , we thus have that ( E ) [ Z ] B ( E ) + . Question 4.
For which E and Γ is E [Γ] ≤ B E + ? EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 12
A slight modification of the argument in 3.5 can be used to show:
Lemma 3.7. ( E ) [ Z ] ≤ B ( E ) [ Z ] p.i. , i.e, Z is Borel reducible to the pairwise inequiv-alent part of Z .Proof. Note that E <ω is hyperfinite, as is any equivalence relation of finite in-dex over E <ω , and hence reducible to E . Thus using Lemma 3.4 we may fixpairwise E -inequivalent { a n : n ∈ Z } and find a reduction f from ( E ) [ Z ]free to( E ↾ X r A ) [ Z ]p.i. , where A = S n ∈ Z [ a n ]. We extend f to the periodic part of ( E ) [ Z ] as follows. With the notation from above, for x ∈ P let g ( x ) = { ( x i , x i +1 , . . . , x i + k ( x ) − ) : 0 ≤ i < k ( x ) } . The relation that such finite sets consist of the same E <ω -classes is then hyperfinite,so there is a reduction h of ( E ) [ Z ] ↾ P to E ↾ X r A . Then for x ∈ P let f ( x )(0) = h ( x ) and f ( x )( n ) = a n for n = 0. (cid:3) The same technique can be applied to the Z -jump of some other equivalencerelations, such as E ∞ and F , but we do not know if this is true for other E .We can now compare the hierarchy Z α with the hierarchy F α . Theorem 3.8.
For all α ≥ , Z α is Borel reducible to F α .Proof. By induction on α . The case of α = 2 follows from Theorem 3.5 since Z ∼ B E [ Z ]0 and F ∼ B E +0 , and each iteration of the Z -jump preserves the propertyof being weakly absorbing so we may again apply Theorem 3.5. (cid:3) As noted earlier, Γ-jumps are gentler than the Friedman–Stanley jump so thereverse of Theorem 3.8 is false.
Proposition 3.9. F is not Borel reducible to J [Γ] α for any α < ω .Proof. J [Γ] α is pinned because the identity relation is pinned, and Γ-jumps andcountable products of pinned relations are pinned. On the other hand, F is notpinned and being pinned is preserved downward under ≤ B . (cid:3) As E +0 ∼ B E + ∞ ∼ B F , we get: Corollary 3.10. E [ Z ]0 < B F and E [ Z ] ∞ < B F . Although none of the Z α ’s are above F , we will see below that they are un-bounded in Borel complexity.The result of Allison–Shani noted above shows that Theorem 3.8 does not holdfor all countable groups Γ, as we do not always have J [Γ]2 ≤ F , e.g. for Γ = Z (since J [ Z ]1 ∼ B E ). Potential complexity bounds do allow us to give a weakercomparison in general. Lemma 3.11. F [Γ] α < B F α +1 .Proof. Theorem 2 of [16] shows that for a Borel equivalence relation E inducedby an action of a closed subgroup of S ∞ we have E ≤ F α iff E ∈ pot(Π ∼ α +1 )for α not a limit, and E ≤ F α iff E ∈ pot(Π ∼ α ) for α a limit. Then F [Γ] α is stillinduced by an action of a closed subgroup of S ∞ and is in pot(Σ ∼ α +2 ) for α not alimit or in pot(Σ ∼ α +1 ) for α a limit. From Theorem 4.1 of [16] we then have that EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 13 F [Γ] α ∈ pot( D (Π ∼ α +1 )) for α not a limit, or in pot(Π ∼ α )) for α a limit, and henceusing Corollary 6.4 of [16] and its notation, we have F [Γ] α ≤ B ∼ = ∗ α +1 < B F α +1 . (cid:3) From this we conclude:
Proposition 3.12.
For any countable group Γ and α < ω we have J [Γ] α ≤ B F α .Proof. By induction on α , noting that J [Γ]1 ≤ B F for all Γ. (cid:3) §
4. Γ -trees
The Friedman–Stanley jump naturally corresponds to the group S ∞ and itsiterates correspond to isomorphism of well-founded trees. Namely, the equivalencerelation F α is bireducible with the isomorphism relation on countable well-foundedtrees of rank at most 2+ α . We will see that Γ-jumps naturally correspond to certaingroup actions, and the iterates J [Γ] α correspond to isomorphism of well-founded Γ-trees, which are trees where the children of each node carry the structure of a subsetof Γ. We make this precise as follows. Definition 4.1.
Let Γ be a countable group. The language L Γ consists of thebinary relation ≺ together with binary relations { R γ : γ ∈ Γ } . A Γ -tree is an L Γ -structure which is a rooted tree with child relation ≺ and satisfies the additional( L Γ ) ω ω -formulas: ◦ R γ ( u, v ) → ∃ t ( u ≺ t ∧ v ≺ t ), for each γ ∈ Γ with γ = 1 Γ ◦ R Γ ( u, v ) ↔ u = v ◦ ∃ t ( u ≺ t ∧ v ≺ t ) → W γ ∈ Γ R γ ( u, v ) ◦ ¬ ( R γ ( u, v ) ∧ R δ ( u, v )), for each γ = δ ∈ Γ ◦ ( R γ ( u, v ) ∧ R δ ( v, w )) → R γδ ( u, w ), for each γ, δ ∈ ΓWe say that a Γ-tree is well-founded if it is well-founded as a tree, and we definerank in the usual way. Next we introduce the infinite Γ-tree, T Γ . The automorphismgroup of T Γ will figure crucially in the next section. Definition 4.2.
Let Γ be a countable group. The infinite Γ -tree , denoted T Γ , isthe non-empty Γ-tree which additionally satisfies for each γ ∈ Γ: ∀ t ∃ u ∃ v ( u ≺ t ∧ v ≺ t ∧ R γ ( u, v )) . This defines the infinite Γ-tree uniquely up to isomorphism. For one model of T Γ , we take the universe to be Γ <ω and interpret each R γ by R γ ( s a α, s a β ) ⇐⇒ αγ = β . Definition 4.3.
We let ∼ = Γ denote the isomorphism relation on countable Γ-trees.For α < ω we let ∼ = Γ α denote the isomorphism relation on countable well-foundedΓ-trees of rank at most 1 + α .Since every Γ-tree is isomorphic to a substructure of the infinite Γ-tree T Γ , wehave that ∼ = Γ is induced by an action of its automorphism group, Aut( T Γ ). Thisgroup is isomorphic to the infinite wreath power Γ ≀ ω , a cli group, and hence ∼ = Γ and any other Aut( T Γ )-action is pinned.We now can relate iterated Γ-jumps to isomorphism of well-founded Γ-trees.Note that nodes of rank 1 in a Γ-tree carry more structure than in a regular tree,where there are only countably many isomorphism types; hence the indexing differsby 1 from the case of the F α and tree isomorphism. EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 14
Proposition 4.4.
For each α < ω , J [Γ] α is Borel bireducible with ∼ = Γ α .Proof. For α = 0, both J [Γ]0 and ∼ = Γ0 are ∆(2). For α = 1, J [Γ]1 is the shift of Γon 2 Γ , E (Γ , x ∈ Γ we naturally associate a Γ-tree T ( x ) ofrank 2 consisting of a root v with children { u α : x ( α ) = 1 } , where we interpret R γ ( u α , u β ) ⇐⇒ αγ = β . If x E (Γ , x ′ , with γ · x = x ′ , then the map f given by f ( v ) = v ′ and f ( u α ) = u ′ ( γ α ) is an isomorphism from T ( x ) to T ( x ′ ).Conversely, let f be an isomorphism from T ( x ) to T ( x ′ ), so f ( v ) = v ′ . Fix u = u α ∈ T ( x ) and suppose f ( u ) = u ′ α ′ = u ′ γ α ∈ T ( x ′ ) for some γ . Then for all β with x ( β ) = 1 we have R α − β ( u α , u β ), so R α − β ( u ′ γ α , f ( u β )); hence f ( u β ) = u ′ γ β . Thus x ( β ) = 1 ⇐⇒ x ′ ( γ β ) = 1, so γ · x = x ′ . Hence x T ( x ) witnesses J [Γ]1 ≤ B ∼ = Γ1 .For the reverse reduction, given a rank 2 Γ-tree T , choose any non-root node v ∈ T and define x ( T ) ∈ Γ by x ( T )( γ ) = 1 ⇐⇒ ∃ v ∈ T R γ ( v , v ). Then T ( x ( T )) ∼ = T ,so the map T T ( x ) witnesses ∼ = Γ1 ≤ B J [Γ]1 .Induction steps are similar. Assuming J [Γ] α ∼ B ∼ = Γ α , given a reduction f : J [Γ] α ≤ B ∼ = Γ α we can send x ∈ X Γ to the tree T ( x ) with children { u γ : γ ∈ Γ } of the root sothat the subtree T ( x ) u γ ∼ = f ( x ( γ )) for each γ to show J [Γ] α +1 ≤ B ∼ = Γ α +1 . The reversereduction is handled in an analogous manner, as are limit stages. (cid:3) Although we will see that in many instances the Γ-jump is proper on the Borelequivalence relations, we will always have that ∼ = Γ is a fixed point of the Γ-jump.This is analogous to the case of tree isomorphism, which is a fixed point of theFriedman–Stanley jump. Proposition 4.5.
For a countable group Γ , ∼ = Γ is a fixed point of the Γ -jump, i.e., ( ∼ = Γ ) [Γ] ∼ B ∼ = Γ .Proof. Given x ∈ X Γ with each x (Γ) coding a Γ-tree T γ , we map x to the Γ-tree T ( x ) with children { u γ : γ ∈ Γ } of the root so that the subtree T ( x ) u γ ∼ = T γ toshow ( ∼ = Γ ) [Γ] ≤ B ∼ = Γ . (cid:3) We also note that none of the ∼ = Γ are of maximal complexity among S ∞ -actions,since they are all pinned. Proposition 4.6.
For every countable group Γ , F B ∼ = Γ , so ∼ = Γ is not Borelcomplete. We will see in the next section that ∼ = Γ is of maximal complexity among Aut( T Γ )-actions. § Reducing actions of
Aut( T Γ ) to iterated Γ -jumps In this section we will establish that if Γ is a countable group, then every Borelequivalence relation induced by a Borel action of a closed subgroup of the automor-phism group of the infinite Γ-tree, Aut( T Γ ), is Borel reducible to some iterate J [Γ] α of the Γ-jump.In the next section we will use this, together with the fact that for certaingroups Γ the power Γ ω has actions of cofinal essential complexity, to show thatfor such groups the Γ-jump is a proper jump operator. As a preview, note that byTheorem 2.3.5 of [5], if H is a closed subgroup of Aut( T Γ ) then every Polish H -spaceis reducible to a Polish Aut( T Γ )-space; similarly, we may reduce a Borel H -spaceto a Polish H -space. Since Γ ω is isomorphic to a closed subgroup of Aut( T Γ ), if we EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 15 know Γ ω has actions of cofinal essential complexity, we will be able to conclude theiterates J [Γ] α are properly increasing in complexity.Recall from the previous section the infinite Γ-tree T Γ may be identified with Γ <ω .We let T Γ ↾ k be the restriction of T Γ to branches of length k . Then Aut( T Γ ↾ k )is the wreath product of k -many copies of Γ, and Aut( T Γ ) is isomorphic the directlimit of the finite wreath powers of Γ.We are now ready to state the main theorem of this section. Theorem 5.1.
Let Γ be a countable group and let E be a Borel equivalence relationinduced by a continuous action of a closed subgroup of Aut( T Γ ) . Let α < ω be anordinal such that E is Π ∼ α . (a) If α = n < ω , n ≥ , then E ≤ B J [Γ] ω · ( n − . (b) If α = λ + 1 , λ a limit, then E ≤ B J [Γ] ω · λ +1 . (c) If α = λ + n , λ a limit, n ≥ , then E ≤ B J [Γ] ω · ( λ + n − . The proof of this theorem is based on some concepts and techniques from [16],and we begin by recalling the relevant portions of this work, adapted slightly toour setting. Fix a countable group Γ and a closed subgroup H of Aut( T Γ ). Let E = E XH be a Π ∼ α orbit equivalence relation given by a continuous action of H ona Polish space X . Fix an open basis { W n } for X .The next definition concerns codes for Π ∼ α subsets of X × X . Definition 5.2.
A Π ∼ α -code is a pair ( T, u ) where T is a well-founded tree of rank α so that if t ∈ T is not terminal then t a n ∈ T for all n , and u : { t ∈ T : t is terminal } → ω × ω . We write u = ( u , u ). For t ∈ T , define the Borel setcoded below t , R t , by R t = ( W u ( t ) × W u ( t ) if t is terminal, T n X × X r R t a n otherwise.The Π ∼ α set coded by ( T, u ) is R ∅ .Let ( T, u ) be a Π ∼ α -code for E . For t ∈ T , let | t | be the rank of t in T . For t ∈ T Γ ,let ht( t ) be its height in T Γ , i.e., its length as a finite sequence.We now define a convenient basis for H . Definition 5.3.
Let s be an enumeration of a finite partial function from T Γ to T Γ given by s = h ( d si , e si ) : i < k i . We define U s = { g ∈ H : ∀ i < k g ( d si ) = e si } .The collection of all such U s is a countable basis for H . Note that U s maybe empty, and the same partial function will have multiple enumerations. Let B consist of all such s , and B k consist of just those s ∈ B with maximum height ofelements of the domain at most k . Definition 5.4.
Given s ∈ B we define the type of s to be the sequence h ht( d si ) : i < k i of heights of elements of the domain of s . Let B denote the set of types.For C ∈ B , let k C be the size of the domain of all s of type C , d Ci = ht( d si ) for i < k C , and j C = P i We next introduce an action of Aut( T Γ ) on this basis. Definition 5.5. Let a ( g, s ) = s g be the action of Aut( T Γ ) on B given by s g = h ( g − ( d si ) , e si ) : i < k i , i.e., the corresponding partial function satisfies s g = s ◦ g ↾ g − [dom( s )]. Then U s · g = U s g . We use the same notation for the action ofAut( T Γ ↾ k ) on B k . For s, t ∈ B we write s ∼ t if s and t are in the same Aut( T Γ )-orbit (equivalently, the same Aut( T Γ ↾ k )-orbit for s, t ∈ B k ). If s ∼ t then s and t have the same type. Note, though, that Aut( T Γ ) acts on the enumerations, not thefunctions themselves, so two enumerations of the same partial function need not be ∼ -equivalent.For R ⊆ X × X and U, V open subsets of H , we define the double Vaughttransform R ∗ U, ∗ V = { ( x, y ) : ∀ ∗ g ∈ U ∀ ∗ h ∈ V ( g · x, h · y ) ∈ R } . The following encodings are adapted from [16], and are used to code the sections R ∗ U s , ∗ Vt ( x ) = { y : ( x, y ) ∈ R ∗ U s , ∗ Vt } . Definition 5.6. For x ∈ X , t ∈ T with | t | ≥ 1, and s ∈ B , define N st ( x ) by N st ( x ) = ( { t a n : x ∈ U − s · W u ( t a n ) } if | t | = 1, { N rt a n ( x ) : s ⊑ r ∈ B ∧ n ∈ ω } if | t | > | t | = 1 this is essentially a real, and in general for | t | = α it is a hereditarilycountable set of rank α . Note that t can be recovered from N st ( x ). As in [16] wehave the following translation property: Lemma 5.7. For g ∈ G , N st ( g · x ) = N s g t ( x ) . Definition 5.8. Let τ be the topology of X . For any x ∈ X and β ≤ | T | definethe topology τ xβ as the one generated by τ and the sets R ∗ U, ∗ Vt ( x ) for 1 ≤ | t | ≤ β and U, V ∈ { U s : s ∈ B } .Then each τ xβ is a Polish topology extending τ . For a set B ⊆ X , let B ∆ = { x : ∃ ∗ g ( g · x ∈ B ) } . We summarize the key properties of these topologies from Sections2 and 3 of [16]. Proposition 5.9 (Hjorth–Kechris–Louveau) . Let B be an open basis for a topologyon X . With the definitions above, we have: ◦ τ xβ = τ g · xβ for g ∈ H . ◦ The action of H on X is continuous for τ xβ . ◦ If x E y then ∀ B ∈ B ( x ∈ B ∆ ⇐⇒ y ∈ B ∆ ) . The latter condition impies [ x ] E = [ y ] E . ◦ If [ x ] E and [ y ] E are G δ then x E y iff ∀ B ∈ B ( x ∈ B ∆ ⇐⇒ y ∈ B ∆ ) iff [ x ] E = [ y ] E . ◦ If E is Π ∼ n for ≤ n < ω , then [ x ] E is G δ in τ xn − , so x E y iff τ xn − = τ yn − ∧ ∀ B ∈ B ( τ xn − )( x ∈ B ∆ ⇐⇒ y ∈ B ∆ ) . ◦ The same holds with τ x<λ in place of τ xn − when E is Π ∼ λ +1 for λ a limit,and with τ xλ + n − in place of τ xn − when E is Π ∼ λ + n for λ a limit and n ≥ . In case E is Π ∼ we have that E is reducible to ∆( R ), so we begin with the casewhere E is Π ∼ n for 3 ≤ n < ω . We show that every Π ∼ n Polish H -space is reducible to EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 17 some iterate of the Γ-jump of the identity relation, J [Γ] α . We note the modificationsfor cases for other α later. Definition 5.10. Define the following hereditarily countable sets: A ( x ) = { N st ( x ) : | t | ≤ n − ∧ s ∈ B } B ( x ) = h m, h r , N s t ( x ) i , . . . , h r k − , N s k − t k − ( x ) ii : | t i | ≤ n − ∧ r i , s i ∈ B ∧ x ∈ " W m ∩ \ i For t ∈ T with | t | ≥ , and s ∈ B k of type C , there is a Borelfunction c st ( x ) and an equivalence relation F Ct , reducible to J [Γ] ω · ( | t |− , so that for g ∈ Aut( T Γ ) and r and s of type C we have: (a) If y = g · x then c s g t ( x ) F Ct c st ( y ) . (b) If c rt ( x ) F Ct c st ( y ) then N rt ( x ) = N st ( y ) .Proof. (a) We use induction on | t | . For | t | = 1, let c st ( x ) = N st ( x ) ∈ R and F Ct = ∆( R ). If y = g · x then c s g t ( x ) = N s g t ( x ) = N st ( g · x ) = N st ( y ) = c st ( y ), and if c rt ( x ) = c st ( y ) then N rt ( x ) = N st ( y ).For | t | > 1, define F Ct by F Ct = Y n ∈ ω Y D ∈B : C ⊑ D (cid:0) F Dt a n (cid:1) [ Γ jD − jC ) ] . Next given C ⊑ D and a sequence v = α k C a β k C a · · · a α k D − a β k D − ∈ Γ j D − j C ) with α i , β i ∈ Γ d Di for j C ≤ i < j D , we let c st ( x )( n )( D )( v ) = c s a vt a n ( x ) , where s a v = s a ( α k C , β k C ) a · · · a ( α k D − , β k D − ). EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 18 If y = g · x , fix n ∈ ω and D ∈ B with C ⊑ D . Then for each v ∈ Γ j D − j C ) wehave c st ( y )( n )( D )( v ) = c s a vt a n ( g · x ) F Dt a n c ( s a v ) g t a n ( x )= c s g a v ′ t a n ( x )= c s g t ( x )( n )( D )( v ′ ) , where v ′ = g − ( α k C ) a β k C a · · · a g − ( α k D − ) a β k D − . Letting ˜ g ∈ Γ j D − j C ) be given by ˜ g = g ↾ d Dk C a id ↾ d Dk C a · · · a g ↾ d Dk D − a id ↾ d Dk D − (where eachconcatenated factor acts on the corresponding coordinates) we have that v ′ = ˜ g − · v for all v ∈ Γ j D − j C ) . Thus ˜ g witnesses c s g t ( x )( n )( D ) (cid:0) F Dt a n (cid:1) [ Γ jD − jC ) ] c st ( y )( n )( D ) , and hence c s g t ( x ) F Ct c st ( y ).(b) If c rt ( x ) F Ct c st ( y ) then for all n ∈ ω and C ⊑ D ∈ B we have c rt ( x )( n )( D ) (cid:0) F Dt a n (cid:1) [ Γ jD − jC ) ] c st ( y )( n )( D ) . Thus there is h = h n,D ∈ Γ j D − j C ) such that for all v ∈ Γ j D − j C ) we have c rt ( x )( n )( j )( h − · v ) F Dt a n c st ( y )( n )( D )( v )and hence c r a h − · vt a n ( x ) F Dt a n c s a vt a n ( y ) , so by inductive assumption we have N r a h − · vt a n ( x ) = N s a vt a n ( y ). Thus { N r a vt a n ( x ) : v ∈ Γ j D − j C ) } = { N s a vt a n ( y ) : v ∈ Γ j D − j C ) } , so N rt ( x ) = [ n [ D ∈B : C ⊑ D { N r a vt a n ( x ) : v ∈ Γ j D − j C ) } = [ n [ D ∈B : C ⊑ D { N s a vt a n ( y ) : v ∈ Γ j D − j C ) } = N st ( y )as required. (cid:3) Definition 5.13. For t ∈ T , let A t ( x ) = { N st ( x ) : s ∈ B } . Lemma 5.14. For t ∈ T , there is a Borel function f t and an equivalence relation F t , reducible to J [Γ] ω ·| t | , so that: (a) If x E y then f t ( x ) F t f t ( y ) . (b) If f t ( x ) F t f t ( y ) then A t ( x ) = A t ( y ) .Proof. (a) Let F t = Q C ∈B (cid:0) F Ct (cid:1) [ Γ jC ] and f t ( x )( C )( v ) = c s ( v ) t ( x ), where s ( v ) = h ( α i , β i ) : i < k C i for v = α a β a · · · a α k C − a β k C − ∈ Γ j C as before.Suppose x E y , so there is g ∈ H with y = g · x . Fix C ∈ B , so we have EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 19 c s g t ( x ) F Ct c st ( y ) for all s ∈ C from the previous lemma. Thus for each v ∈ Γ j C wehave f t ( y )( C )( v ) = c s ( v ) t ( g · x ) F Ct c ( s ( v )) g t ( x )= c s ( v ′ ) t ( x )= f t ( x )( C )( v ′ ) , where v ′ = g − ( α ) a β a · · · a g − ( α k C − ) a β k C − . Letting ˜ g ∈ Γ j C be givenby ˜ g = g ↾ d C a id ↾ d C a · · · a g ↾ d Ck C − a id ↾ d Ck C − (where each concatenatedfactor acts on the corresponding coordinates) we have that v ′ = ˜ g − · v for all v ∈ Γ j C . Thus ˜ g witnesses f t ( x )( C ) (cid:0) F Ct (cid:1) [ Γ jC ] f t ( y )( C ) , and hence f t ( x ) F t f t ( y ).(b) Suppose f t ( x ) F t f t ( y ), so for each C ∈ B we have f t ( x )( C ) (cid:0) F Ct (cid:1) [Γ jC ] f t ( y )( C ). Thus there is h = h C ∈ Γ j C so that for all v ∈ Γ j C we have f t ( x )( C )( h − · v ) F Ct f t ( y )( C )( v )and hence c s ( h − · v ) t ( x ) F Ct c s ( v ) t ( y ) , so we have N s ( h − · v ) t ( x ) = N s ( v ) t ( y ). Thus { N st ( x ) : s ∈ C } = { N st ( y ) : s ∈ C } , so A t ( x ) = [ C ∈B { N st ( x ) : s ∈ C } = [ C ∈B { N st ( y ) : s ∈ C } = A t ( y )as required. (cid:3) Lemma 5.15. There is a Borel function f A and an equivalence relation F A , re-ducible to J [Γ] ω · ( n − , so that: (a) If x E y then f A ( x ) F A f A ( y ) . (b) If f A ( x ) F A f A ( y ) then A ( x ) = A ( y ) .Proof. Since t can be recovered from N st ( x ) we have A ( x ) = A ( y ) if and onlyif A t ( x ) = A t ( y ) for each t . Letting F A = Q t : | t |≤ n − F t and f ( x )( t ) = f t ( x )suffices. (cid:3) Definition 5.16. For m, k ∈ ω , ¯ r ∈ B k , and ¯ t ∈ T k , let B m,k ¯ r, ¯ t ( x ) = h N s t ( x ) , . . . , N s k − t k − ( x ) i : s i ∈ B ∧ x ∈ " W m ∩ \ i Lemma 5.17. There is a Borel function f B and an equivalence relation F B , re-ducible to J [Γ] ω · ( n − , so that: (a) If x E y then f B ( x ) F B f B ( y ) . (b) If f B ( x ) F B f B ( y ) then B ( x ) = B ( y ) .Proof. For m , k , ¯ r ∈ B k , and ¯ t ∈ T k with | t i | ≤ n − F m,k ¯ r, ¯ t = Y ¯ C ∈B k Y i Proof of Theorem 5.1. Let E be Π ∼ α .(a) For α = n ≥ 3, the function f ( x ) = ( f A ( x ) , f B ( x )) is a reduction of E to F A × F B as shown above, so E ≤ B J [Γ] ω · ( n − × J [Γ] ω · ( n − ≤ B J [Γ] ω · ( n − .(b) For α = λ + 1, λ a limit, we repeat the preceding argument using thetopology τ x<λ in place of τ xn − .(c) For α = λ + n , λ a limit and n ≥ 2, we use the topology τ xλ + n − . (cid:3) Corollary 5.18. Let Γ be a countable group and let E be a Borel equivalencerelation induced by a continuous action of Γ ω . Then E ≤ B J [Γ] α for some α < ω . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 21 In particular, the above shows that a Π ∼ Polish Aut( T Γ )-space is reducible to J [Γ] ω . We can improve this slightly in the case of Σ ∼ . Proposition 5.19. Let Γ be a countable group and let E be a Borel equivalencerelation induced by a continuous action of a closed subgroup of Aut( T Γ ) . If E is Σ ∼ then E ≤ B L n ∈ ω J [Γ] n .Proof. Since Aut( T Γ ) is a closed subgroup of S ∞ , E is essentially countable, so let f be a reduction of E to E ∞ . For each x , there is some z ∈ [ f ( x )] E ∞ so that theset { g ∈ Aut( T Γ ) : f ( g · x ) = z } is nonmeager, and hence comeager in some U s .Define the relation P ( x, C ) on X × B by P ( x, C ) ⇐⇒ ∃ s ∈ C ∃ z ∀ ∗ g ∈ U s f ( g · x ) = z. Then ∀ x ∃ CP ( x, C ) and P is E -invariant, so there is an E -invariant Borel functionΨ : X → B with P ( x, Ψ( x )) for all x . Let X C = Ψ − ( { C } ). On X C define thereduction ϕ : E ↾ X C ≤ B ∆( R )[ Γ jC ] by ϕ ( x )( h ) = ( z if ∀ ∗ g ∈ U s ( h ) f ( g · x ) = z ∗ if no such z exists , where s ( h ) is defined as before and ∗ is some new element E ∞ -inequivalent to all z . Suppose x E y , so there is g with y = g · x . Then as before there is ˜ g ∈ Γ j C so that for all h ∈ Γ j C we have s (˜ g − · h ) = s ( h ) ˜ g − , so that ˜ g witnesses ϕ ( x ) ∆( R )[ Γ jC ] ϕ ( y ). Conversely, if ϕ ( x ) ∆( R )[ Γ jC ] ϕ ( y ), then there are some h , h ′ , and z with ϕ ( x )( h ) = z = ∗ and ϕ ( y )( h ′ ) = z . Then f ( x ) E ∞ z E ∞ f ( y ), so x E y . (cid:3) Corollary 5.20. Let E be a countable Borel equivalence relation. If E ≤ B J [Γ] α forsome α < ω then E ≤ B L n ∈ ω J [Γ] n .Proof. Let ϕ be a reduction of E to J [Γ] α . Then the J [Γ] α -saturation B of the rangeof ϕ is Borel, and E ∼ B J [Γ] α ↾ B . We can refine the topology of B so that J [Γ] α ↾ B becomes a Polish Aut( T Γ )-space which is Σ ∼ . Thus J [Γ] α ↾ B , and hence E , isreducible to L n ∈ ω J [Γ] n . (cid:3) Note that, e.g., when Γ = F we already have that J [Γ]1 ∼ B E ∞ , so the abovebound is not always optimal, but we do not know if this is true for other Γ, suchas Γ = Z . The following seems quite optimistic: Question 5. If E is a countable Borel equivalence relation reducible to some J [Γ] α ,is E reducible to J [Γ]2 or even to J [Γ]1 ?The same techniques also show that isomorphism of Γ-trees is maximal amongAut( T Γ )-actions. By a labelled Γ -tree we mean the infinite Γ-tree T Γ with eachnode labelled with a real. It is straightforward to see that isomorphism of labelledΓ-trees, ∼ = Γlab , is reducible to isomorphism of Γ-trees. Theorem 5.21. Let G = Aut( T Γ ) . For any Polish G -space E XG we have E XG ≤ B ∼ = Γ . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 22 Proof. It will suffice to show that E XG ≤ B ∼ = Γlab . Fix an enumeration { e i : i ∈ ω } of T Γ , and let C n be the type h ht( e ) , . . . , ht( e n − ) i for each n . For t ∈ T Γ withht( t ) = j C n we let s t ∈ B be such that t = d s a · · · a d sn − and e si = e i for i < n .Given x , we let T x be the labelled Γ-tree defined as follows. For t with ht( t ) = j C n for some n we let T x ( t ) = { ℓ : x ∈ U − s t · W ℓ } , and let T x ( t ) = ∅ for other t . Weclaim that the map x T x is a reduction from E XG to ∼ = Γlab .Suppose first that g · x = y . Then for t with ht( t ) = j C n we have T y ( t ) = { ℓ : y ∈ U − s t · W ℓ } = { ℓ : g · x ∈ U − s t · W ℓ } = { ℓ : x ∈ g − U − s t · W ℓ } = { ℓ : x ∈ ( U s t g ) − · W ℓ } = { ℓ : x ∈ U − s gt · W ℓ } = T x ( ϕ ( t )) , where ϕ ∈ Aut( T Γ ) is induced by sending each node of the form d s a · · · a d sn − to g − ( d s ) a · · · a g − ( d sn − ). Then ϕ is an isomorphism from T y to T x .Conversely, suppose T x ∼ = T y via ϕ ∈ Aut( T Γ ). Let T ′ x be the subtree of T x consisting of initial segments of those t with T x ( t ) = ∅ (equivalently, U s t = ∅ ).Then [ T ′ x ] is a Polish subspace of Γ ω , and the set of branches A x = { α ∈ [ T ′ x ] : S n s α ↾ j Cn is an automorphism of T Γ } is comeager in [ T ′ x ]. Since ϕ induces a home-omorphism from [ T ′ x ] to [ T ′ y ], there are α ∈ A x and β ∈ A y with ϕ ( α ) = β (i.e., ϕ ( α ↾ n ) = β ↾ n for all n ). Let g α , g β ∈ Aut( T Γ ) be the induced automorphisms,so T n U s α ↾ jCn = { g α } and T n U s β ↾ jCn = { g β } . Since { ℓ : x ∈ U − s α ↾ j Cn · W ℓ } = { ℓ : y ∈ U − s β ↾ j Cn · W ℓ } for all n we must then have g α · x = g β · y . (cid:3) The group Aut( T Γ ) corresponds to the Γ-jump in an analogous way to that ofthe group S ∞ with respect to the Friedman–Stanley jump, so it is natural to ask ifit satisfies similar properties. For instance, Friedman showed: Theorem 5.22 (Friedman, Theorem 1.5 of [11]) . If E is a Borel equivalence rela-tion reducible to some S ∞ -action, then E is reducible to F α for some α < ω . Allison has noted that the analogous result holds for Aut( T Γ )-actions: Corollary 5.23 (Allison) . If E is a Borel equivalence relation reducible to some Aut( T Γ ) -action, then E is reducible to J [Γ] α for some α < ω .Proof. Since Aut( T Γ ) is a closed subgroup of S ∞ , Friedman’s theorem implies that E is reducible to some F α , and hence to some Π ∼ α orbit equivalence relation of S ∞ . By Theorem 2.4 of [1], E is then reducible to an action of Aut( T Γ ) with apotentially Π ∼ α equivalence relation. By Theorem 2.7 of [1], E is then reducible toa continuous action of Aut( T Γ ) with a Π ∼ α orbit equivalence relation, and hence tosome J [Γ] α by Theorem 5.1. (cid:3) Friedman and Stanley asked in [10] if every E given by an S ∞ -action whichsatisfies F α ≤ B E for all α < ω must be Borel complete; this remains open. Wecan similarly ask: Question 6. If E is given by an Aut( T Γ )-action which satisfies J [Γ] α ≤ B E for all α < ω , is ∼ = Γ reducible to E ?We note that there are Borel equivalence relations induced by actions of Aut( T Γ )which are not reducible to one induced by an action of Γ ω . This follows from theresult of Allison and Panagiotopoulos that E [ Z ]0 is generically ergodic with respect EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 23 to any orbit equivalence relation of a TSI Polish group (Corollary 2.3 of [2]) andthe fact that Γ ω is TSI for every countable Γ.We do not know if there is a canonical obstruction to reducibility to Aut( T Γ )-actions, in the way that turbulence is an obstruction for S ∞ -actions. Question 7. Is there a dynamical characterization of when a Borel equivalencerelation is reducible to an Aut( T Γ )-action?We close this section by noting that Shani has observed that Γ-jumps give “nat-ural” examples of equivalence relations at intermediate levels of the Borel hierarchy,defined in Section 6 of [16]. Namely, the Z -jump of F (which is ∼ = in the notationof [16]) is reducible to the relation ∼ = ∗ , defined there, and has potential complexityprecisely D (Π ∼ ). We do not know if this holds for other equivalence relations orgroups other than Z since the relations ∼ = ∗ α, require pairwise-inequivalent successorsat each node, and we do not know in general when E [Γ] is reducible to E [Γ]p.i. . § Properness of the Γ -jump In this section we consider the question of when the Γ-jump is a proper jump. Tobegin, we use our results of the previous section together with a result of Solecki toestablish that the Γ-jump is proper for a large class of countable groups, includingthe Z -jump. Theorem 6.1. Let Γ be a countable group such that Z or Z <ωp for a prime p is aquotient of a subgroup of Γ . Then the Γ -jump is a proper jump operator.Proof. Suppose towards a contradiction that E is a Borel equivalence relation suchthat ∆(2) ≤ B E and E [Γ] ∼ B E . Then by induction on α we have that J [Γ] α ( E ) ≤ B E for all α < ω , successor stages following from our assumption and limit stagesfrom Proposition 2.2 and the fact that J [Γ] λ ( E ) = (cid:16)L α<λ J [Γ] α ( E ) (cid:17) [Γ] ≤ B ( E ω ) [Γ] for a limit ordinal λ . Hence J [Γ] α ≤ B E for all α , so by Theorem 5.1, every Borelorbit equivalence relation induced by an action of Aut( T Γ ) is reducible to E . Itis a theorem of Solecki [24, Theorem 1] that if ∆ is one of the groups Z or Z <ωp for a prime p then ∆ ω admits non-Borel orbit equivalence relations, and by [14,Theorem 5.11] it follows that ∆ ω and hence Aut( T ∆ ) induces Borel orbit equivalencerelations of cofinal essential complexity. By our hypothesis and Proposition 2.5it then follows that Aut( T Γ ) induces Borel orbit equivalence relations of cofinalessential complexity, contradicting that they are all reducible to E . (cid:3) Although the Γ-jump has no Borel fixed points for such Γ, there are alwaysanalytic equivalence relations E with E [Γ] ∼ B E . Of course, the universal analyticequivalence relation is an example. Moreover, by Proposition 4.5, the isomorphismrelation ∼ = Γ on countable Γ-trees is a fixed point too. Thus we have the following: Corollary 6.2. Let Γ be a countable group such that Z or Z <ωp for a prime p is aquotient of a subgroup of Γ . Then ∼ = Γ is not Borel. For such Γ, ∼ = Γ gives a new example of a non-Borel, non-Borel complete isomor-phism relation. Other known examples include for instance isomorphism of abelian p -groups [10], as well as several given in [25]. EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 24 Friedman and Stanley’s proof that E < B E + utilized a theorem of Friedman’son the non-existence of Borel diagonalizers. We do not know if an analogous resultholds for Γ-jumps. Definition 6.3. A Borel diagonalizer for E [Γ] is a Borel E [Γ] -invariant mapping f : X Γ → X such that for all γ ∈ Γ we have f ( x ) E x ( γ ).The following is the analogue of Friedman–Stanley’s application of diagonalizersto the jump: Lemma 6.4. If E [Γ] ≤ B E then there is a Borel diagonalizer for (cid:0) E [Γ] (cid:1) [Γ] .Proof. Let f be a Borel reduction from E [Γ] to E . We may find z ∈ X Γ so that z α E z β for all α, β ∈ Γ and f ( z ) E z α for all α . Define F : (cid:0) X Γ (cid:1) Γ → X Γ by F (¯ x )( γ ) = ( f ( x γ ) if ∀ δf ( x γ ) E x γδ f ( z ) otherwise . Then F is (cid:0) E [Γ] (cid:1) [Γ] -invariant, and we claim that for all ¯ x and γ we have F (¯ x ) E [Γ] x γ . Suppose instead that there is γ with F (¯ x ) E [Γ] x γ . Let A = { x α : ∀ δf ( x α ) Ex αδ } ∪ { z : ∃ γ ∃ δf ( x γ ) E x γδ } , so that ran( F (¯ x )) = { f ( y ) : y ∈ A } .If there is y ∈ A with f ( x γ ) E f ( y ), then x γ E [Γ] y , so ∀ δf ( x γ ) E x γ δ (notingthis is true for z from its choice). Thus x γ ∈ A , so f ( x γ ) ∈ ran( F (¯ x )). But F (¯ x ) E [Γ] x γ , so there would be δ with f ( x γ ) E x γ δ , a contradiction. Hence thereis no y ∈ A with f ( x γ ) E f ( y ), so there is no δ with f ( x γ ) E x γ δ . But then wealso have x γ / ∈ A , and there is hence some δ with f ( x γ ) E x γ δ , again producing acontradiction. (cid:3) Question 8. Is it the case that E [Γ] doesn’t admit a Borel diagonalizer when theΓ-jump is proper? Note that this statement is at least as strong as Friedman’stheorem, since a diagonalizer for E + easily produces a diagonalizer for E [Γ] .We now turn to the question of finding Γ-jumps that are not proper. Recall that agroup satisfies the descending chain condition if it does not have an infinite properlydescending chain of subgroups. For example, the Pr¨ufer p -groups Z ( p ∞ ) (also calledquasi-cyclic groups) satisfy the descending chain condition. More generally if Γ is quasi-finite , meaning it is infinite with no infinite proper subgroups, then Γ satisfiesthe descending chain condition.We use the descending chain condition to obtain the condition that a descend-ing chain of cosets of subgroups has nonempty intersection. One may verify thatfor countable groups, the descending chain condition is equivalent to this lattercondition. Lemma 6.5. If Γ satisfies the descending chain condition, then J [Γ] λ ( E ) is Borelbireducible with Q α<λ J [Γ] α ( E ) for λ a limit.Proof. First we show Q α<λ J [Γ] α ( E ) ≤ B J [Γ] λ ( E ) for any infinite Γ. Let Γ = { γ n : n ∈ ω } and λ = { α n : n ∈ ω } . Define f by f ( x )( γ n ) = x ( α n ); then f is a reductionfrom Q α<λ J [Γ] α ( E ) to (cid:16)L α<λ J [Γ] α ( E ) (cid:17) [Γ] = J [Γ] λ ( E ). EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 25 For the other direction, we define a reduction from J [Γ] λ ( E ) = (cid:16)L α<λ J [Γ] α ( E ) (cid:17) [Γ] to Q α<λ (cid:16)L β<α J [Γ] β ( E ) (cid:17) [Γ] , which is bireducible with Q α<λ J [Γ] α ( E ). Fix some a ∈ X and let f ( x )( α )( γ ) = ( x ( γ ) if x ( γ ) ∈ L β<α J [Γ] β ( E ) a otherwise . Then f is easily a homomorphism. Suppose that f ( x ) Q α<λ (cid:16)L β<α J [Γ] β ( E ) (cid:17) [Γ] f ( x ′ ). For each α < λ let H α = (cid:26) γ : γ · f ( x )( α ) (cid:16)L β<α J [Γ] β ( E ) (cid:17) Γ f ( x ′ )( α ) (cid:27) , sothat the H α ’s form a nonempty descending chain of cosets of subgroups of Γ. SinceΓ satisfies the descending chain condition, there is some γ ∈ T α<λ H α , and this γ satisfies γ · x (cid:16)L α<λ J [Γ] α ( E ) (cid:17) Γ x ′ , so x J [Γ] λ ( E ) x ′ . (cid:3) Using Lemma 3.4, one can show that the above result also holds when Γ = Z ,but we do not know whether it holds for other groups Γ. Theorem 6.6. If Γ satisfies the descending chain condition, then J [Γ] ω +1 ∼ B J [Γ] ω .In particular, for such Γ the Γ -jump is not a proper jump operator.Proof. From the previous lemma we have J [Γ] ω +1 ∼ B (cid:16)Q n ∈ ω J [Γ] n (cid:17) [Γ] and J [Γ] ω ∼ B Q n ∈ ω (cid:16)Q k ∈ n J [Γ] k (cid:17) [Γ] , so it suffices to define a Borel reduction ϕ from (cid:16)Q n ∈ ω J [Γ] n (cid:17) [Γ] to Q n ∈ ω (cid:16)Q k ∈ n J [Γ] k (cid:17) [Γ] . For this we let ϕ ( x )( n )( γ ) = x ( γ ) ↾ n .To see that ϕ is a homomorphism, we calculate: x Y n ∈ ω J [Γ] n ! [Γ] y ⇐⇒ ∃ γ ∀ α ∀ n x ( γ − α )( n ) J [Γ] n y ( α )( n )= ⇒ ∀ n ∃ γ ∀ α ∀ k ∈ n x ( γ − α )( k ) J [Γ] k y ( α )( k ) ⇐⇒ ∀ n ∃ γ ∀ α ϕ ( x )( n )( γ − α ) Y k ∈ n J [Γ] k ! ϕ ( y )( n )( α ) ⇐⇒ ϕ ( x ) Y n ∈ ω Y k ∈ n J [Γ] k ! [Γ] ϕ ( y ) . Now suppose conversely that ϕ ( x ) Q n ∈ ω (cid:16)Q k ∈ n J [Γ] k (cid:17) [Γ] ϕ ( y ). For each n , let H n = ( γ : ∀ α x ( γ − α ) ↾ n Y k ∈ n J [Γ] k ! y ( α ) ↾ n ) . Then the sequence H n is a descending chain of cosets of subgroups of Γ. It followsfrom the descending chain condition that T n H n = ∅ . If γ ∈ T n H n , then γ witnesses that x (cid:16)Q n ∈ ω J [Γ] n (cid:17) [Γ] y , completing the proof. (cid:3) EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 26 We do not know if the statement of Theorem 6.6 is tight in the sense that J [Γ] ω is the least fixed-point. If Γ is a group such that the Γ-jump is improper, one mayask what is the least α such that J [Γ] α = J [Γ] α +1 . In the next section we will showthat for all Γ we have J [Γ]0 < B J [Γ]1 < B J [Γ]2 .The following is a consequence of Theorem 6.6 together with the proofs of The-orem 6.1 and Corollary 5.18. Corollary 6.7. If Γ satisfies the descending chain condition, then the family ofBorel orbit equivalence relations induced by continuous actions of Aut( T Γ ) is boundedwith respect to ≤ B . For abelian groups G , having non-Borel orbit equivalence relations is equivalentto the family of Borel orbit equivalence relations induced by continuous actionsof G being unbounded with respect to ≤ B , but we do not know if this holds forAut( T Γ ). It suffices to ask about ∼ = Γ : Question 9. If Γ satisfies the descending chain condition, is ∼ = Γ Borel?From Lemma 6.4 we also have: Corollary 6.8. If Γ satisfies the descending chain condition then there is a Boreldiagonalizer for J [Γ] ω +2 . We conclude this section by exploring the gap between our results on proper andimproper Γ-jumps. In light of Theorem 6.1, it is natural to ask for which countablegroups Γ we have non-Borel orbit equivalence relations of Γ ω or of Aut( T Γ ). Forabelian groups Γ, the answer to this question is known in the case of Γ ω -actions. Definition 6.9. Let Γ be a group and p a prime number. Then Γ is said to be p -compact if, for any descending chain of subgroups G k ≤ Z /p Z × Γ such that( ∀ k ) π [ G k ] = Z /p Z , we have π [ T k ∈ ω G k ] = Z /p Z . (Here π : Z /p Z × Γ → Z /p Z denotes the projection.)If Γ satisfies the descending chain condition, then Γ is p -compact for all primes p . Indeed, given G k as above, for each a ∈ Z p let H ak = { g ∈ Γ : ( a, g ) ∈ G k } .Then H ak is a descending chain of cosets of subgroups of Γ, so their intersection isnon-empty, meaning a ∈ π [ T k ∈ ω G k ].An example of a group which is p -compact for all p , but does not satisfy thedescending chain condition, is Γ = L p prime Z p . Moreover Γ is a group for whichthe hypotheses of both Theorem 6.1 and Theorem 6.6 are not satisfied.It is natural to ask whether Theorem 6.6 can be generalized to all groups whichare p -compact for all p . Indeed, it is shown in [24, Theorem 2] that if Γ is p -compactfor all p , then Γ ω does not have a non-Borel orbit equivalence relation. It is furthershown in [24] that for abelian groups Γ, this is a complete characterization. Thequestion of which non-abelian Γ admit non-Borel Γ ω orbit equivalence relations isopen. Of course it may also be possible that some Γ-jump is proper without thiscondition holding. Question 10. Which countable groups Γ give rise to proper jump operators? If Γis p -compact for all primes p , is the Γ-jump improper? Does the group L p prime Z p give rise to a proper jump operator? EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 27 § Bounds on potential complexities The bounds in the statement of Theorem 5.1 are not always tight, that is, some-times it is possible to reduce an equivalence relation E to fewer iterates of the jump.As a conseqence the lower bounds on the potential complexity of J [Γ] α that it givesare also not always tight. The bounds provided in Theorem 5.1 are derived fromthe definitions of the equivalence relations introduced in the proof, together withProposition 2.2, Lemma 2.6, and the fact that Q α<λ J [Γ] α ≤ B J [Γ] λ for limit λ (seethe proof of Lemma 6.5). In general this technique requires about ω -many iteratesof the Γ-jump to ensure the potential complexity has increased by one level in theBorel hierarchy.In this section we provide a more direct proof that the iterated jumps have cofinalpotential complexity, in the special case when Γ = Z <ω , with tighter bounds onpotential complexity of the iterates of the Z <ω -jump. As a consequence we obtainan alternate proof that the Z <ω -jump is proper. Proposition 7.1. For Γ = Z <ω we have the following: ◦ For n ≥ , J [Γ] n ∈ pot(Π ∼ n +2 ) r pot(Σ ∼ n +1 ) . ◦ For λ a limit, J [Γ] λ ∈ pot(Π ∼ λ ) r pot(Σ ∼ λ ) . ◦ For λ a limit and n ≥ , J [Γ] λ + n ∈ pot(Π ∼ λ + n +1 ) r pot(Σ ∼ λ + n +1 ) .Proof. The upper bounds follow from Propositions 3.12. To establish the lowerbounds, we use the properly increasing tower A α of equivalence relations definedby Hjorth–Kechris–Louveau in § E F α there). To define A α , weadopt the following notation. For any x ∈ ω we may regard x as an element of(2 ω ) ω , and we write x n for the n th element of 2 ω in x . Also for any x ∈ ω we let¯ x ( n ) = 1 − x ( n ).We now define A = E and x A α +1 y if and only if:(a) for all n , either x n A α y n or x n A α y n , and(b) for all but finitely many n , x n A α y n .We can think of A as ∆(2). At limits we code a disjoint union. Hjorth–Kechris–Louveau showed (Theorem 5.8 of [16]) that A α < B A α +1 for all α and that A n / ∈ pot(Σ ∼ n +1 ) for n ≥ α ). We show that the each A α ≤ B J [Γ] α .We claim that A α +1 ≤ B (( A α ) ω ) [Γ] for each α . Admitting this claim, wehave by hypothesis that A = E ∼ B J [Γ]1 . Since Z <ω ∼ = Z <ω × Z <ω , we have( E ω ) [Γ] ≤ B E [Γ] for any E by Proposition 2.7. Hence if A α ≤ B J [Γ] α then A α +1 ≤ B (cid:16) ( J [Γ] α ) ω (cid:17) [Γ] ≤ B (cid:16) J [Γ] α (cid:17) [Γ] = J [Γ] α +1 . Since we take disjoint unions at limits in bothcases, continuing inductively we have that A α ≤ B J [Γ] α for all α , establishing thedesired lower bounds on potential complexity.To establish the claim, let x ∈ ω be given and define α x ( n, s ) for n ∈ ω and s ∈ <ω by α x ( n, s ) = ( x n s ( n ) = 0 x n s ( n ) = 1 . Then if x A α +1 y , we define γ ∈ Γ by γ ( n ) = 0 if x n A α y n and γ ( n ) = 1 otherwise.Then γ witnesses that α x is equivalent to α y . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 28 On the other hand, if α x is equivalent to α y , let γ ∈ Γ witness this. Then thecoordinates of γ witness that x A α +1 y . (cid:3) Note that the upper bounds in the previous proposition still leave a gap forfinite iterates of the jump. For n = 2 we have that J [Γ]2 ∈ pot(Σ ∼ ), and hence J [Γ]2 ∈ pot( D (Π ∼ )) by [16]. On the other hand, results of Allison–Shani ([3]) showthat J [Γ]2 / ∈ pot(Π ∼ ), so J [Γ]2 ∈ pot( D (Π ∼ )) r pot(Π ∼ ), but this leaves the questionopen for higher finite iterates.We also do not know the optimal bounds in the case of the Z -jump. Question 11. What are the exact potential Wadge degrees of the Z α ’s? § Generic E -ergodicity of Γ -jumps In this section we will show that Γ-jumps of countable Borel equivalence relationsproduce new examples of equivalence relations intermediate between E ω and F .Furthermore, we will see that although some Γ-jumps admit Borel fixed points, anysuch fixed point must be strictly above E . The key notion in establishing theseresults is the following. Definition 8.1. We say that E is generically F -ergodic if for every Borel homo-morphism ϕ from E to F there is a y so that ϕ − [ y ] F is comeager. We say E is generically ergodic when it is generically ∆( R )-ergodic.Note that if E is generically F -ergodic, and E does not have a comeager equiv-alence class, then E B F . Furthermore observe that each E [Γ]0 -class is meager.Our main result will be to show that E [Γ]0 is generically E ∞ -ergodic for anycountable group Γ. Throughout this section we let Γ = { γ n : n ∈ ω } be a countablyinfinite group and fix an increasing sequence of finite subsets Γ n with Γ = S n Γ n . Definition 8.2. Let G = Z Γ induce the orbit equivalence relation E Γ0 in the naturalway. Then G together with the shift action of Γ on (2 ω ) Γ generates E [Γ]0 . For each n and each ¯ k ∈ Z Γ n , let U n ¯ k = { g ∈ G : g ↾ Γ n = ¯ k } , so that { U n ¯ k : n, ¯ k } is acountable basis for G .The following lemma is derived from Theorem 7.3 of [15]: Lemma 8.3. Let f : E [Γ]0 → E ∞ be a Borel homomorphism. Then there is a Borel ˜ f such that ˜ f ( x ) E ∞ f ( x ) for all x , and there are E Γ0 -invariant sets X n with (2 ω ) Γ = ∐ n X n such that for x, x ′ ∈ X n with x E Γ0 x ′ and x ↾ Γ n = x ′ ↾ Γ n we have ˜ f ( x ) = ˜ f ( x ′ ) .Proof. For each x , [ f ( x )] E ∞ = { y i : i ∈ ω } is countable, so there is some i so that G i = { g ∈ G : f ( g · x ) = y i } is nonmeager, and hence comeager in some U n ¯ k . Hence: ∀ x ∃ n ∃ ¯ k ∃ y ∈ [ f ( x )] E ∞ ∀ ∗ g ∈ U n ¯ k ( f ( g · x ) = y ) . Set P ( x, n ) ⇔ ∃ ¯ k ∃ y ∈ [ f ( x )] E ∞ ∀ ∗ g ∈ U n ¯ k ( f ( g · x ) = y ), so ∀ x ∃ nP ( x, n ). Since P is E Γ0 -invariant (if P ( x, n ) and x E Γ0 x ′ then P ( x ′ , n )), there is a E Γ0 -invariant Borelfunction s : (2 ω ) Γ → N with P ( x, s ( x )) for all x . Let X n = s − [ { n } ].For x ∈ X n , let ¯ k ( x ) be the least ¯ k (in some fixed enumeration of Z Γ n ) sothat ∃ y ∈ [ f ( x )] E ∞ ∀ ∗ g ∈ U n ¯ k ( f ( g · x ) = y ). Note that if x E Γ0 x ′ and x ↾ EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 29 Γ n = x ′ ↾ Γ n then ¯ k ( x ) = ¯ k ( x ′ ). We can now let ˜ f ( x ) be the unique y so that ∀ ∗ g ∈ U n ¯ k ( x ) ( f ( g · x ) = y ). (cid:3) Now X n is nonmeager for some n , and since E Γ0 is induced by a continuousgroup action with dense orbits it must be comeager. Let n be this n . Let Y be acomeager set with ˜ f continuous on Y . Let Y = T n ∈ ω γ n [ Y ∗ G ∩ X n ] so that Y iscomeager and E [Γ]0 -invariant. Lemma 8.4. For all x, x ′ ∈ Y with x ↾ Γ n = x ′ ↾ Γ n we have ˜ f ( x ) = ˜ f ( x ′ ) .Proof. Let x, x ′ ∈ Y with x ↾ Γ n = x ′ ↾ Γ n . We can then choose g ∈ G with g · x, g · x ′ ∈ Y and g · x ↾ Γ n = g · x ′ ↾ Γ n = x ↾ Γ n . Hence ˜ f ( g · x ) = ˜ f ( x ) and˜ f ( g · x ′ ) = ˜ f ( x ′ ), so, replacing x and x ′ by g · x and g · x ′ , we may assume x, x ′ ∈ Y .Suppose ˜ f ( x ) = ˜ f ( x ′ ), and let k be least such that ˜ f ( x )( k ) = ˜ f ( x ′ )( k ). We canthen find neighborhoods U and U ′ in (2 ω ) Γ with x ∈ U and x ′ ∈ U ′ , so that if y ∈ U ∩ Y then ˜ f ( y )( k ) = ˜ f ( x )( k ) and if y ′ ∈ U ′ ∩ Y then ˜ f ( y ′ )( k ) = ˜ f ( x ′ )( k ).Since G -orbits are dense, the set { g ∈ G : g · x ∈ U ′ ∧ g · x ↾ Γ n = x ↾ Γ n } is non-empty and open, so it contains some g with g · x ∈ Y . But then ˜ f ( g · x ) = ˜ f ( x ) bythe previous lemma, whereas ˜ f ( g · x )( k ) = ˜ f ( x ′ )( k ) = ˜ f ( x )( k ), a contradiction. (cid:3) Lemma 8.5. For all x, x ′ ∈ Y , if there is γ ∈ Γ with x ↾ γ Γ n = x ′ ↾ γ Γ n then ˜ f ( x ) E ∞ ˜ f ( x ′ ) .Proof. We have γ − · x ↾ Γ n = γ − · x ′ ↾ Γ n , and since Y is E [Γ]0 -invariant we have˜ f ( γ − · x ) = ˜ f ( γ − · x ′ ). Since x E [Γ]0 γ − · x and ˜ f is a homomorphism we have˜ f ( x ) E ∞ ˜ f ( γ − · x ), and similarly for x ′ , so the result follows. (cid:3) Now we are ready for the main result of this section: Theorem 8.6. E [Γ]0 is generically E ∞ -ergodic.Proof. Let f : E [Γ]0 → E ∞ be a Borel homomorphism, and let ˜ f , n , and Y be asabove. By the Kuratowski–Ulam theorem we have ∀ ∗ z ∈ (2 ω ) Γ n ∀ ∗ w ∈ (2 ω ) Γ r Γ n ( z a w ∈ Y ) , so the set Y ′ = { x ∈ Y : ∀ ∗ w ∈ (2 ω ) Γ r Γ n ( x ↾ Γ n a w ∈ Y ) } is comeager. It suffices to show that ˜ f maps Y ′ into a single E ∞ -class. Fix x, x ′ ∈ Y ′ , and let γ ∈ Γ so that Γ n and γ Γ n are disjoint. We can then find w ∈ (2 ω ) Γ r Γ n so that y = x ↾ Γ n a w ∈ Y and y ′ = x ′ ↾ Γ n a w ∈ Y . We have ˜ f ( x ) E ∞ ˜ f ( y )and ˜ f ( x ′ ) E ∞ ˜ f ( y ′ ) by agreement on Γ n , and ˜ f ( y ) E ∞ ˜ f ( y ′ ) by agreement on γ Γ n , so ˜ f ( x ) E ∞ ˜ f ( x ′ ). (cid:3) When E is generically F -ergodic it is also generically F ω -ergodic, so we imme-diately get: Theorem 8.7. E [Γ]0 is generically E ω ∞ -ergodic. Corollary 8.8. E [ Z ]0 is generically E ω -ergodic. EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 30 Since ( E ) ω is Borel reducible to E [ Z ]0 , we have in particular that ( E ) ω is properlybelow E [ Z ]0 in complexity.Allison and Panagiotopoulos have since strengthened the last result to showthat E [ Z ]0 is generically ergodic with respect to any orbit equivalence relation of aTSI Polish group (Corollary 2.3 of [2]). More recently, Allison has shown that anequivalence relation E is generically ergodic with respect to any orbit equivalencerelation E XG with G non-Archimedean abelian if and only if E is generically E -ergodic (Theorem 6.6 of [1]), and E is generically ergodic with respect to any orbitequivalence relation E XG with G non-Archimedean TSI if and only if E is generically E ∞ -ergodic (Theorem 6.5 of [1]).From the above, we can see that Γ-jumps do not have fixed points at the firsttwo levels. Corollary 8.9. For any countable group Γ we have J [Γ]0 < B J [Γ]1 < B J [Γ]2 .Proof. For any countable Γ, J [Γ]0 = ∆(2), and J [Γ]1 is the shift action of Γ on 2 Γ ,which is countable and generically ergodic. (cid:3) We now proceed to show that E [ Z ]0 and E [ Z ] ∞ are genuinely new levels in the Borelcomplexity order. We begin by considering which countable Borel equivalence rela-tions are reducible to E [ Z ]0 . We give a partial characterization of when a countableBorel equivalence relation admits Borel assignments of scattered linear orders toeach equivalence class. Scattered linear orders are discussed in more detail in Sec-tion 9, and a general study of structurable equivalence relations may be found in[6]. Lemma 8.10. If a countable Borel equivalence relation E admits a Borel as-signment of scattered linear orders of rank α to each equivalence class, then E ≤ B Z α .Proof. Suppose E admits a Borel assignment of scattered linear orders of rank 1+ α to each equivalence class,. Using that the shift action of Z on R is reducible to theshift of Z on 2, we can reduce E to the isomorphism relation of scattered linearorders of rank 1 + α , ∼ = α , by sending x to the scattered linear order obtainedby replacing each point corresponding to y ∈ [ x ] E in the ordering of [ x ] E with anordering encoding y . From Theorem 9.7 below we then have E ≤ B ∼ = α ≤ Z α . (cid:3) Lemma 8.11. Let E be a countable Borel equivalence relation. Then E ≤ B E [ Z ]0 if and only if E is Z ∗ Z -orderable, i.e., E admits a Borel assignment of subordersof Z ∗ Z to each equivalence class.Proof. Suppose first that E is a countable Borel equivalence relation with E ≤ B E [ Z ]0 . By Lemma 3.7 we have ( E ) [ Z ] ≤ B ( E ) [ Z ]p.i. , so let f be a reduction of E to( E ) [ Z ]p.i. . Recall that E Z denotes the product of Z -many copies of E , so E Z is acountable index subequivalence relation of E [ Z ]0 and E Z ∼ = E ω . Because of pairwiseinequivalence, on the range of f there is a Borel ordering ≺ of E -classes with theproperty that the E Z -classes within each E [ Z ]0 -class are isomorphic to the ordering Z . Let ˜ E be the pullback of E Z under f , that is, x ˜ E x ′ iff f ( x ) E Z f ( x ′ ). Then ˜ E is a subequivalence relation of E , and hence countable. Since ˜ E ≤ B E ω , it follows EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 31 from the Sixth Dichotomy Theorem [15, Theorem 7.1] that ˜ E is hyperfinite. Hencethere is an assignment of suborders of Z to each ˜ E -class. Moreover, letting ≺ bethe pullback of ≺ , we have that the ˜ E -classes within each E -class are isomorphicto a suborder of Z . It follows that there is a Borel assignment of suborders of Z ∗ Z to the equivalence classes of E .The converse follows from the previous lemma. (cid:3) Note that this is analogous to the result that a countable Borel equivalencerelation is reducible to E if and only if it admits a Borel assignment of subordersof Z to each equivalence class. We do not know whether analogous results for theforward direction hold for higher iterates of the Z -jump. Question 12. For a countable Borel equivalence relation E , does E admit anassignment of scattered linear orders of rank 1 + α iff E ≤ B J [ Z ] α ?As observed in Corollary 5.20, if E is a countable Borel equivalence relationwith E ≤ B J [ Z ] α for some α < ω , then E ≤ B L n ∈ ω J [ Z ] n . Here it may be worthinvestigating a restricted form of the Γ-jump using finite supported wreath productsto preserve countability of the equivalence relations, instead of the full wreathproducts considered here.We can now establish the following: Corollary 8.12. E ∞ B E [ Z ]0 .Proof. Suppose towards a contradiction that E ∞ ≤ B E [ Z ]0 . From the previouslemma, there would be a Borel assignment of suborders of Z ∗ Z to the equivalenceclasses of E ∞ , and therefore by [18] E ∞ would be amenable. Since E ∞ is notamenable, this contradiction completes the proof. (cid:3) We may now complete the picture to see that E [ Z ]0 and E [ Z ] ∞ are new examplesof natural equivalence relations below F . Theorem 8.13. We have the following: ◦ E ω < B E [ Z ]0 < B E [ Z ] ∞ < B F . ◦ E ω < B E ω ∞ < B E [ Z ] ∞ < B F . ◦ E [ Z ]0 and E ω ∞ are ≤ B -incomparable. Previously, the only known examples of equivalence relations between E ω ∞ and F were the examples constructed in [26], which are non-pinned. Shani has alsoproduced a new example of an intermediate pinned equivalence relation in [22],denoted E Π , which is also incomparable to E [ Z ]0 and E [ Z ] ∞ . Theorem 8.14 (Shani, Theorem 1.7 of [22]) . E Π is pinned and satisfies: ◦ E ω ∞ < B E Π < B F . ◦ E Π B E [Γ] ∞ and E [Γ]0 B E Π for any infinite countable group Γ . Note that these results give several other equivalence relations strictly between E ω ∞ and F , such as E ω ∞ × E [ Z ]0 and E [ F ] ∞ (where F is the free group on 2 generators).Shani has shown, as consequences of Corollary 5.6 and Proposition 5.14 of [22]: Theorem 8.15 (Shani) . The following hold: ◦ E [ Z ] ∞ B E [ Z ]0 × E ω ∞ . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 32 ◦ E [ Z ]0 × E ω B E [ Z ] ∞ . ◦ E [ Z ] ∞ and E [ Z ]0 × E ω ∞ are ≤ B -incomparable. ◦ E [ Z ] < B (cid:0) E [ Z ] (cid:1) for any generically ergodic countable Borel equivalencerelation E . One can also ask if there are any equivalence relations strictly between E ω and E [ Z ]0 . Question 13. If E ≤ B E [ Z ]0 , does either E ∼ B E [ Z ]0 or E ≤ B E ω ?From the Hjorth–Kechris dichotomy for E ω , we can obtain a weak partial result. Lemma 8.16. If E ≤ B E [ Z ]0 then either E ω ≤ B E or E ≤ B E ∞ .Proof. Let f be a reduction of E to E [ Z ]0 . Set x F x ′ iff f ( x ) E Z f ( x ′ ), so that F is of countable index below E and F ≤ B E Z ∼ B E ω . Hence either E ω ≤ B F or F ≤ B E . In the first case, let g be a reduction of E ω to F , and set x E ′ x ′ iff g ( x ) E g ( x ′ ). Then E ′ ≤ B E and E ′ is of countable index over E ω , so that E ω ≤ B E ′ (see Corollary 8.32 of [8]) and hence E ω ≤ B E . In the second case,let h be a reduction of F to E , and define the equivalence relation E ′ by y E ′ y ′ iff y E y ′ ∨ ∃ x, x ′ ( x E x ′ ∧ h ( x ) E y ∧ h ( x ′ ) E y ′ ). Then E ′ is a countableanalytic equivalence relation with E ≤ B E ′ , so E is essentially countable (see, e.g.,Proposition 7.1 of [9]); hence E ≤ B E ∞ . (cid:3) Question 14. If E ≤ B E [ Z ] ∞ , does either E [ Z ]0 ≤ B E or E ≤ B E ω ∞ ? § Z -trees and scattered linear orders In this section we give an application of our results about the Z -jump to theclassification of countable scattered linear orders. We then give several applications.First we establish a complexity bound on countable Borel equivalence relationswhich admit an assignment of scattered linear orders to its equivalence classes.Next we study the classification of countable complete linear orders, and give anapplication to classifying more general classes of countable models. Definition 9.1. A linear order L is said to be scattered if there does not exist anembedding (a one-to-one order-preserving map) from Q to L .Recall the scattered linear orders admit a derivative or collapse operation as wellas a rank function. To begin, define an equivalence relation on L by x ∼ y if theinterval between x, y is finite. The ∼ -equivalence classes are convex, so we mayform a quotient ordering L/ ∼ . Next we define equivalence relations ∼ α for anyordinal α as follows. If ∼ β has been defined, we let x ∼ β +1 y iff [ x ] β ∼ [ y ] β , where[ x ] β , [ y ] β denote the ∼ β -equivalence classes of x, y . If λ is a limit ordinal and ∼ β have been defined for all β < λ , we let x ∼ λ y iff there exists β < λ such that x ∼ β y .Now a linear order L is scattered if and only if there exists α such that L/ ∼ α isa single point (see [19, Exercise 34.18]). Thus we may define the rank of a scatteredlinear order L as the least α such that L/ ∼ α is a single point. Let S denote theclass of countable scattered linear orders, S α denote the class of countable scatteredlinear orders of rank α , and S ≤ α for the class of countable scattered linear ordersof rank at most α . EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 33 Proposition 9.2 (Exercises 33.2, 34.18 of [19]) . The set S is a Π ∼ -complete setand the subsets S α and S ≤ α are Borel sets. Moreover, the function which maps acountable scattered linear order to its rank is a Π ∼ -rank on S . We will write ∼ = α to abbreviate ∼ = S α . Furthermore, although S is a non-Borelclass of countable structures, we will write ∼ = S for the isomorphism relation on allcountable scattered orderings. Proposition 9.3. There is no absolutely ∆ ∼ reduction from ∼ = S to a Borel equiva-lence relation. On the other hand, no Borel complete equivalence relation is Borelreducible to ∼ = S .Proof. For the first statement we recall from [14] (see remarks following Corol-lary 3.3) that there is no absolutely ∆ ∼ reduction from E ω to a Borel equivalencerelation, where E ω is the isomorphism relation on countable well-orders. Moreoverwe have that E ω is absolutely ∆ ∼ -reducible to ∼ = S (see, e.g., Theorem 3.5 of [7]).Finally compositions of absolutely ∆ ∼ reductions are again absolutely ∆ ∼ .For the second statement, it will follow from the results below that F is notreducible to ∼ = S , but we can give a more basic proof. Suppose there exists a Borelreduction f from some Borel complete equivalence relation to ∼ = S . Then by theBoundedness Theorem, the range of f is contained in some S α . Since Borel com-plete equivalence relations are not Borel, it follows that ∼ = α is not a Borel equiv-alence relation. But it is not difficult to see that ∼ = α is Borel, for instance seeProposition 9.7 below. This is a contradiction. (cid:3) The classification of scattered linear orders is closely related to the classificationof Z -trees. We use a modification of the definition of Γ-trees given earlier, which isslightly simpler and more natural in the case of orders. Definition 9.4. A Z -tree is a tree together with, for each node x , a linear orderingon the set of immediate successors of x which is isomorphic to a subordering of Z .If T is a Z -tree, we will use the < symbol for the order relation on the immediatesuccessors of any node of T . We let ZT denote the space of countable Z -trees, and ZT α the space of countable Z -trees of rank α . Lemma 9.5. The isomorphism relation ∼ = α on countable scattered linear orders ofrank α is Borel bireducible with the isomorphism relation on ZT α .Proof. We first show ∼ = α is Borel reducible to the isomorphism relation on ZT α .Given a scattered linear order L of rank α we define a tree T consisting of the ∼ β equivalence classes for β ≤ α , with the ordering t ≤ t ′ iff t ⊂ t ′ . It is not difficultto see that L T is a Borel reduction as desired.To show that the isomorphism relation on ZT α is Borel reducible to ∼ = α , weproceed by induction. For the base case, it is clear that both the isomorphismrelation on ZT and ∼ = are bireducible with ∆(1).For the inductive step, let α > β < α there existsa Borel reduction f β from the isomorphism relation on ZT β to ∼ = β . Furtherassume that for all β and all T ∈ dom( f β ) we have that f β ( T ) does not containany ∼ -class of size 1. (We may do so without loss of generality, by inserting animmediate successor to each point of f β ( T ).) EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 34 Given a Z -tree T of rank 1 + α , let N denote the order type of the children of theroot of T , so that N is a suborder of Z . For each n ∈ N let T n denote the subtreeof T rooted at the n th child of T , and let β n be the rank of T n . Finally let L = X n ∈ N f β n ( T n ) + Z + 1 + Z Then the mapping T L is a Borel reduction as desired. To see one can recover T from L , note that a separator widget Z + 1 + Z may be identified by its ∼ -classof size 1. Thus one can determine each f β n ( T n ) up to isomorphism, and then usethe reductions f β n to recover each T n and thus T up to isomorphism. (cid:3) Note that this notion of Z -tree carries less information at rank 2 than our earlierdefinition, as the isomorphism relation on ZT is bireducible with ∆( ω ). Since(∆(2)) [ Z ] ∼ B (∆( ω )) [ Z ] ∼ B E , we easily see that the isomorphism relations for thetwo notions of Z -tree are equivalent with ranks adjusted appropriately. Lemma 9.6. Isomorphism on ZT α is Borel bireducible with ∼ = Z α . (cid:3) As seen above, the isomorphism relation ∼ = ZT on all Z -trees is not Borel, butalso not Borel complete. Theorem 9.7. The isomorphism relation ∼ = α on countable scattered linear ordersof rank α is Borel bireducible with Z α = J [ Z ] α .Proof. By Lemma 9.5, it suffices to prove that isomorphism on ZT α is Borelbireducible with Z α , which follows since ∼ = Z α is bireducible with Z α by Proposi-tion 4.4. (cid:3) It follows using Corollary 5.18 that every Borel Z ω -action is reducible to some ∼ = α . Also, it follows using Theorem 6.1 that the complexity of the classificationof countable scattered linear orders increases strictly with the rank. We note thatAlvir and Rossegger have shown in [4] that the complexity of Scott sentences alsoincreases strictly with rank.We may ask about the relationship between isomorphism of scattered linear or-ders and assignments of scattered linear orderings to equivalence classes of countableBorel equivalence relations. Kechris has shown in [18] that every countable Borelequivalence relation which admits a Borel assignment of scattered linear orders toeach equivalence class is amenable, and has asked whether the converse is true.It is also a long-standing open question whether every countable amenable Borelequivalence relation is hyperfinite. In light of Lemma 8.10, Corollary 5.20, andboundedness, we have: Corollary 9.8. If E is a countable Borel equivalence relation which admits a Borelassignment of scattered linear orders to each equivalence class, then E ≤ B ∼ = ω . We may then ask: Question 15. If E is a countable Borel equivalence relation with E ≤ B ∼ = ω , is E hyperfinite?Next we consider a class of linear orders that is very closely related to the scat-tered linear orders, and show that this together with our results above leads to amodel-theoretic corollary. EW JUMP OPERATORS ON EQUIVALENCE RELATIONS 35 Definition 9.9. A linear order L is said to be complete if for every A ⊂ L , if A has an upper bound then A has a least upper bound.Every countable complete linear order is scattered. Indeed, if L is countable andcomplete then it has just countably many Dedekind cuts. It follows that there isno embedding of Q into L , and that L is scattered. On the other hand, one mayverify that if L is a countable scattered linear order then its Dedekind completion¯ L is a countable complete linear order. In the following, we let C denote the subsetof S consisting of the countable complete linear orders. Theorem 9.10. The statements of Lemma 9.5 holds with ∼ = α replaced with ∼ = α ↾ C .In particular for all α we have that the relations Z α , ∼ = α , and ∼ = α ↾ C are Borelbireducible with one another.Proof. In the proof of Lemma 9.5, given a sequence f β n ( T n ) we defined a scatteredorder L using the separator Z + 1 + Z . Assuming the f β n ( T n ) are complete, we canensure that L will be complete by using the separator 1 + Z + 1 + Z + 1. (cid:3) We close this section by mentioning an application to the classification of count-able models of certain theories T , which was pointed out to us by Ali Enayat. Let T be a theory with built-in Skolem functions, and with a unary predicate P anda linear ordering < of P . Further suppose that there is a model M of T with aproper elementary end extension N , that is, N is an elementary extension of M and P M < P N r P M . 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