Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension

The algebraic dimension of a Polish permutation group $Q\leq \mathrm{Sym}(\mathbb{N})$ is the smallest $n\in\omega$, so that for all $A\subseteq \mathbb{N}$ of size $n+1$, the orbit of every $a\in A$ under the pointwise stabilizer of $A\setminus\{a\}$ is finite. We study the Bernoulli shift $P\curvearrowright \mathbb{R}^{\mathbb{N}}$ for various Polish permutation groups $P$ and we provide criteria under which the $P$-shift is generically ergodic relative to the injective part of the $Q$-shift, when $Q$ has algebraic dimension $\leq n$. We use this to show that the sequence of pairwise $*$-reduction-incomparable equivalence relations defined in [KP21] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal $\aleph_1^{+}$ which strongly resembles the equivalence relation of pinned cardinal $\aleph_1^{+}$ from [Zap11], but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group $Q$. We show that when $Q$ is"locally finite"-- e.g. when $Q=\mathrm{Aut}(\mathcal{M})$, where $\mathcal{M}$ is a locally finite countable structure with no algebraicity -- the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.


Introduction
Let G be a Polish group acting continuously G X on a Polish space X.A fundamental question which motivates several threads of research, both in descriptive set theory and in ergodic theory, is how much information does the associated orbit equivalence relation E G X preserve from the original action.In particular, one would like to know which structural properties of the group G can produce dynamical phenomena which are strong and distinct enough, so as to have an impact on the "complexity" of E G X ; the later being measured by the position of E G X in the Borel reduction hierarchy ≤ B , as well as by its strong ergodic properties.
When it comes to Polish permutation groups P ≤ Sym(N), an action which often serves as bridge between structural and dynamical properties of P is the Bernoulli shift P R N of P .For example, if E(P ) is its orbit equivalence relation, then: (1) E(P ) is smooth if and only if P is compact, see [KMPZ20]; (2) E(P ) is essentially countable if and only if P is locally compact, [KMPZ20]; (3) E(P ) is classifiable by CLI-group actions if and only if P is CLI, [KMPZ20]; (4) E(P ) B E(P ≀ Γ) for various countable groups Γ, [CC20]; 2000 Mathematics Subject Classification.03E15, 54H05, 03E40, 03E75.
(5) E(Z ≀ Z) is not classifiable by TSI-group actions, [AP21].(6) E(Γ ≀ Γ) ≤ B E(∆ ≀ ∆) for various countable Γ, ∆, [Sha19].In fact, the above results still hold if we replace all shifts with their injective parts.By the injective part E inj (P ) of E(P ) we mean the restriction of E(P ) to the P -invariant subset Inj(N, R) of R N , consisting of all injective sequences.
In this paper we study how the algebraic dimension of a Polish permutation group affects the ergodic behavior of its Bernoulli shift.For every Polish permutation group P ≤ Sym(N) and every A ⊆ N, we denote the pointwise stabilizer of A by P A .The algebraic closure of A with respect to P is the set: [A] P := {a ∈ N : the orbit P A • a is finite}.
The algebraic dimension of the permutation group P is simply the dimension of the closure operator A → [A] P , where A ranges over finite subsets of N.More precisely: Definition 1.The algebraic dimension of a permutation group P ≤ Sym(N) is the smallest n ∈ ω ∪ {∞}, so that for every A ⊆ N, with |A| = n + 1, there is a ∈ A with a ∈ [A\ {a}] P .If n = ∞, we say that P is of bounded algebraic dimension.
When P := Aut(N ) is the automorphism group of a countable structure N , it is often then case that the dynamical definition of algebraic dimension above coincides with the model theoretic one; see [TZ12].In this context, algebraic dimension has extensively been studied when A → [A] P forms a pregeometry, i.e., when it additionally satisfies the exchange principle.Examples include the case where N is the n-dimensional Q-vector space or the complete n-branching tree.It is easy to see that this additional assumption forces P = Aut(N ) to be locally compact and, in turn, E(P ) to be essentially countable [Kec92].From this point of view, bounded algebraic dimension can be seen as a partial weakening of local compactness.
The following theorem shows that the strong ergodic properties of the Bernoulli shift of P , as well as the Borel reduction complexity of E(P ), are both sensitive to the algebraic dimension of P .We say that P is locally finite if [A] P is finite for all finite A. We say that P is n-free, if for all finite A ⊆ N, there are g 0 , . . ., g n−1 ∈ P so that for all i ∈ {0, . . ., n − 1}, [g i • A] P and [ j:j =i (g j • A)] P are disjoint.
Theorem 2. Let P, Q ≤ Sym(N) be Polish permutation groups and let n ∈ ω.If the algebraic dimension of Q is at most n and P is locally finite and (n + 1)-free, then E(P ) are generically E inj (Q)-ergodic.In particular, we have that: While it is tempting to conjecture that one should be able to replace E inj (Q) with the full shift E(Q) above, one of three "red herrings" from [BFKL15] suggests that Theorem 2 could actually be sharp; see Remark 7 below.
Baldwin, Koerwien and Laskowski defined for each n ≥ 2 a countable, locally finite, ultrahomogeneous structure N BKL n whose L ω 1 ω -theory has models of size ℵ n−1 but not of size ℵ n ; see [BKL17].The construction of the ℵ n−1 -sized model was based on the fact that the category Age(N BKL n ), of all finite substructures of N BKL n , has mdisjoint amalgamation for all m ≤ n, and the lack of an ℵ n -sized models was based on the fact that the algebraic dimension of N BKL n is n.Using the amalgamation properties of N BKL n it was shown in [KP21] that the collection: consists of pairwise incomparable equivalence relations under * -reductions, i.e., under reductions which preserve comeager sets.However, the question of how they compare with respect to ≤ B remained open.Since for every countable structure N , for which Age(N ) satisfies the disjoint m-amalgamation for all m ≤ n+ 1, the group Aut(N BKL n+1 ) satisfies the assumption of P in Theorem 2, we have the following.
Then, E is generically F -ergodic.In particular: In contrast to [KP21], which uses a classical argument based on the amalgamation properties of N BKL n , our arguments here employ set-theoretic symmetric model techniques, as developed in [Sha21] and [Sha19], and they directly utilize the "algebraicity" properties of N BKL n .When it comes to the second statement of Corollary 3 (the Borel irreducibility conclusion), there is yet a third way to approach it.Namely, using a set-theoretic pinned cardinality argument, as in [Zap11,LZ21], and the fact that the L ω 1 ,ω -theory of N BKL n has models of size ℵ n−1 but not of size ℵ n ; see Section 5.However, the analysis we provide here is sharper in a few ways.First, while pinned cardinality methods suffice for proving Borel irreducibility, Theorem 2 establishes stronger generic ergodicity results.Second, using our methods we are able to distinguish between two equivalence relations which have the same pinned cardinals.

Amalgamation properties −→
Pinned equivalence relations form a bounded complexity class within the Borel reduction hierarchy, which has a forcing-theoretic definition.The most central example of a non-pinned equivalence relation is = + , which can be identified to E(Sym(N)).Answering a question of Kechris, Zapletal showed that = + is not the minimal unpinned Borel equivalence relation [Zap11].Essential in his proof was the notion of pinned cardinality, which is a measure of how far an equivalence relation is from being pinned.In particular, he provided a transfinite sequence of unpinned equivalence relations below = + that is linearly ordered under ≤ B .The first ω examples in this sequence share many properties with the sequence of the E(Aut(N BKL n )) in Corollary 3 above.The minimal example in Zapletal's sequence is defined as follows: first, [Zap11] considers a certain Borel graph on R, which has cliques of size ℵ 1 but no larger, and then defines Z ⊂ R N as the set of all injective sequences of reals which enumerate a clique in this graph.The restricted equivalence relation = + ↾ Z is then unpinned and strictly below = + in Borel reducibility.The key point distinguishing = + ↾ Z is that its pinned cardinality is ℵ + 1 , while the pinned cardinality of = + is c + .The equivalence relation E(Aut(N BKL 2 )) has pinned cardinality ℵ + 1 as well.Hence, arguments based on pinned cardinality cannot separate it from = + ↾ Z.However, as a consequence of Theorem 2 we prove the following.)) is generically = + ↾ Z-ergodic and hence: ))?If not, do they form a minimal basis for non-pinned equivalence relation?Question 6. Can we replace E inj (Q) with the full shift E(Q) in Theorem 2?
To appreciate the subtleties in addressing Question 6, notice that, despite its bounded algebraic dimension, Q can potentially admit as complex actions as Sym(N): ), for some n ≥ 2. While by Corollary 3 we have that E inj (Sym(N)) ≤ B E inj (Q), it turns out that any orbit equivalence relation that is induced by an action of Sym(N) can be Borel reduced to some action of Q.This follows from a classical result of Mackey [Gao08, Theorem 3.5.2]and the fact that Q contains a closed subgroup which admits a continuous homomorphism onto Sym(N).The latter constitutes, essentially, one of the three "red herrings" from [BFKL15].Indeed, N BKL n is the Fraïssé limit of a Fraïssé class which satisfies the disjoint amalgamation property; see Example 10.Hence, by Theorem 1.10.(1) in [BFKL15], the domain N of the structure N BKL n can be endowed with an equivalence relation E which partitions it into countably many disjoint pieces so that: every permutation of Sym(N/E) lifts to some element of the subgroup Q E of Q, consisting of all E-preserving elements of Q, i.e., g ∈ Q E if and only if aEb ⇐⇒ gaEgb.
Acknowledgments.We would like to thank A. Kruckman and S. Allison for numerous enlightening and inspiring conversations.

Definitions and preliminaries
2.1.Borel reductions and strong ergodicity.Let E and F be analytic equivalence relation on the Polish spaces X and Y , respectively.A Borel reduction from E to F is any Borel function f : X → Y , with x E x ′ ⇐⇒ f (x) F f (x ′ ), for all x, x ′ ∈ X.If there exists a Borel reduction from E to F , we say that E is Borel reducible to F and we denote this by ≤ B .We say that E is smooth if E ≤ B = R , where = R is equality on R. We say that E is essentially countable if E ≤ B F , where F is a Borel equivalence relation with countable F -classes.We say that E is classifiable by countable structures if E ≤ B ≃ iso , where ≃ iso is the isomorphism equivalence relation on the Polish space of all L-structures on N for some first order language L; see [Kan08b,12.3], [Gao08], [Hjo00].
Often the non-existence of a Borel reduction from E to F is a consequence of a strong ergodicity phenomenon.A Baire-measurable homomorphism from E to F is any Baire measurable function f : X → Y , with xEx ′ =⇒ f (x)F f (x ′ ), for all x, x ′ ∈ X.We say that E is generically F -ergodic if any Baire measurable homomorphism from E to F maps a comeager subset of X into a single F -class.For example, E is generically = R -ergodic if and only if E is generically ergodic, that is, every E-invariant subset of X is either meager or comeager.Notice that if every E-class is meager in X and E is generically If G X is a continuous action of a Polish group on a Polish space X, then the associated orbit equivalence relation is the relation E G X on X defined by: 2.2.Bernoulli shifts and automorphism groups.Let Sym(N) be the Polish group of all permutations g : N → N of N endowed with the pointwise convergence topology.A Polish permutation group P is any closed subgroup of Sym(N).The (left) action of P on N, by (g, a) → g(a), induces a (left) action of P on the space R N , of all maps x : N → R, by (g, x) → gx, where (gx)(a) = x(g −1 (a)).Following [KMPZ20], we call P R N the Bernoulli shift of P and we denote by E(P ) the associated orbit equivalence relation on R N .The injective part of the Bernoulli shift is the restriction of the above action to the Several connections between the Borel reduction complexity of E(P ) and the dynamical properties of P are summarized in the introduction.Often these dynamical properties of P are reflections of combinatorial and model-theoretic properties of first order structures.The main observation here is that P can always be expressed as an automorphism group Aut(N ) of some countable structure N .A first order language L is relational if it only contains relation symbols.Let N be an L-structure with domain N.For every A ⊆ N we denote by A N the smallest substructure of N containing every element of A in its domain.We say that N is ultrahomogeneous if for every bijection f : A → B between finite subsets of N which induces an isomorphism from A N to B N , we have that f extends to an automorphism of N .The L-algebraic closure acl N (A) of A ⊆ N in N is the set of all b ∈ N for which there is an L-formula ϕ with parameters from A so that N |= ϕ(b) and {c ∈ N : N |= ϕ(c)} is finite.In general, if P := Aut(N ), then the algebraic closure A → [A] P defined in the introduction does not coincide with its model-theoretic counterpart A → acl N (A).However, we can always enrich the structure N and make the two notions equal: Lemma 8.If P is a Polish permutation group, then there is exists ultrahomogeneous L-structure N = (N, . ..), on a countable relational language L, so that P = Aut(N ), and with [A] P = acl N (A), for every finite A ⊆ N.
Proof.This follows from the usual "orbit completion" argument; e.g.see [Hod93,Theorem 4.1.4]or [BK96, Section 1.5]: for every n ∈ 1, 2, . .., and for every orbit Clearly N is ultrahomogeneous and P = Aut(N ) follows from the fact that P a closed subgroup of Sym(N).Assume now that b has a finite orbit under the stabilizer P ā of ā = (a 1 , . . ., a n−1 ).Notice that there are only finitely many relations which are satisfied by sub-tuples of (ā, b).Hence, the conjunction of them is an Lformula ϕ(x, y).Notice that ϕ(x, y) completely captures the quantifier-free type of ϕ(ā, b).By ultrahomogeneity there are only finitely many c so that N |= ϕ(ā, c).
Here we will be interested in the following examples.
Example 9.If P = Sym(N) is the full symmetric group, then E(Sym(N)) is Borel bireducible with the equivalence relation = + of countable sets of reals, where Example 10.Fix n ≥ 2 and consider the language L n which contains countably many n-ary relation symbols R 0 , R 1 , R 2 , . . .and countably many n-ary function symbols f 0 , f 1 , f 2 , . ... It follows by [BKL17], that there is a unique up to isomorphism ultrahomogenous structure N which satisfies the following properties: (1) N is locally finite, i.e., dom( (5) every finite L n -structure which satisfies the above (1)-(4) embeds in N .
We denote this structure by N BKL ) also n-free.By considering the Bernoulli shifts of these automorphism groups we have the following collection of equivalence relations:

Labelled logic actions.
Let L be a countable language and let Str(L, N) be the Polish space of all L-structures on domain N; see [Kec95,Gao08].The logic action is the action (g, N ) → gN of Sym(N) on Str(L, N), which is defined by setting for all relations R ∈ L and all functions f ∈ L: If σ is an L ω 1 ,ω -sentence then we will denote by Str(σ, N) the Sym(N)-invariant Borel subset of Str(L, N) which consists of all N with N |= σ; see [Kec95,Gao08].In both cases we denote by ≃ iso the associated orbit equivalence relation.
As in [KP21], we also consider the labelled logic action Sym(N) Str R (L, N), where Str R (L, N) := R N × Str(L, N) and (g, (x, N )) → (gx, gN ).The injective part of the labelled logic action is the restriction of the above action to Sym(N)- Similarly we define the labelled logic action Sym(N) Str R (σ, N) on models of an L ω 1 ,ω -sentence σ, together with its injective part Str inj R (σ, N).We will denote by ≃ iso the associated orbit equivalence relation.For every countable L-structure N we have an Aut(N )-equivariant embedding of the Bernoulli shift with g → g and x → (x, N ).Notice that this induces a Borel reduction: Notice that the above map embeds the injective part of the Bernoulli shift to the injective part of the labelled logic action.

A symmetric model argument
One of the many early consequences of forcing was the independence of the axiom of choice (AC) from the Zermelo-Fraenkel (ZF) axioms.To construct a model of ZF+¬AC, Cohen started with the Cohen forcing P which adds a countable sequence of Cohen reals x n : n ∈ N , but instead of taking the full forcing extension V [G], he considered the substructure of V [G] generated by all P-names which are invariant under the natural action of Sym(N) on P by index permutation; see [Kan08a].The resulting basic Cohen model coincides with the smallest extension V ({x n : n ∈ N}) of V in V [G] which contains the set {x n : n ∈ N} and satisfies ZF.The failure of AC in this model comes from the fact that V ({x n : n ∈ N}) contains no well-ordering of {x n : n ∈ N}.The latter is established by developing a theory of supports for definable sets in V ({x n : n ∈ N}).
Since {x n : n ∈ N} can be alternatively thought as the classifying = + -invariant of the generic point (x n ) ∈ R N , one would hope that the ergodic theoretic properties of = + are reflected to the structure of definable sets in V ({x n : n ∈ N}).This point of view is adopted in [Sha21,Sha19], where a general translation principle is established between strong ergodic properties of equivalence relations which are classifiable by countable structures and the structure of definable sets in the associated symmetric models; see Lemma 13 below.
In this section we build further on the symmetric model techniques developed in [Sha21,Sha19] and use the resulting theory to prove Theorem 2 and Corollary 3 from the introduction.In the process we develop a robust theory of supports, similar to the one in the basic Cohen model, for symmetric models which are associated to locally finite permutation groups; see Lemma 19.
3.1.Symmetric models.For the rest of this section, and through the paper, we work over the universe V = L.This can be done as all the equivalence relations we talk about are in L, and the questions of Borel reducibility and strong ergodicity between them are absolute.This restriction to L is not necessary, and will never in fact be used.For this reason we always use the notation V instead of L. We only do this to simplify the introduction below to ordinal definability and minimal extensions, and provide a more concrete context for the reader who is less familiar with the subject.
Let V [G] be a generic extension of V with respect to some forcing notion P and let A ∈ V [G].Let V (A) be the minimal transitive substructure W of V [G] such that W satisfies ZF, W extends V , and W contains A. This is precisely the Hajnal relativized L-construction L(A); see [Jec03,p. 193].1 Recall the notion of ordinal definability [Jec03,p. 194]: if W is a model of ZF and A is a set in W , then say that S is ordinal definable over A (in W ), denoted S ∈ OD(A) W , if S ∈ W and there is a formula ϕ, ordinal parameters b and parameters ā from the transitive closure of A, so that S = {s ∈ W : W |= ϕ(s, A, ā, b)}.Here we will be working with models W of ZF which are generic extensions of V , or intermediate extensions of such, and therefore they all have the same ordinals with V .
Let HOD(A) W be the subclass of W consisting of all sets S ∈ W so that every element of the transitive closure tc({S}) of {S} is in OD(A) W .If W is any model of ZF and A a set in W , then HOD(A) W is a transitive model of ZF.See the proof of [Jec03, Theorem 13.26] and the following discussion.
Fact 11.For every S ∈ V (A) there is some formula ϕ, parameters ā from the transitive closure of A and v ∈ V such that S = {s ∈ V (A) : In this case we say that ā is a definable support for S.
Proof.Work inside V (A), and consider HOD(A) V (A) , the class of sets which are hereditarily ordinal definable over A, as calculated in V (A).Then HOD(A) V (A) is a transitive model of ZF containing A and all the ordinals.By the minimality of V (A), it follows that V (A) is equal to HOD(A) V (A) .So every set in V (A) is definable as in the statement of the fact.2

Absolute classification.
In what follows, we study the Borel complexity of various analytic equivalence relations E, such as the orbit equivalence relations of Bernoulli shifts E(P ), by analysing models of the form V (A) where A is a classifying invariant for E. This is possible when E admits a "reasonable enough" classification, as follows.
Let X ∈ V be a Polish space and E ∈ V an analytic equivalence relation on X.A complete classification of (X, E) in V is any assignment x → A x in V given by a first-order formula ϕ(x, A x ) of set theory, so that for all x, y ∈ X we have: We write x → ϕ A x when the assignment x → A x is given by ϕ.The most "canonical" such assignment would be to map each x to its E-equivalence class [x] E .While this assignment is canonical, it does not give us any new information about the classification problem (X, E) and it does not behave nicely when passing to forcing extensions.For example, even when x ∈ X ∩ V , [x] E as computed in V [G] may not coincide with [x] E as computed in V , since the forcing may add new reals in the E-class of x.A complete classification x → ϕ A x is an absolute classification if whenever W ⊇ V is a model of ZF, the assignment x → ϕ A x as computed in W , is still a complete classification for (X, E) as computed in W , and for all x ∈ X ∩ V we have that A x as computed in W , is equal to A x , as computed in V .
Example 12. Recall E(Sym(N)) from Example 9.The assignment defined on R N is a complete classification when restricted to the comeager subset Inj(N, R) of R N , consisting of all injective functions from N to R. It is easy to check that it is also an absolute classification.
More generally, if (X, E) is classifiable by countable structures, then it also admits an absolute classification x → A x , where A x is a hereditarily countable set.Such absolute classification is attained simply by post-composing the Borel reduction which witnesses classifiability by countable structures with the usual Scott analysis procedure; see [Gao08,Hjo00].Being classifiable by countable structures is an absolute condition.For the absoluteness of the Scott analysis see also [Fri00,Lemma 2.4].
The following theorem establishes the main connection between Borel reduction complexity and symmetric models which we are going to use later on.Notice that if A x below is computed with respect to the assignment of Example 12, where x ∈ R N is Cohen generic, then V (A x ) will coincide with the basic Cohen model.

Lemma 13 ([Sha19]
). Suppose E and F are Borel equivalence relations on X and Y respectively and x → A x and y → B y are absolute classifications of E and F respectively.Then, the following are equivalent.
(1) For every Borel homomorphism f : (X 0 , E) → (Y, F ), where X 0 ⊆ X is non-meager, f maps a non-meager set into a single F -class; (2) If x ∈ X is Cohen-generic over V and B is a potential F -invariant in V (A x ) which is definable only from A x and parameters in V , then B ∈ V .
Here, by "B is a potential F -invariant" we simply mean that there is a y in some further generic extension, such that B = B y .For example, any set of reals in V (A x ) is a potential = + -invariant for the assignment from Example 12, since we can always move to a forcing extension which collapses it to a countable set.
Remark 14. Suppose E is generically ergodic.Then condition (1) in Lemma 13 is equivalent to: E is generically F -ergodic.In our examples below E will be generically ergodic and Lemma 13 will be used to conclude generic F -ergodicity.
Proof sketch of Lemma 13 (2) =⇒ (1).We briefly sketch the proof of the direction which we are going to use in this paper.Assume (2), and let f be as in (1).Let x be a generic Cohen real in the domain of f , and let A = A x and B = B f (x) .By absoluteness of x → A x and y → B y , the set B can be defined in V (A) from A as follows: B is the unique set satisfying that in any generic extension of . By assumption (2), B is in V .By the forcing theorem, there is a condition p in the Cohen poset (a non-meager set), forcing that B f ( ẋ) = B.In particular, all generics x extending p are mapped to the same F -equivalence class.Now the set of all x ∈ p which are generic, over some large enough countable model, is a non-meager set sent to the same F -class by f .Next we develop absolute classifications for (a comeager part of) the Bernoulli shift P R N .By Lemma 8, it will suffice to consider permutation groups P of the form Aut(N ), where N is some countable L-structure.
Definition 15.Let L be a countable language.A countable L-structure on R is any L-structure A = (A, . ..) whose domain A is a countable subset of R.
Example 16.Let N be a countable L-structure and let the Aut(N ) Inj(N, R) be the injective part of the Bernoulli shift.Consider the assignment: x → A x mapping every x ∈ Inj(N, R) to the countable L-structure A x = (A x , . ..) on R, where A x = {x n : n ∈ N}, and for all relations R ∈ L and functions f ∈ L we have: x is an absolute classification for E(Aut(N )) restricted to Inj(N, R).
We similarly we have the following absolute classification for the labelled logic action from Section 2.3.
Example 17.Let L be a countable language and let Sym(N) Str inj R (L, N) be the injective part of the labelled logic action.Consider the assignment: which maps every (x, N ) ∈ Inj(N, R) × Str(L, N) to the countable L-structure A x,N on R, which is computed as A x in Example 16 with respect to N .For every g ∈ Sym(N) and all (x, N ) ∈ Inj(N, R) × Str(L, N), we have that: and since (gx) g(n) = x n , for all n ∈ N, it follows that A gx,gN |= R(x n 0 , . . ., x n k−1 ).Hence, (x, N ) → A x,N is a complete classification of ≃ iso on Str inj R (L, N).It is clear that it is also an absolute classification.
3.3.Proofs of the main results.We may now formulate and prove our main theorem which is a common generalization of both Theorem 2 and Theorem 4.
Theorem 18.Let n ∈ ω and let P ≤ Sym(N) be a locally finite, (n + 1)-free Polish permutation group.Let also σ be an L ω 1 ,ω -sentence so that for every N ∈ Str(σ, N) and every b 0 , . . ., b n ∈ dom(N ), there is some i ≤ n so that b i is in the L-algebraic closure acl N ({b j : j = i}) of the remaining points.Then, E(P ) is generically ergodic with respect to Str inj R (σ, N), ≃ iso .Before we proceed to the proof of Theorem 18, we recover from it Theorem 2.
Proof of Theorem 2 from Theorem 18.Let P and Q be as in the statement of Theorem 2. By Lemma 8 we have that Q = Aut(N ) for some countable ultrahomogeneous L-structure.with the property that [A] Q = acl N (A), for every finite A ⊆ N. Let σ be the Scott sentence of N .Since every N ′ ∈ Str(σ, N) is isomorphic to N , we have that Aut(N ′ ) is of algebraic dimension less than or equal to n.By the conclusion of Theorem 18 and since, by Section 2.3, we have that We now turn to the proof of Theorem 18.We first fix some notation regarding Cohen forcing on R N .Consider the poset P whose elements are finite partial maps p from N such that p(n) ⊆ R is an open interval with rational endpoints for every n in the domain of p.A condition p extends q, denoted by p ≤ q, if dom(p) ⊇ dom(q) and p(n) ⊆ q(n) for every n ∈ dom(q).We fix some P-generic G ⊆ P over V .Working in V [G], define x = (x n ) to be the unique element of R N such that x n ∈ p(n), for every p ∈ G with n ∈ dom(p).We will refer to x as the Cohen generic point of R N .In what follows we also fix some countable L-structure M in V and let be the image of (x, M) under the absolute classification defined in Examples 16,17.We finally let We will need the following lemma which is interesting on its own right.The main point is that if M has enough symmetries, then the associated symmetric model V (A) admits an analysis of the definable supports similar to the one in [HL71], for the basic Cohen model.
Lemma 19.Assume that Aut(M) is locally finite.Take any set S ∈ V (A) such that S ⊆ V , and let ā be a definable support for S in V (A).
Proof.By local finiteness we may assume without any loss of generality that ā enumerates [ā] Aut(A) .Fix a formula φ and fix some parameter v ∈ V such that . By reflection, let ξ be a large enough ordinal so that S ⊆ V ξ and so that for any p ∈ P we have: Since the forcing relation is definable in V , S ′ is in V (ā).We will conclude the proof of the lemma by showing that S = S ′ .
To see that S ′ ⊆ S, let s ∈ S ′ and take any condition p ∈ G such that n 0 , ..., n k−1 ∈ dom(p).By definition of S ′ and by the choice of ξ, p ϕ V ( Ȧ) (š, v, ā, Ȧ).Since p ∈ G we have that ϕ V (A) (s, v, ā, A) holds in V [G], and therefore, ϕ(s, v, ā, A) holds in V (A), and so s ∈ S. The converse direction S ⊆ S ′ follows from the next claim.
Claim.For any p, q ∈ P, if a i ∈ p(n i ) ∩ q(n i ) for i < k, then for any s ∈ V , Proof.Assume towards a contradiction that p, q are as in the claim but we additionally have that p ϕ V (A) (x, v, ā, Ȧ) and q ¬ϕ V (A) (x, v, ā, Ȧ).Without loss of generality we may also assume that p ∈ G. Let L := {m 0 , . . ., m ℓ−1 } ⊆ N = dom(M) be disjoint from K := {n 0 , ..., n k−1 }, so that L ∪ K contains the domains of p and q.Since ā = [ā] Aut(A) , we have that K = [K] Aut(M) .Hence, for every ℓ ∈ L the orbit of ℓ under the stabilizer Aut(M) K of K is infinite.But then, by the Neumann lemma [Neu76, Lemma 2.3], there is an infinite sequence (π j ) j in Aut(M) so that: (1) π j fixes K pointwise; and (2) π i (L) ∩ π j (L) = ∅ for all i = j.By genericity of G there is some j such that x • π j extends q.Set π := π j and let G ′ := {p • π : p ∈ G} and x ′ := x • π.We have that: By (ii),(iii), and since x ′ extends q, we have that ¬ϕ On the other hand, since Ȧ[G] = A and x extends p, we have that ϕ . We conclude that V (A) satisfies both ¬ϕ(s, v, ā, A) and ϕ(s, v, ā, A), a contradiction.
Let S ∈ V (A) with S ⊆ V and let F ⊆ A = dom(A) be a set.We say that F is a support for S if S ∈ V (F ).Notice that by mutual genericity, for any two We define the support supp(S) of S to be the intersection of all F ⊆ A which are supports for S.Under the assumptions of the previous lemma, every S as above has a finite support supp(S) with S ∈ V (supp(S)).
We may now conclude with the proof of Theorem 18.
Proof of Theorem 18.We will apply Lemma 13 (2) =⇒ (1).For that, let x ∈ Inj(N, R) be the generic associated to G and set A := A x the corresponding invariant for the M-shift.Let also B = (B, ...) be a potential invariant for Str inj R (σ, N), ≃ iso in V (A) which is definable only from A and parameters from V .Recall by potential invariant we mean that there are y ∈ Inj(N, R) and N ∈ Str(σ, N), in some further generic extension of V (A), so that B := B y,N .
Since the definable support of B is ∅, the rest follows from the next claim, which together with Lemma 19 implies that that B ∈ V and the desired generic ergodicity follows from Lemma 13 (2) =⇒ (1).
Claim.Set B := dom(B).We have that B ⊆ V .Proof of Claim.Assume towards a contradiction that there is some b ∈ B not in V .It follows that the support of F := supp({b}) ⊆ A of b is not empty.Let m 0 , . . ., m k−1 ∈ N with F = {x m i : i < k}.Fix a condition p forcing the above.By local finiteness of M we may choose some finite K ⊆ N = dom(M) which contains dom(p) ∪ {m 0 , . . ., m k−1 } and it is algebraically closed, i.e., [K] Aut(M) = K.Since M is (n + 1)-free, there are π 0 , . . ., π n ∈ Aut(M) so that for all i ≤ n we have that Notice now that [∅] Aut(M) = ∅.Indeed, any (n + 1)-free permutation group is in particular 1-free; and any locally finite 1-free permutation group P satisfies [∅] P = ∅.Hence, the orbit of each ℓ ∈ [π i (K)] Aut(M) is infinite for all i ≤ n.By the Neumann lemma [Neu76, Lemma 2.3], for every finite E ⊆ N there is some π E ∈ Aut(M), so that (π E • π i )(K) ∩ E = ∅ for all i ≤ n.Hence, by the genericity of x ∈ Inj(N, R) we may assume that for all i ≤ n we have that x i := x • π i extends p.Let G i := {q • π i : q ∈ G} be the associated filters.Since all π i above can be chosen in V , we have that each G i is generic and For each i ≤ n, working in V [G i ], since p ∈ G i and x i E(P ) x, we have that: the realization of Ȧ is the same A; the realization of Ḃ is the same B; the interpretation of Ḟ is some F i ⊆ A such that {F i : i = 0, ..., n} are pairwise disjoint; and F i is the support of the interpretation b i of ḃ in V [G i ].Since p ∈ G i , we have that b i ∈ B for all i ≤ n.By assumption, there is some i such that, in B, b i is in the L-algebraic closure acl B ({b j : j = i}).Here we are using that "for all N ∈ Str(σ, N) and every b 0 , . . ., b n ∈ dom(N ), there is some i ≤ n so that b i ∈ acl N ({b j : j = i})" is absolute.
It follows that b i is definable in V (A) from {b j : j = i} and B. This is because there is a finite set of reals, definable from {b j : j = i} and B, which contains b i .Recall that B is definable from A, by assumption.Since b j : j = i is definable from F j : j = i , it follows that j : j =i F j is a definable support for b i in V (A) and by Lemma 19 we have that b i ∈ V ([ j : j =i F j ] Aut(A) ).By mutual genericity we have ))-ergodic.
In particular, E inj (Aut(N BKL k+1 )) is not Borel reducible to E inj (Aut(N BKL k )). Finally we note that these equivalence relations increase in Borel complexity.
Proposition 20.There is a Borel reduction from E(Aut(N

This map sends injective sequences to injective sequences, thus reducing
)) as well.
Proof.Let (A, R j , s j ) j∈ω be a BKL n structure.Define a structure (A, Rj , sj ) j∈ω as follows, where Rj are n + 1-ary relations and sj are n + 1-ary function symbols.
It is easy to see that (A, Rj , sj ) j∈ω is a BKL n+1 -structure and that the map sending a BKL n -invariant (A, R j , s j ) j∈ω to the BKL n+1 -invariant (A, Rj , sj ) j∈ω gives the desired reduction.

Unpinned equivalence relations
We recall the definition of pinned equivalence relations and pinned cardinals and explain the results in [Zap11] about the equivalence relation = + ↾ Z.We then present the equivalence relations E inj (Aut(N BKL n+1 )) in this context and give another proof that E inj (Aut(N BKL n+1 ) ≤ B E inj (Aut(N BKL n ), a weak form of Corollary 3. Finally, we prove Theorem 4.

Definition 21 ([LZ21]
).Let E be an analytic equivalence relation on a Polish space X.Let P be a poset and τ a P-name.
• The name τ is E-pinned if P × P forces that τ l is E-equivalent to τ r , where τ l and τ r are the interpretation of τ using the left and right generics respectively.• If τ is E-pinned the pair P, τ is called an E-pin.
• An E-pin P, σ is trivial if there is some x ∈ X such that P σ E x.
• E is pinned if all E-pins are trivial.
• Given two E-pins P, σ and Q, τ , say that they are Ē-equivalent, P, σ Ē Q, τ , if P × Q σ E τ .• The pinned cardinal of E, κ(E), is the smallest κ such that every E-pin is Ē-equivalent to an E-pin with a post of size < κ.
For equivalence relations which are classifiable by countable structures, the pinned cardinal can be calculated more easily by the size of "potential invariants", as explained below.
Remark 23.Ulrich, Rast, and Laskowski [URL17] have independently developed the notion of pinned cardinality in the special case of isomorphism relations.The presentation below is essentially equivalent to the one in [URL17].
Assume that E is a Borel equivalence relation which admits an absolute classifiable x → A x .Say that a set A is a potential E-invariant if in some forcing extension there is an x in the domain of E such that A = A x .If A is a potential invariant for E, say that A is trivial if there is an x in the ground model such that A = A x .
Proposition 24.There is a one-to-one correspondence between • E pins P, τ , and • potential invariants A, such that P, τ is trivial if and only if A is trivial.Specifically, a potential invariant A corresponds to P, τ if and only if P A τ = Ǎ.Assume first that P, τ is an E-pin.Let G l × G r be P × P-generic and let x l , x r be the interpretations of τ according to G l ,G r respectively.Since x l and x r are Erelated, it follows that A = A x l = A xr .Furthermore, this set A is in the intersection V [G l ] ∩ V [G r ], which is equal to V by mutual genericity.If P, τ is trivial there is x ∈ V such that x E x l .In particular, A x = A x l = A. Conversely, if there is x ∈ V with A = A x then x witnesses that P, τ is trivial: given any P-generic G over V , A τ [G] = A = A x , so τ [G] is E-related to x.
Now let A be a potential invariant for E. By assumption there is a poset Q, a generic G and x ∈ V [G] such that A = A x .Let τ be a Q-name such that τ [G] = x.Fix a condition q ∈ Q such that q forces that A τ = A and define P = Q ↾ p.Now P, τ is an E-pin such that P A τ = A. To find an unpinned equivalence relation strictly below = + , Zapletal [Zap11] restricted = + to an invariant subset, in the following way, precisely to limit the size of its potential invariants.
Example 27 ([Zap11]).Fix Borel functions f n : R → R, n ∈ N, such that the graph G on R, defined by x G y ⇐⇒ ∃n(f n (x) = y ∨ f n (y) = x), has cliques of size ℵ 1 , but no cliques of size greater than ℵ 1 (see [Zap11, Fact 2.2]).Let Z ⊆ R ω be the G δ set of all injective countable sequences of reals which enumerate a clique in G, and consider the equivalence relation = + ↾ Z on Z.The potential invariants of = + ↾ Z are precisely all sets of reals which form a clique in G, and those have cardinality ≤ ℵ 1 .Therefore the pinned cardinal of = + ↾ Z is ℵ + 1 .Corollary 28 ([Zap11]).= + is not Borel reducible to = + ↾ Z.
Proof.By the absoluteness of Borel reducibility, we may work in some forcing extension where the continuum hypothesis fails, that is, c > ℵ 1 .In this model the pinned cardinal of = + is strictly greater than that of = + ↾ Z, so the corollary follows from Lemma 22.
Example 29.By Example 16, a potential invariant for E inj (Aut(N BKL n+1 )) is a set of reals together with a BKL n+1 -structure on it.Since BKL n+1 models are of size at most ℵ n , and there is a model of size ℵ n , it follows that the pinned cardinal of )) and = + ↾ Z have the same pinned cardinal ℵ + 1 .Using Theorem 18 we are able to separate them.
Proof of Theorem 4. Consider a language with countably many function symbols h i and the theory σ asserting that for any x and y there is some i such that either h i (x) = y or h i (y) = x.Notice that every structure in Str(σ, N) has algebraic dimension less than or equal to 1.By Theorem 18 and the observation in Example

n.
It is easy to see that for all finite A ⊆ N we have that dom( A N ) = [A] Aut(N BKL n ) = acl(A).Hence, properties (1),(4) above imply that the permutation group Aut(N BKL n ) is locally finite and of algebraic dimension at most n.Clearly acl(∅) = ∅.Finally, by ultrahomogeneity of N BKL n and the fact that Age(N BKL n ) has the disjoint m-amalgamation for all m ≤ n-see [BKL17]we have that Aut(N BKL n and therefore ȧ[G ′ ] = ā, by (1) above; (iii) Ȧ[G ′ ] = Ȧ[G] = A, since x E(Aut(M)) x ′ ; see Example 16.
contradicting the assumption that supp(b) = ∅.4. Proof of Corollary 3 Recall from Example 10 that for k ≥ 2, Aut(N BKL k ) is locally finite, k-free, and has algebraic dimension precisely k.It follows from Theorem 2 that E inj (Aut(N BKL k+1 )) is generically E inj (Aut(N BKL k

Corollary 25 .
The pinned cardinal of E is κ if and only if any potential E-invariant is trivial in a generic extension by a poset of size < κ.Example 26.Consider the equivalence relation = + on R N with the complete classification R N ∋ x → {x(n) : n ∈ N}.The potential invariants of = + are precisely all sets of reals.Therefore the pinned cardinal of = + is c + .
in a model where |R| ≥ ℵ n .It follows from Lemma 22 that E inj (Aut(N BKL n+1 )) is not Borel reducible to E inj (Aut(N BKL