#  Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic
dimension

Aristotelis Panagiotopoulos  Institut für Mathematische Logik und
Grundlagenforschung, Westfalische Wilhelms-Universität Münster, Einsteinstr.
62, 48149 Münster, Germany<EMAIL_ADDRESS>http://apanagiotopoulos.org and  Assaf Shani  Department of Mathematics,
Harvard University, Cambridge, MA 02138, USA<EMAIL_ADDRESS>http://people.math.harvard.edu/ shani/

###### Abstract.

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 [KP] 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.

###### 2000 Mathematics Subject Classification:

03E15, 54H05, 03E40, 03E75

##  1\. Introduction

Let $G$ be a Polish group acting continuously $G\curvearrowright X$ 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_ $\leq_{B}$, as well as by its _strong ergodic
properties_.

When it comes to Polish permutation groups $P\leq\mathrm{Sym}(\mathbb{N})$, an
action which often serves as bridge between structural and dynamical
properties of $P$ is the Bernoulli shift
$P\curvearrowright\mathbb{R}^{\mathbb{N}}$ of $P$. For example, if $E(P)$ is
its orbit equivalence relation, then:

  1. (1)

$E(P)$ is _smooth_ if and only if $P$ is compact, see [KMPZ];

  2. (2)

$E(P)$ is _essentially countable_ if and only if $P$ is locally compact,
[KMPZ];

  3. (3)

$E(P)$ is _classifiable by CLI-group actions_ if and only if $P$ is CLI,
[KMPZ];

  4. (4)

$E(P)\lneq_{B}E(P\wr\Gamma)$ for various countable groups $\Gamma$, [CC20];

  5. (5)

$E(\mathbb{Z}\wr\mathbb{Z})$ is not _classifiable by TSI-group actions_ ,
[AP20].

  6. (6)

$E(\Gamma\wr\Gamma)\not\leq_{B}E(\Delta\wr\Delta)$ for various countable
$\Gamma,\Delta$, [Sha19].

In fact, the above results still hold if we replace all shifts with their
_injective parts_. By the injective part $E_{\mathrm{inj}}(P)$ of $E(P)$ we
mean the restriction of $E(P)$ to the $P$-invariant subset
$\mathrm{Inj}(\mathbb{N},\mathbb{R})$ of $\mathbb{R}^{\mathbb{N}}$, 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 $P\leq\mathrm{Sym}(\mathbb{N})$ and every
$A\subseteq\mathbb{N}$, we denote the pointwise stabilizer of $A$ by $P_{A}$.
The algebraic closure of $A$ with respect to $P$ is the set:

$[A]_{P}:=\\{a\in\mathbb{N}\colon\text{the orbit }P_{A}\cdot a\text{ is
finite}\\}.$

The algebraic dimension of the permutation group $P$ is simply the dimension
of the closure operator $A\mapsto[A]_{P}$, where $A$ ranges over finite
subsets of $\mathbb{N}$. More precisely:

######  Definition 1.

The algebraic dimension of a permutation group $P\leq\mathrm{Sym}(\mathbb{N})$
is the smallest $n\in\omega\cup\\{\infty\\}$, so that for every
$A\subseteq\mathbb{N}$, with $|A|=n+1$, there is $a\in A$ with
$a\in[A\setminus\\{a\\}]_{P}$. If $n\not=\infty$, we say that $P$ is of
bounded algebraic dimension.

When $P:=\mathrm{Aut}(\mathcal{N})$ is the automorphism group of a countable
structure $\mathcal{N}$, 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
$A\mapsto[A]_{P}$ forms a pregeometry, i.e., when it additionally satisfies
the _exchange principle_. Examples include the case where $\mathcal{N}$ is the
$n$-dimensional $\mathbb{Q}$-vector space or the complete $n$-branching tree.
It is easy to see that this additional assumption forces
$P=\mathrm{Aut}(\mathcal{N})$ 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
$\boldsymbol{n}$-free, if for all finite $A\subseteq\mathbb{N}$, there are
$g_{0},\ldots,g_{n-1}\in P$ so that for all $i\in\\{0,\ldots,n-1\\}$,
$[g_{i}\cdot A]_{P}$ and $[\bigcup_{j:j\neq i}(g_{j}\cdot A)]_{P}$ are
disjoint.

######  Theorem 2.

Let $P,Q\leq\mathrm{Sym}(\mathbb{N})$ be Polish permutation groups and let
$n\in\omega$. If the algebraic dimension of $Q$ is at most $n$ and $P$ is
locally finite and $(n+1)$-free, then $E(P)$ are generically
$E_{\mathrm{inj}}(Q)$-ergodic. In particular, we have that:

$E(P)\not\leq_{B}E_{\mathrm{inj}}(Q).$

While it is tempting to conjecture that one should be able to replace
$E_{\mathrm{inj}}(Q)$ 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 $n\geq 2$ a countable,
locally finite, ultrahomogeneous structure $\mathcal{N}^{\mathrm{BKL}}_{n}$
whose $\mathcal{L}_{\omega_{1}\omega}$-theory has models of size
$\aleph_{n-1}$ but not of size $\aleph_{n}$; see [BKL17]. The construction of
the $\aleph_{n-1}$-sized model was based on the fact that the category
$\mathrm{Age}(\mathcal{N}^{\mathrm{BKL}}_{n})$, of all finite substructures of
$\mathcal{N}^{\mathrm{BKL}}_{n}$, has _$m$ -disjoint amalgamation_ for all
$m\leq n$, and the lack of an $\aleph_{n}$-sized models was based on the fact
that the algebraic dimension of $\mathcal{N}^{\mathrm{BKL}}_{n}$ is $n$. Using
the amalgamation properties of $\mathcal{N}^{\mathrm{BKL}}_{n}$ it was shown
in [KP] that the collection:

$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2})),E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{3})),\ldots,E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n})),\ldots,E_{\mathrm{inj}}(\mathrm{Sym}(\mathbb{N}))$

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 $\leq_{B}$ remained open.
Since for every countable structure $\mathcal{N}$, for which
$\mathrm{Age}(\mathcal{N})$ satisfies the _disjoint $m$-amalgamation_ for all
$m\leq n+1$, the group $\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1})$
satisfies the assumption of $P$ in Theorem 2, we have the following.

######  Corollary 3.

Assume that $F=E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{m}))$
and either $E=E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n}))$
with $n>m$, or $E=E_{\mathrm{inj}}(\mathrm{Sym}(\mathbb{N}))$. Then, $E$ is
generically $F$-ergodic. In particular:

$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2}))\lneq_{B}E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{3}))\lneq_{B}\ldots\ldots\lneq_{B}E_{\mathrm{inj}}(\mathrm{Sym}(\mathbb{N})).$

In contrast to [KP], which uses a classical argument based on the amalgamation
properties of $\mathcal{N}^{\mathrm{BKL}}_{n}$, our arguments here employ set-
theoretic symmetric model techniques, as developed in [Sha18] and [Sha19], and
they directly utilize the “algebraicity” properties of
$\mathcal{N}^{\mathrm{BKL}}_{n}$. 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, LZ], and the fact that the $\mathcal{L}_{\omega_{1},\omega}$-theory of
$\mathcal{N}^{\mathrm{BKL}}_{n}$ has models of size $\aleph_{n-1}$ but not of
size $\aleph_{n}$; 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.

$\displaystyle\text{Amalgamation properties}\longrightarrow$ [KP]
$\displaystyle\longleftarrow\text{Classical dynamics}$
$\displaystyle\text{Spectrum of
}\mathcal{L}_{\omega_{1}\omega}(\mathcal{N})\longrightarrow$ [Zap11, LZ]
$\displaystyle\longleftarrow\text{Pinned cardinality}$
$\displaystyle\text{Algebraic dimension}\longrightarrow$ Theorem 2
$\displaystyle\longleftarrow\text{Symmetric models}$

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 $E(\mathrm{Sym}(\mathbb{N}))$. 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 $\leq_{B}$. The first
$\omega$ examples in this sequence share many properties with the sequence of
the $E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n}))$ in Corollary 3 above.
The minimal example in Zapletal’s sequence is defined as follows: first,
[Zap11] considers a certain Borel graph on $\mathbb{R}$, which has cliques of
size $\aleph_{1}$ but no larger, and then defines
$Z\subset\mathbb{R}^{\mathbb{N}}$ as the set of all injective sequences of
reals which enumerate a clique in this graph. The restricted equivalence
relation $=^{+}\restriction Z$ is then unpinned and strictly below $=^{+}$ in
Borel reducibility. The key point distinguishing $=^{+}\restriction Z$ is that
its pinned cardinality is $\aleph_{1}^{+}$, while the pinned cardinality of
$=^{+}$ is $\mathfrak{c}^{+}$. The equivalence relation
$E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2}))$ has pinned cardinality
$\aleph_{1}^{+}$ as well. Hence, arguments based on pinned cardinality cannot
separate it from $=^{+}\restriction Z$. However, as a consequence of Theorem 2
we prove the following.

######  Theorem 4.

$E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2}))$ is generically
$=^{+}\restriction Z$-ergodic and hence:

$E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2})){\not\leq_{B}}=^{+}\restriction
Z$

######  Question 5.

Is $Z$ Borel reducible to $E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2}))$?
If not, do they form a minimal basis for non-pinned equivalence relation?

######  Question 6.

Can we replace $E_{\mathrm{inj}}(Q)$ 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 $\mathrm{Sym}(\mathbb{N})$:

######  Remark 7.

Let $Q:=\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n})$, for some $n\geq 2$.
While by Corollary 3 we have that
$E_{\mathrm{inj}}(\mathrm{Sym}(\mathbb{N}))\not\leq_{B}E_{\mathrm{inj}}(Q)$,
it turns out that any orbit equivalence relation that is induced by an action
of $\mathrm{Sym}(\mathbb{N})$ can be Borel reduced to some action of $Q$. This
follows from a classical result of Mackey [Gao09]*Theorem 3.5.2 and the fact
that $Q$ contains a closed subgroup which admits a continuous homomorphism
onto $\mathrm{Sym}(\mathbb{N})$. The latter constitutes, essentially, one of
the three “red herrings” from [BFKL15]. Indeed,
$\mathcal{N}^{\mathrm{BKL}}_{n}$ 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 $\mathbb{N}$ of the structure
$\mathcal{N}^{\mathrm{BKL}}_{n}$ can be endowed with an equivalence relation
$E$ which partitions it into countably many disjoint pieces so that: every
permutation of $\mathrm{Sym}(\mathbb{N}/E)$ lifts to some element of the
subgroup $Q_{E}$ of $Q$, consisting of all $E$-preserving elements of $Q$,
i.e., $g\in Q_{E}$ if and only if $aEb\iff gaEgb$.

### Acknowledgments

We would like to thank A. Kruckman and S. Allison for numerous enlightening
and inspiring conversations.

##  2\. 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
$f\colon X\to Y$, with $x\mathrel{E}x^{\prime}\iff
f(x)\mathrel{F}f(x^{\prime})$, for all $x,x^{\prime}\in X$. If there exists a
Borel reduction from $E$ to $F$, we say that $E$ is Borel reducible to $F$ and
we denote this by $\leq_{B}$. We say that $E$ is smooth if
$E\leq_{B}\;=_{\mathbb{R}}$, where $=_{\mathbb{R}}$ is equality on
$\mathbb{R}$. We say that $E$ is essentially countable if $E\leq_{B}F$, where
$F$ is a Borel equivalence relation with countable $F$-classes. We say that
$E$ is classifiable by countable structures if
$E\leq_{B}\;\simeq_{\mathrm{iso}}$, where $\simeq_{\mathrm{iso}}$ is the
isomorphism equivalence relation on the Polish space of all
$\mathcal{L}$-structures on $\mathbb{N}$ for some first order language
$\mathcal{L}$; see [Kan08]*12.3, [Gao09], [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 $f\colon X\to Y$, with
$xEx^{\prime}\implies f(x)Ff(x^{\prime})$, for all $x,x^{\prime}\in X$. 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 $=_{\mathbb{R}}$-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 $F$-ergodic, then $E\not\leq_{B}F$.

If $G\curvearrowright X$ 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:

$x\mathrel{E^{G}_{X}}x^{\prime}\iff\exists g\in G\;(gx=x^{\prime}).$

###  2.2. Bernoulli shifts and automorphism groups

Let $\mathrm{Sym}(\mathbb{N})$ be the Polish group of all permutations
$g\colon\mathbb{N}\to\mathbb{N}$ of $\mathbb{N}$ endowed with the pointwise
convergence topology. A Polish permutation group $P$ is any closed subgroup of
$\mathrm{Sym}(\mathbb{N})$. The (left) action of $P$ on $\mathbb{N}$, by
$(g,a)\mapsto g(a),$ induces a (left) action of $P$ on the space
$\mathbb{R}^{\mathbb{N}}$, of all maps $x\colon\mathbb{N}\to\mathbb{R}$, by
$(g,x)\mapsto gx$, where $(gx)(a)=x(g^{-1}(a))$. Following [KMPZ], we call
$P\curvearrowright\mathbb{R}^{\mathbb{N}}$ the Bernoulli shift of $P$ and we
denote by $E(P)$ the associated orbit equivalence relation on
$\mathbb{R}^{\mathbb{N}}$. The injective part of the Bernoulli shift is the
restriction of the above action to the $P$-invariant subset
$\mathrm{Inj}(\mathbb{N},\mathbb{R})$ of $\mathbb{R}^{\mathbb{N}}$ which
consists of all injective functions of $\mathbb{R}^{\mathbb{N}}$. Notice that
$\mathrm{Inj}(\mathbb{N},\mathbb{R})$ is a dense $G_{\delta}$ subset of
$\mathbb{R}^{\mathbb{N}}$.

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
$\mathrm{Aut}(\mathcal{N})$ of some countable structure $\mathcal{N}$. A first
order language $\mathcal{L}$ is relational if it only contains relation
symbols. Let $\mathcal{N}$ be an $\mathcal{L}$-structure with domain
$\mathbb{N}$. For every $A\subseteq\mathbb{N}$ we denote by $\langle
A\rangle_{\mathcal{N}}$ the smallest substructure of $\mathcal{N}$ containing
every element of $A$ in its domain. We say that $\mathcal{N}$ is
ultrahomogeneous if for every bijection $f\colon A\to B$ between finite
subsets of $\mathbb{N}$ which induces an isomorphism from $\langle
A\rangle_{\mathcal{N}}$ to $\langle B\rangle_{\mathcal{N}}$, we have that $f$
extends to an automorphism of $\mathcal{N}$. The $\mathcal{L}$-algebraic
closure $\mathrm{acl}_{\mathcal{N}}(A)$ of $A\subseteq\mathbb{N}$ in
$\mathbb{N}$ is the set of all $b\in\mathbb{N}$ for which there is an
$\mathcal{L}$-formula $\varphi$ with parameters from $A$ so that
$\mathcal{N}\models\varphi(b)$ and
$\\{c\in\mathbb{N}\colon\mathcal{N}\models\varphi(c)\\}$ is finite. In
general, if $P:=\mathrm{Aut}(\mathcal{N})$, then the algebraic closure
$A\mapsto[A]_{P}$ defined in the introduction does not coincide with its
model-theoretic counterpart $A\mapsto\mathrm{acl}_{\mathcal{N}}(A)$. However,
we can always enrich the structure $\mathcal{N}$ and make the two notions
equal:

######  Lemma 8.

If $P$ is a Polish permutation group, then there is exists ultrahomogeneous
$\mathcal{L}$-structure $\mathcal{N}=(\mathbb{N},\ldots)$, on a countable
relational language $\mathcal{L}$, so that $P=\mathrm{Aut}(\mathcal{N})$, and
with $[A]_{P}=\mathrm{acl}_{\mathcal{N}}(A)$, for every finite
$A\subseteq\mathbb{N}$.

###### Proof.

This follows from the usual “orbit completion” argument; e.g. see
[Hodges]*Theorem 4.1.4 or [BK96]*Section 1.5: for every $n\in 1,2,\ldots$, and
for every orbit $O\subseteq\mathbb{N}^{n}$ in the diagonal action
$g\cdot(a_{1},\ldots,a_{n})=(g\cdot a_{1},\ldots,g\cdot a_{n})$ of $P$ on
$\mathbb{N}^{n}$, we introduce a $n$-ary relation symbol $R^{n,O}$ and we set
$\mathcal{N}\models R^{n,O}(\bar{a})$ if and only if $\bar{a}\in O$. Clearly
$\mathcal{N}$ is ultrahomogeneous and $P=\mathrm{Aut}(\mathcal{N})$ follows
from the fact that $P$ a closed subgroup of $\mathrm{Sym}(\mathbb{N})$. Assume
now that $b$ has a finite orbit under the stabilizer $P_{\bar{a}}$ of
$\bar{a}=(a_{1},\ldots,a_{n-1})$. Notice that there are only finitely many
relations which are satisfied by sub-tuples of $(\bar{a},b)$. Hence, the
conjunction of them is an $\mathcal{L}$-formula $\varphi(\bar{x},y)$. Notice
that $\varphi(\bar{x},y)$ completely captures the quantifier-free type of
$\varphi(\bar{a},b)$. By ultrahomogeneity there are only finitely many $c$ so
that $\mathcal{N}\models\varphi(\bar{a},c)$. ∎

Here we will be interested in the following examples.

######  Example 9.

If $P=\mathrm{Sym}(\mathbb{N})$ is the full symmetric group, then
$E(\mathrm{Sym}(\mathbb{N}))$ is Borel bireducible with the equivalence
relation $=^{+}$ of countable sets of reals, where $(x_{n})=^{+}(y_{n})$ if
and only if $\\{x_{n}:n\in\mathbb{N}\\}=\\{y_{n}:n\in\mathbb{N}\\}$, for all
$(x_{n}),(y_{n})\in\mathbb{R}^{\mathbb{N}}$.

######  Example 10.

Fix $n\geq 2$ and consider the language $\mathcal{L}_{n}$ which contains
countably many $n$-ary relation symbols $R_{0},R_{1},R_{2},\ldots$ and
countably many $n$-ary function symbols $f_{0},f_{1},f_{2},\ldots$. It follows
by [BKL17], that there is a unique up to isomorphism ultrahomogenous structure
$\mathcal{N}$ which satisfies the following properties:

  1. (1)

$\mathcal{N}$ is locally finite, i.e., $\mathrm{dom}(\langle
A\rangle_{\mathcal{N}})$ is finite if so is $A\subseteq\mathbb{N}$;

  2. (2)

the collection $\\{R^{\mathcal{N}}_{j}\colon j\in\mathbb{N}\\}$ partitions
$\mathbb{N}^{n}$;

  3. (3)

if $R^{\mathcal{N}}_{j}(a_{1},\cdots,a_{n})$, then for all $i>j$ we have
$f^{\mathcal{N}}_{i}(a_{1},\cdots,a_{n})=a_{1}$;

  4. (4)

if $A\subseteq\mathbb{N}$ is of size $n+1$, then there is $a\in A$ so that
$a\in\mathrm{dom}(\langle A\setminus\\{a\\}\rangle_{\mathcal{N}})$.

  5. (5)

every finite $\mathcal{L}_{n}$-structure which satisfies the above (1)–(4)
embeds in $\mathcal{N}$.

We denote this structure by $\mathcal{N}^{\mathrm{BKL}}_{n}$. It is easy to
see that for all finite $A\subseteq\mathbb{N}$ we have that
$\mathrm{dom}(\langle
A\rangle_{\mathcal{N}})=[A]_{\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n})}=\mathrm{acl}(A)$.
Hence, properties (1),(4) above imply that the permutation group
$\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n})$ is locally finite and of
algebraic dimension at most $n$. Clearly $\mathrm{acl}(\emptyset)=\emptyset$.
Finally, by ultrahomogeneity of $\mathcal{N}^{\mathrm{BKL}}_{n}$ and the fact
that $\mathrm{Age}(\mathcal{N}^{\mathrm{BKL}}_{n})$ has the disjoint
$m$-amalgamation for all $m\leq n$—see [BKL17]—we have that
$\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n})$ also $n$-free. By considering
the Bernoulli shifts of these automorphism groups we have the following
collection of equivalence relations:

$E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2})),E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{3})),\ldots,E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n})),\ldots,$

###  2.3. Labelled logic actions

Let $\mathcal{L}$ be a countable language and let
$\mathrm{Str}(\mathcal{L},\mathbb{N})$ be the Polish space of all
$\mathcal{L}$-structures on domain $\mathbb{N}$; see [Kec95, Gao09]. The logic
action is the action $(g,\mathcal{N})\mapsto g\mathcal{N}$ of
$\mathrm{Sym}(\mathbb{N})$ on $\mathrm{Str}(\mathcal{L},\mathbb{N})$, which is
defined by setting for all relations $R\in\mathcal{L}$ and all functions
$f\in\mathcal{L}$:

$\displaystyle g\mathcal{N}\models R(n_{0},\ldots,n_{k-1})\quad$
$\displaystyle\iff\quad\mathcal{N}\models
R(g^{-1}(n_{0}),\ldots,g^{-1}(n_{k-1}))$ $\displaystyle g\mathcal{N}\models
f(n_{0},\ldots,n_{k-1})=n\quad$ $\displaystyle\iff\quad\mathcal{N}\models
f(g^{-1}(n_{0}),\ldots,g^{-1}(n_{k-1}))=g^{-1}(n)$

If $\sigma$ is an $\mathcal{L}_{\omega_{1},\omega}$-sentence then we will
denote by $\mathrm{Str}(\sigma,\mathbb{N})$ the
$\mathrm{Sym}(\mathbb{N})$-invariant Borel subset of
$\mathrm{Str}(\mathcal{L},\mathbb{N})$ which consists of all $\mathcal{N}$
with $\mathcal{N}\models\sigma$; see [Kec95, Gao09]. In both cases we denote
by $\simeq_{\mathrm{iso}}$ the associated orbit equivalence relation.

As in [KP], we also consider the labelled logic action
$\mathrm{Sym}(\mathbb{N})\curvearrowright\mathrm{Str}_{\mathbb{R}}(\mathcal{L},\mathbb{N})$,
where
$\mathrm{Str}_{\mathbb{R}}(\mathcal{L},\mathbb{N}):=\mathbb{R}^{\mathbb{N}}\times\mathrm{Str}(\mathcal{L},\mathbb{N})$
and $(g,(x,\mathcal{N}))\mapsto(gx,g\mathcal{N})$. The injective part of the
labelled logic action is the restriction of the above action to
$\mathrm{Sym}(\mathbb{N})$-invariant subset
$\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\mathcal{L},\mathbb{N})$ of
$\mathrm{Str}_{\mathbb{R}}(\mathcal{L},\mathbb{N})$, given by

$\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\mathcal{L},\mathbb{N}):=\mathrm{Inj}(\mathbb{N},\mathbb{R})\times\mathrm{Str}(\mathcal{L},\mathbb{N}),$

where $\mathrm{Inj}(\mathbb{N},\mathbb{R})$ consists of all injective
functions in $\mathbb{R}^{\mathbb{N}}$. Notice that
$\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\mathcal{L},\mathbb{N})$ is a dense
$G_{\delta}$ subset of $\mathrm{Str}_{\mathbb{R}}(\mathcal{L},\mathbb{N})$.

Similarly we define the labelled logic action
$\mathrm{Sym}(\mathbb{N})\curvearrowright\mathrm{Str}_{\mathbb{R}}(\sigma,\mathbb{N})$
on models of an $\mathcal{L}_{\omega_{1},\omega}$-sentence $\sigma$, together
with its injective part
$\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\sigma,\mathbb{N})$. We will denote
by $\simeq_{\mathrm{iso}}$ the associated orbit equivalence relation. For
every countable $\mathcal{L}$-structure $\mathcal{N}$ we have an
$\mathrm{Aut}(\mathcal{N})$-equivariant embedding of the Bernoulli shift

$\big{(}\mathrm{Aut}(\mathcal{N})\curvearrowright\mathbb{R}^{\mathbb{N}}\big{)}\hookrightarrow\big{(}\mathrm{Sym}(\mathbb{N})\curvearrowright\mathrm{Str}_{\mathbb{R}}(\mathcal{L},\mathbb{N})\big{)},$

with $g\mapsto g$ and $x\mapsto(x,\mathcal{N})$. Notice that this induces a
Borel reduction:

$E(\mathrm{Aut}(\mathcal{N}))\leq_{B}\big{(}\mathrm{Str}_{\mathbb{R}}(\mathcal{L},\mathbb{N}),\simeq_{\mathrm{iso}}\big{)}.$

Notice that the above map embeds the injective part of the Bernoulli shift to
the injective part of the labelled logic action.

##  3\. 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+$\neg$AC, Cohen started with the Cohen forcing $\mathbb{P}$ which
adds a countable sequence of Cohen reals $\left<x_{n}\colon
n\in\mathbb{N}\right>$, but instead of taking the full forcing extension
$V[G]$, he considered the substructure of $V[G]$ generated by all
$\mathbb{P}$-names which are invariant under the natural action of
$\mathrm{Sym}(\mathbb{N})$ on $\mathbb{P}$ by index permutation; see [Ka08].
The resulting basic Cohen model coincides with the smallest extension
$V(\\{x_{n}\colon n\in\mathbb{N}\\})$ of $V$ in $V[G]$ which contains the set
$\\{x_{n}\colon n\in\mathbb{N}\\}$ and satisfies ZF. The failure of AC in this
model comes from the fact that $V(\\{x_{n}\colon n\in\mathbb{N}\\})$ contains
no well-ordering of $\\{x_{n}\colon n\in\mathbb{N}\\}$. The latter is
established by developing a _theory of supports_ for definable sets in
$V(\\{x_{n}\colon n\in\mathbb{N}\\})$.

Since $\\{x_{n}\colon n\in\mathbb{N}\\}$ can be alternatively thought as the
classifying $=^{+}$-invariant of the generic point
$(x_{n})\in\mathbb{R}^{\mathbb{N}}$, one would hope that the ergodic theoretic
properties of $=^{+}$ are reflected to the structure of definable sets in
$V(\\{x_{n}\colon n\in\mathbb{N}\\})$. This point of view is adopted in
[Sha18, 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 [Sha18, 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
$\mathbb{P}$ and let $A\in V[G]$. 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. 193.111Without the assumption $V=L$, $V(A)$ can be
defined as the union of all models of the form $L(A,v)$ for some $v\in V$.
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 $S\in\mathrm{OD}(A)^{W}$, if $S\in W$ and there is a formula
$\varphi$, ordinal parameters $\bar{b}$ and parameters $\bar{a}$ from the
transitive closure of $A$, so that $S=\\{s\in
W:W\models\varphi(s,A,\bar{a},\bar{b})\\}$. 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 $\mathrm{HOD}(A)^{W}$ be the subclass of $W$ consisting of all sets $S\in
W$ so that every element of the transitive closure $\mathrm{tc}(\\{S\\})$ of
$\\{S\\}$ is in $\mathrm{OD}(A)^{W}$. If $W$ is any model of ZF and $A$ a set
in $W$, then $\mathrm{HOD}(A)^{W}$ is a transitive model of ZF. See the proof
of [Jec03]*Theorem 13.26 and the following discussion.

######  Fact 11.

For every $S\in V(A)$ there is some formula $\varphi$, parameters $\bar{a}$
from the transitive closure of $A$ and $v\in V$ such that $S=\\{s\in
V(A):V(A)\models\varphi(s,A,\bar{a},v)\\}$. In this case we say that $\bar{a}$
is a definable support for $S$.

###### Proof.

Work inside $V(A)$, and consider $\mathrm{HOD}(A)^{V(A)}$, the class of sets
which are hereditarily ordinal definable over $A$, as calculated in $V(A)$.
Then $\mathrm{HOD}(A)^{V(A)}$ 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 $\mathrm{HOD}(A)^{V(A)}$. So every set in $V(A)$ is definable as in the
statement of the fact.222If we were not to assume that $V=L$, the same
arguments hold when $\mathrm{HOD}(A)$ is changed to $\mathrm{HOD}(A,V)$, all
the sets hereditarily definable using parameters from the transitive closure
of $A$ and arbitrary parameters in $V$. ∎

###  3.2. 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 $X\in V$ be a Polish space and $E\in V$ an analytic equivalence relation
on $X$. A complete classification of $(X,E)$ in $V$ is any assignment
$x\mapsto A_{x}$ in $V$ given by a first-order formula $\varphi(x,A_{x})$ of
set theory, so that for all $x,y\in X$ we have:

$x\mathrel{E}y\iff A_{x}=A_{y}.$

We write $x\mapsto_{\varphi}A_{x}$ when the assignment $x\mapsto A_{x}$ is
given by $\varphi$. 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 $x\in X\cap V$, $[x]_{E}$ 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
$x\mapsto_{\varphi}A_{x}$ is an absolute classification if whenever
$W\supseteq V$ is a model of ZF, the assignment $x\mapsto_{\varphi}A_{x}$ as
computed in $W$, is still a complete classification for $(X,E)$ as computed in
$W$, and for all $x\in X\cap V$ we have that $A_{x}$ as computed in $W$, is
equal to $A_{x}$, as computed in $V$.

######  Example 12.

Recall $E(\mathrm{Sym}(\mathbb{N}))$ from Example 9. The assignment

$\displaystyle\left<x_{n}:\,n\in\mathbb{N}\right>$
$\displaystyle\mapsto\\{x_{n}:\,n\in\mathbb{N}\\}$

defined on $\mathbb{R}^{\mathbb{N}}$ is a complete classification when
restricted to the comeager subset $\mathrm{Inj}(\mathbb{N},\mathbb{R})$ of
$\mathbb{R}^{\mathbb{N}}$, consisting of all injective functions from
$\mathbb{N}$ to $\mathbb{R}$. 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 $x\mapsto A_{x}$, 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 [Gao09,
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 $x\in\mathbb{R}^{\mathbb{N}}$ 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 $x\mapsto A_{x}$ and $y\mapsto B_{y}$ are absolute
classifications of $E$ and $F$ respectively. Then, the following are
equivalent.

  1. (1)

For every Borel homomorphism $f\colon(X_{0},E)\to(Y,F)$, where $X_{0}\subseteq
X$ is non-meager, $f$ maps a non-meager set into a single $F$-class;

  2. (2)

If $x\in X$ 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
$B\in V$.

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)$\implies$(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
$x\mapsto A_{x}$ and $y\mapsto B_{y}$, 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 $V(A)$, if $x^{\prime}$ is such that $A=A_{x^{\prime}}$, then
$B=B_{f(x^{\prime})}$. 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 $B_{f(\dot{x})}=\check{B}$. In particular, all generics $x$
extending $p$ are mapped to the same $F$-equivalence class. Now the set of all
$x\in p$ 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 $P\curvearrowright\mathbb{R}^{\mathbb{N}}$. By Lemma 8, it
will suffice to consider permutation groups $P$ of the form
$\mathrm{Aut}(\mathcal{N})$, where $\mathcal{N}$ is some countable
$\mathcal{L}$-structure.

######  Definition 15.

Let $\mathcal{L}$ be a countable language. A countable $\mathcal{L}$-structure
on $\mathbb{R}$ is any $\mathcal{L}$-structure $\mathcal{A}=(A,\ldots)$ whose
domain $A$ is a countable subset of $\mathbb{R}$.

######  Example 16.

Let $\mathcal{N}$ be a countable $\mathcal{L}$-structure and let the
$\mathrm{Aut}(\mathcal{N})\curvearrowright\mathrm{Inj}(\mathbb{N},\mathbb{R})$
be the injective part of the Bernoulli shift. Consider the assignment:

$x\mapsto\mathcal{A}_{x}$

mapping every $x\in\mathrm{Inj}(\mathbb{N},\mathbb{R})$ to the countable
$\mathcal{L}$-structure $\mathcal{A}_{x}=(A_{x},\ldots)$ on $\mathbb{R}$,
where $A_{x}=\\{x_{n}\colon n\in\mathbb{N}\\}$, and for all relations
$R\in\mathcal{L}$ and functions $f\in\mathcal{L}$ we have:

$\displaystyle\mathcal{A}_{x}\models R(x_{n_{0}},\ldots,x_{n_{k-1}})\quad$
$\displaystyle\iff\quad\mathcal{N}\models R(n_{0},\ldots,n_{k-1})$
$\displaystyle\mathcal{A}_{x}\models
f(x_{n_{0}},\ldots,x_{n_{k-1}})=x_{n}\quad$
$\displaystyle\iff\quad\mathcal{N}\models f(n_{0},\ldots,n_{k-1})=n$

Then $x\mapsto\mathcal{A}_{x}$ is an absolute classification for
$E(\mathrm{Aut}(\mathcal{N}))$ restricted to
$\mathrm{Inj}(\mathbb{N},\mathbb{R})$.

We similarly we have the following absolute classification for the labelled
logic action from Section 2.3.

######  Example 17.

Let $\mathcal{L}$ be a countable language and let
$\mathrm{Sym}(\mathbb{N})\curvearrowright\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\mathcal{L},\mathbb{N})$
be the injective part of the labelled logic action. Consider the assignment:

$(x,\mathcal{N})\mapsto\mathcal{A}_{x,\mathcal{N}}$

which maps every
$(x,\mathcal{N})\in\mathrm{Inj}(\mathbb{N},\mathbb{R})\times\mathrm{Str}(\mathcal{L},\mathbb{N})$
to the countable $\mathcal{L}$-structure $\mathcal{A}_{x,\mathcal{N}}$ on
$\mathbb{R}$, which is computed as $\mathcal{A}_{x}$ in Example 16 with
respect to $\mathcal{N}$. For every $g\in\mathrm{Sym}(\mathbb{N})$ and all
$(x,\mathcal{N})\in\mathrm{Inj}(\mathbb{N},\mathbb{R})\times\mathrm{Str}(\mathcal{L},\mathbb{N})$,
we have that:

$\displaystyle\mathcal{A}_{x,\mathcal{N}}\models
R(x_{n_{0}},\ldots,x_{n_{k-1}})\implies$ $\displaystyle\mathcal{N}\models
R(n_{0},\ldots,n_{k-1})\implies$ $\displaystyle g\mathcal{N}\models
R(g(n_{0}),\ldots,g(n_{k-1}))\implies$
$\displaystyle\mathcal{A}_{z,g\mathcal{N}}\models
R(z_{g(n_{0})},\ldots,z_{g(n_{k-1})})\text{ for all
$z\in\mathrm{Inj}(\mathbb{N})$},$

and since $(gx)_{g(n)}=x_{n}$, for all $n\in\mathbb{N}$, it follows that
$\mathcal{A}_{gx,g\mathcal{N}}\models R(x_{n_{0}},\ldots,x_{n_{k-1}})$. Hence,
$(x,\mathcal{N})\mapsto\mathcal{A}_{x,\mathcal{N}}$ is a complete
classification of $\simeq_{\mathrm{iso}}$ on
$\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\mathcal{L},\mathbb{N})$. 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 $n\in\omega$ and let $P\leq\mathrm{Sym}(\mathbb{N})$ be a locally finite,
$(n+1)$-free Polish permutation group. Let also $\sigma$ be an
$\mathcal{L}_{\omega_{1},\omega}$-sentence so that for every
$\mathcal{N}\in\mathrm{Str}(\sigma,\mathbb{N})$ and every
$b_{0},\ldots,b_{n}\in\mathrm{dom}(\mathcal{N})$, there is some $i\leq n$ so
that $b_{i}$ is in the $\mathcal{L}$-algebraic closure
$\mathrm{acl}_{\mathcal{N}}(\\{b_{j}\colon j\neq i\\})$ of the remaining
points. Then,

$E(P)\text{ is generically ergodic with respect to
}\big{(}\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\sigma,\mathbb{N}),\simeq_{\mathrm{iso}}\big{)}.$

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
$Q=\mathrm{Aut}(\mathcal{N})$ for some countable ultrahomogeneous
$\mathcal{L}$-structure. with the property that
$[A]_{Q}=\mathrm{acl}_{\mathcal{N}}(A)$, for every finite
$A\subseteq\mathbb{N}.$ Let $\sigma$ be the Scott sentence of $\mathcal{N}$.
Since every $\mathcal{N}^{\prime}\in\mathrm{Str}(\sigma,\mathbb{N})$ is
isomorphic to $\mathcal{N}$, we have that $\mathrm{Aut}(\mathcal{N}^{\prime})$
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

$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}))\leq_{B}\big{(}\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\sigma,\mathbb{N}),\simeq_{\mathrm{iso}}\big{)},$

we conclude that $E(\mathrm{Aut}(\mathcal{M}))$ is generically ergodic with
respect to $E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}))$. ∎

We now turn to the proof of Theorem 18. We first fix some notation regarding
Cohen forcing on $\mathbb{R}^{\mathbb{N}}$. Consider the poset $\mathbb{P}$
whose elements are finite partial maps $p$ from $\mathbb{N}$ such that
$p(n)\subseteq\mathbb{R}$ is an open interval with rational endpoints for
every $n$ in the domain of $p$. A condition $p$ extends $q$, denoted by $p\leq
q$, if $\mathrm{dom}(p)\supseteq\mathrm{dom}(q)$ and $p(n)\subseteq q(n)$ for
every $n\in\mathrm{dom}(q)$. We fix some $\mathbb{P}$-generic
$G\subseteq\mathbb{P}$ over $V$. Working in $V[G]$, define $x=(x_{n})$ to be
the unique element of $\mathbb{R}^{\mathbb{N}}$ such that $x_{n}\in p(n)$, for
every $p\in G$ with $n\in\mathrm{dom}(p)$. We will refer to $x$ as the Cohen
generic point of $\mathbb{R}^{\mathbb{N}}$. In what follows we also fix some
countable $\mathcal{L}$-structure $\mathcal{M}$ in $V$ and let

$\mathcal{A}:=\mathcal{A}_{x,\mathcal{M}}$

be the image of $(x,\mathcal{M})$ under the absolute classification defined in
Examples 16,17. We finally let $V(\mathcal{A})\subseteq V[G]$ be the symmetric
model associated to $\mathcal{A}\in V[G]$.

We will need the following lemma which is interesting on its own right. The
main point is that if $\mathcal{M}$ has enough symmetries, then the associated
symmetric model $V(\mathcal{A})$ admits an analysis of the definable supports
similar to the one in [HL64], for the basic Cohen model.

######  Lemma 19.

Assume that $\mathrm{Aut}(\mathcal{M})$ is locally finite. Take any set $S\in
V(\mathcal{A})$ such that $S\subseteq V$, and let $\bar{a}$ be a definable
support for $S$ in $V(\mathcal{A})$. If $[\bar{a}]\subseteq A$ is the
algebraic closure of $\bar{a}$ with respect to $\mathrm{Aut}(\mathcal{A})$,
then $S\in V([\bar{a}])$.

###### Proof.

By local finiteness we may assume without any loss of generality that
$\bar{a}$ enumerates $[\bar{a}]_{\mathrm{Aut}(\mathcal{A})}$. Fix a formula
$\phi$ and fix some parameter $v\in V$ such that $S=\big{\\{}s\in V\mid
V(\mathcal{A})\models\varphi(s,v,\bar{a},\mathcal{A})\big{\\}}$. By
reflection, let $\xi$ be a large enough ordinal so that $S\subseteq V_{\xi}$
and so that for any $p\in\mathbb{P}$ we have:

$p\Vdash\varphi^{V(\mathcal{\dot{A}})}(\check{s},v,\dot{\bar{a}},\dot{\mathcal{A}})\text{
if and only if
}V_{\xi}\models\big{(}p\Vdash\varphi^{V(\dot{\mathcal{A}})}(\check{s},v,\dot{\bar{a}},\dot{\mathcal{A}})\big{)}.$

Let $n_{0},\ldots,n_{k-1}\in\mathbb{N}$ so that
$\bar{a}=\\{x_{n_{0}},...,x_{n_{k-1}}\\}$. Working in $V(\bar{a})$ we define

$S^{\prime}=\\{s\in V_{\xi}\mid\text{ for any }p,\text{ if }a_{i}\in
p(n_{i})\text{ for }i<k,\text{ then
}V_{\xi}\models(p\Vdash\varphi^{V(\mathcal{\dot{A}})}(\check{s},\check{v},\bar{a},\dot{\mathcal{A}}))\\},$

Since the forcing relation is definable in $V$, $S^{\prime}$ is in
$V(\bar{a})$. We will conclude the proof of the lemma by showing that
$S=S^{\prime}$.

To see that $S^{\prime}\subseteq S$, let $s\in S^{\prime}$ and take any
condition $p\in G$ such that $n_{0},...,n_{k-1}\in\mathrm{dom}(p)$. By
definition of $S^{\prime}$ and by the choice of $\xi$,
$p\Vdash\varphi^{V(\mathcal{\dot{A}})}(\check{s},\check{v},\bar{a},\dot{\mathcal{A}})$.
Since $p\in G$ we have that
$\varphi^{V(\mathcal{A})}(s,v,\bar{a},\mathcal{A})$ holds in $V[G]$, and
therefore, $\varphi(s,v,\bar{a},\mathcal{A})$ holds in $V(\mathcal{A})$, and
so $s\in S$. The converse direction $S\subseteq S^{\prime}$ follows from the
next claim.

######  Claim.

For any $p,q\in\mathbb{P}$, if $a_{i}\in p(n_{i})\cap q(n_{i})$ for $i<k$,
then for any $s\in V$,

$p\Vdash\varphi^{V(\dot{\mathcal{A}})}(\check{s},\check{v},\dot{\bar{a}},\dot{\mathcal{A}})\iff
q\Vdash\varphi^{V(\dot{\mathcal{A}})}(\check{s},\check{v},\dot{\bar{a}},\dot{\mathcal{A}}).$

###### Proof.

Assume towards a contradiction that $p,q$ are as in the claim but we
additionally have that
$p\Vdash\varphi^{V(\mathcal{A})}(\check{x},\check{v},\bar{a},\dot{\mathcal{A}})$
and
$q\Vdash\neg\varphi^{V(\mathcal{A})}(\check{x},\check{v},\bar{a},\dot{\mathcal{A}})$.
Without loss of generality we may also assume that $p\in G$. Let
$L:=\\{m_{0},\ldots,m_{\ell-1}\\}\subseteq\mathbb{N}=\mathrm{dom}(\mathcal{M})$
be disjoint from $K:=\\{n_{0},...,n_{k-1}\\}$, so that $L\cup K$ contains the
domains of $p$ and $q$. Since $\bar{a}=[\bar{a}]_{\mathrm{Aut}(\mathcal{A})}$,
we have that $K=[K]_{\mathrm{Aut}(\mathcal{M})}$. Hence, for every $\ell\in L$
the orbit of $\ell$ under the stabilizer $\mathrm{Aut}(\mathcal{M})_{K}$ of
$K$ is infinite. But then, by the Neumann lemma [neumann1976structure]*Lemma
2.3, there is an infinite sequence $(\pi_{j})_{j}$ in
$\mathrm{Aut}(\mathcal{M})$ so that:

  1. (1)

$\pi_{j}$ fixes $K$ pointwise; and

  2. (2)

$\pi_{i}(L)\cap\pi_{j}(L)=\emptyset$ for all $i\neq j$.

By genericity of $G$ there is some $j$ such that $x\circ\pi_{j}$ extends $q$.
Set $\pi:=\pi_{j}$ and let $G^{\prime}:=\\{p\circ\pi\colon p\in G\\}$ and
$x^{\prime}:=x\circ\pi$. We have that:

  1. (i)

$G^{\prime}$ is $\mathbb{P}$-generic and $V[G^{\prime}]=V[G]$, since $\pi\in
V$;

  2. (ii)

$x^{\prime}(n_{i})=x(n_{i})=a_{i}$ and therefore
$\dot{\bar{a}}[G^{\prime}]=\bar{a}$, by (1) above;

  3. (iii)

$\dot{\mathcal{A}}[G^{\prime}]=\dot{\mathcal{A}}[G]=\mathcal{A}$, since
$x\mathrel{E(\mathrm{Aut}(\mathcal{M}))}x^{\prime}$; see Example 16.

By (ii),(iii), and since $x^{\prime}$ extends $q$, we have that
$\neg\varphi^{V(\mathcal{A})}(s,v,\bar{a},\mathcal{A})$ holds in
$V[G^{\prime}]$. On the other hand, since $\dot{\mathcal{A}}[G]=\mathcal{A}$
and $x$ extends $p$, we have that
$\varphi^{V(\mathcal{A})}(s,v,\bar{a},\mathcal{A})$ holds in $V[G]$. We
conclude that $V(\mathcal{A})$ satisfies both
$\neg\varphi(s,v,\bar{a},\mathcal{A})$ and $\varphi(s,v,\bar{a},\mathcal{A})$,
a contradiction. ∎

∎

Let $S\in V(\mathcal{A})$ with $S\subseteq V$ and let $F\subseteq
A=\mathrm{dom}(\mathcal{A})$ be a set. We say that $F$ is a support for $S$ if
$S\in V(F)$. Notice that by mutual genericity, for any two
$F_{1},F_{2}\subseteq A$ we have $V(F_{1})\cap V(F_{2})=V(F_{1}\cap F_{2})$.
We define the support $\mathrm{supp}(S)$ of $S$ to be the intersection of all
$F\subseteq A$ which are supports for $S$. Under the assumptions of the
previous lemma, every $S$ as above has a finite support $\mathrm{supp}(S)$
with $S\in V(\mathrm{supp}(S))$.

We may now conclude with the proof of Theorem 18.

###### Proof of Theorem 18.

We will apply Lemma 13 $(2)\implies(1)$. For that, let
$x\in\mathrm{Inj}(\mathbb{N},\mathbb{R})$ be the generic associated to $G$ and
set $\mathcal{A}:=\mathcal{A}_{x}$ the corresponding invariant for the
$\mathcal{M}$-shift. Let also $\mathcal{B}=(B,...)$ be a potential invariant
for
$\big{(}\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\sigma,\mathbb{N}),\simeq_{\mathrm{iso}}\big{)}$
in $V(\mathcal{A})$ which is definable only from $\mathcal{A}$ and parameters
from $V$. Recall by potential invariant we mean that there are
$y\in\mathrm{Inj}(\mathbb{N},\mathbb{R})$ and
$\mathcal{N}\in\mathrm{Str}(\sigma,\mathbb{N})$, in some further generic
extension of $V(\mathcal{A})$, so that
$\mathcal{B}:=\mathcal{B}_{y,\mathcal{N}}$.

Since the definable support of $B$ is $\emptyset$, the rest follows from the
next claim, which together with Lemma 19 implies that that $B\in V$ and the
desired generic ergodicity follows from Lemma 13 $(2)\implies(1)$.

######  Claim.

Set $B:=\mathrm{dom}(\mathcal{B})$. We have that $B\subseteq V$.

###### Proof of Claim.

Assume towards a contradiction that there is some $b\in B$ not in $V$. It
follows that the support of $F:=\mathrm{supp}(\\{b\\})\subseteq A$ of $b$ is
not empty. Let $m_{0},\ldots,m_{k-1}\in\mathbb{N}$ with $F=\\{x_{m_{i}}\colon
i<k\\}$. Fix a condition $p$ forcing the above. By local finiteness of
$\mathcal{M}$ we may choose some finite
$K\subseteq\mathbb{N}=\mathrm{dom}(\mathcal{M})$ which contains

$\mathrm{dom}(p)\cup\\{m_{0},\ldots,m_{k-1}\\}$

and it is algebraically closed, i.e., $[K]_{\mathrm{Aut}(\mathcal{M})}=K$.
Since $\mathcal{M}$ is $(n+1)$-free, there are
$\pi_{0},\ldots,\pi_{n}\in\mathrm{Aut}(\mathcal{M})$ so that for all $i\leq n$
we have that

$[\pi_{i}(K)]_{\mathrm{Aut}(\mathcal{M})}\;\text{ is disjoint from
}\;[\bigcup_{j:j\neq i}\pi_{j}(K)]_{\mathrm{Aut}(\mathcal{M})}$

Notice now that $[\emptyset]_{\mathrm{Aut}(\mathcal{M})}=\emptyset$. Indeed,
any $(n+1)$-free permutation group is in particular $1$-free; and any locally
finite $1$-free permutation group $P$ satisfies $[\emptyset]_{P}=\emptyset$.
Hence, the orbit of each $\ell\in[\pi_{i}(K)]_{\mathrm{Aut}(\mathcal{M})}$ is
infinite for all $i\leq n$. By the Neumann lemma [neumann1976structure]*Lemma
2.3, for every finite $E\subseteq\mathbb{N}$ there is some
$\pi_{E}\in\mathrm{Aut}(\mathcal{M})$, so that $(\pi_{E}\circ\pi_{i})(K)\cap
E=\emptyset$ for all $i\leq n$. Hence, by the genericity of
$x\in\mathrm{Inj}(\mathbb{N},\mathbb{R})$ we may assume that for all $i\leq n$
we have that $x^{i}:=x\circ\pi_{i}$ extends $p$. Let
$G_{i}:=\\{q\circ\pi_{i}\colon q\in G\\}$ be the associated filters. Since all
$\pi_{i}$ above can be chosen in $V$, we have that each $G_{i}$ is generic and
$V[G]=V[G_{i}]$.

For each $i\leq n$, working in $V[G_{i}]$, since $p\in G_{i}$ and
$x^{i}\mathrel{E(P)}x$, we have that: the realization of $\dot{\mathcal{A}}$
is the same $\mathcal{A}$; the realization of $\dot{\mathcal{B}}$ is the same
$\mathcal{B}$; the interpretation of $\dot{F}$ is some $F_{i}\subseteq A$ such
that $\left\\{F_{i}:\,i=0,...,n\right\\}$ are pairwise disjoint; and $F_{i}$
is the support of the interpretation $b_{i}$ of $\dot{b}$ in $V[G_{i}]$. Since
$p\in G_{i}$, we have that $b_{i}\in B$ for all $i\leq n$. By assumption,
there is some $i$ such that, in $\mathcal{B}$, $b_{i}$ is in the
$\mathcal{L}$-algebraic closure
$\mathrm{acl}_{\mathcal{B}}(\left\\{b_{j}:\,j\neq i\right\\})$. Here we are
using that “for all $\mathcal{N}\in\mathrm{Str}(\sigma,\mathbb{N})$ and every
$b_{0},\ldots,b_{n}\in\mathrm{dom}(\mathcal{N})$, there is some $i\leq n$ so
that $b_{i}\in\mathrm{acl}_{\mathcal{N}}(\\{b_{j}\colon j\neq i\\})$” is
absolute.

It follows that $b_{i}$ is definable in $V(\mathcal{A})$ from
$\left\\{b_{j}:\,j\neq i\right\\}$ and $\mathcal{B}$. This is because there is
a finite set of reals, definable from $\left\\{b_{j}:\,j\neq i\right\\}$ and
$\mathcal{B}$, which contains $b_{i}$. Recall that $\mathcal{B}$ is definable
from $\mathcal{A}$, by assumption. Since $\left\langle b_{j}:\,j\neq
i\right\rangle$ is definable from $\left\langle F_{j}:\,j\neq i\right\rangle$,
it follows that $\bigcup_{j\colon j\neq i}F_{j}$ is a definable support for
$b_{i}$ in $V(\mathcal{A})$ and by Lemma 19 we have that $b_{i}\in
V([\bigcup_{j\colon j\neq i}F_{j}]_{\mathrm{Aut}(\mathcal{A})})$. By mutual
genericity we have

$V(F_{i})\bigcap V([\bigcup_{j\colon j\neq
i}F_{j}]_{\mathrm{Aut}(\mathcal{A})})=V(F_{i}\;\bigcap\;[\bigcup_{j\colon
j\neq i}F_{j}]_{\mathrm{Aut}(\mathcal{A})})=V$

But then, $\mathrm{supp}(b_{i})=\emptyset$, contradicting the assumption that
$\mathrm{supp}(b)\neq\emptyset$. ∎

∎

##  4\. Proof of Corollary 3

Recall from Example 10 that for $k\geq 2$,
$\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{k})$ is locally finite, $k$-free,
and has algebraic dimension precisely $k$. It follows from Theorem 2 that

$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{k+1}))\textrm{ is
generically
}E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{k}))\textrm{-ergodic}.$

In particular,
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{k+1}))$ is not
Borel reducible to
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{k}))$.

Finally we note that these equivalence relations increase in Borel complexity.

######  Proposition 20.

There is a Borel reduction from
$E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n}))$ to
$E(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1}))$. This map sends injective
sequences to injective sequences, thus reducing
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n}))$ to
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1}))$ as well.

###### Proof.

Let $(A,R_{j},s_{j})_{j\in\omega}$ be a $\mathrm{BKL}_{n}$ structure. Define a
structure $(A,\bar{R}_{j},\bar{s}_{j})_{j\in\omega}$ as follows, where
$\bar{R}_{j}$ are ${n+1}$-ary relations and $\bar{s}_{j}$ are $n+1$-ary
function symbols.

  * •

$\bar{R}_{j}(a_{0},...,a_{n})\iff R_{j}(a_{0},...,a_{n-1})$;

  * •

$\bar{s}_{j}(a_{0},...,a_{n})=s_{j}(a_{0},...,a_{n-1})$.

It is easy to see that $(A,\bar{R}_{j},\bar{s}_{j})_{j\in\omega}$ is a
$\mathrm{BKL}_{n+1}$-structure and that the map sending a
$\mathrm{BKL}_{n}$-invariant $(A,R_{j},s_{j})_{j\in\omega}$ to the
$\mathrm{BKL}_{n+1}$-invariant $(A,\bar{R}_{j},\bar{s}_{j})_{j\in\omega}$
gives the desired reduction.

∎

##  5\. Unpinned equivalence relations

We recall the definition of pinned equivalence relations and pinned cardinals
and explain the results in [Zap11] about the equivalence relation
$=^{+}\restriction Z$. We then present the equivalence relations
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1}))$ in this
context and give another proof that
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1})\not\leq_{B}E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n})$,
a weak form of Corollary 3. Finally, we prove Theorem 4.

######  Definition 21 ([LZ]).

Let $E$ be an analytic equivalence relation on a Polish space $X$. Let
$\mathbb{P}$ be a poset and $\tau$ a $\mathbb{P}$-name.

  * •

The name $\tau$ is $E$-pinned if $\mathbb{P}\times\mathbb{P}$ forces that
$\tau_{l}$ is $E$-equivalent to $\tau_{r}$, where $\tau_{l}$ and $\tau_{r}$
are the interpretation of $\tau$ using the left and right generics
respectively.

  * •

If $\tau$ is $E$-pinned the pair $\left<\mathbb{P},\tau\right>$ is called an
$E$-pin.

  * •

An $E$-pin $\left<\mathbb{P},\sigma\right>$ is trivial if there is some $x\in
X$ such that $\mathbb{P}\Vdash\sigma\mathrel{E}\check{x}$.

  * •

$E$ is pinned if all $E$-pins are trivial.

  * •

Given two $E$-pins $\left<\mathbb{P},\sigma\right>$ and
$\left<\mathbb{Q},\tau\right>$, say that they are $\bar{E}$-equivalent,
$\left<\mathbb{P},\sigma\right>\mathrel{\bar{E}}\left<\mathbb{Q},\tau\right>$,
if $\mathbb{P}\times\mathbb{Q}\Vdash\sigma\mathrel{E}\tau$.

  * •

The pinned cardinal of $E$, $\kappa(E)$, is the smallest $\kappa$ such that
every $E$-pin is $\bar{E}$-equivalent to an $E$-pin with a post of size
$<\kappa$.

######  Lemma 22 ([LZ]).

If $E$ is Borel reducible to $F$ then $\kappa(E)\leq\kappa(F)$.

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 $x\mapsto A_{x}$. 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 $\left<\mathbb{P},\tau\right>$, and

  * •

potential invariants $A$,

such that $\left<\mathbb{P},\tau\right>$ is trivial if and only if $A$ is
trivial. Specifically, a potential invariant $A$ corresponds to
$\left<\mathbb{P},\tau\right>$ if and only if $\mathbb{P}\Vdash
A_{\tau}=\check{A}$.

Assume first that $\left<\mathbb{P},\tau\right>$ is an $E$-pin. Let
$G_{l}\times G_{r}$ be $\mathbb{P}\times\mathbb{P}$-generic and let
$x_{l},x_{r}$ be the interpretations of $\tau$ according to $G_{l}$,$G_{r}$
respectively. Since $x_{l}$ and $x_{r}$ are $E$-related, it follows that
$A=A_{x_{l}}=A_{x_{r}}$. Furthermore, this set $A$ is in the intersection
$V[G_{l}]\cap V[G_{r}]$, which is equal to $V$ by mutual genericity. If
$\left<\mathbb{P},\tau\right>$ is trivial there is $x\in V$ such that
$x\mathrel{E}x_{l}$. In particular, $A_{x}=A_{x_{l}}=A$. Conversely, if there
is $x\in V$ with $A=A_{x}$ then $x$ witnesses that
$\left<\mathbb{P},\tau\right>$ is trivial: given any $\mathbb{P}$-generic $G$
over $V$, $A_{\tau[G]}=A=A_{x}$, so $\tau[G]$ is $E$-related to $x$.

Now let $A$ be a potential invariant for $E$. By assumption there is a poset
$\mathbb{Q}$, a generic $G$ and $x\in V[G]$ such that $A=A_{x}$. Let $\tau$ be
a $\mathbb{Q}$-name such that $\tau[G]=x$. Fix a condition $q\in\mathbb{Q}$
such that $q$ forces that $A_{\tau}=A$ and define
$\mathbb{P}=\mathbb{Q}\restriction p$. Now $\left<\mathbb{P},\tau\right>$ is
an $E$-pin such that $\mathbb{P}\Vdash A_{\tau}=A$.

######  Corollary 25.

The pinned cardinal of $E$ is $\kappa$ if and only if any potential
$E$-invariant is trivial in a generic extension by a poset of size $<\kappa$.

######  Example 26.

Consider the equivalence relation $=^{+}$ on $\mathbb{R}^{\mathbb{N}}$ with
the complete classification $\mathbb{R}^{\mathbb{N}}\ni
x\mapsto\left\\{x(n):\,n\in\mathbb{N}\right\\}$. The potential invariants of
$=^{+}$ are precisely all sets of reals. Therefore the pinned cardinal of
$=^{+}$ is $\mathfrak{c}^{+}$.

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 $f_{n}\colon\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$,
such that the graph $\mathcal{G}$ on $\mathbb{R}$, defined by

$x\mathrel{\mathcal{G}}y\iff\exists n(f_{n}(x)=y\vee f_{n}(y)=x),$

has cliques of size $\aleph_{1}$, but no cliques of size greater than
$\aleph_{1}$ (see [Zap11]*Fact 2.2). Let $Z\subseteq\mathbb{R}^{\omega}$ be
the $G_{\delta}$ set of all injective countable sequences of reals which
enumerate a clique in $\mathcal{G}$, and consider the equivalence relation
$=^{+}\restriction Z$ on $Z$. The potential invariants of $=^{+}\restriction
Z$ are precisely all sets of reals which form a clique in $\mathcal{G}$, and
those have cardinality $\leq\aleph_{1}$. Therefore the pinned cardinal of
$=^{+}\restriction Z$ is $\aleph_{1}^{+}$.

######  Corollary 28 ([Zap11]).

$=^{+}$ is not Borel reducible to $=^{+}\restriction Z$.

###### Proof.

By the absoluteness of Borel reducibility, we may work in some forcing
extension where the continuum hypothesis fails, that is,
$\mathfrak{c}>\aleph_{1}$. In this model the pinned cardinal of $=^{+}$ is
strictly greater than that of $=^{+}\restriction Z$, so the corollary follows
from Lemma 22. ∎

######  Example 29.

By Example 16, a potential invariant for
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1}))$ is a set of
reals together with a $\mathrm{BKL}_{n+1}$-structure on it. Since
$\mathrm{BKL}_{n+1}$ models are of size at most $\aleph_{n}$, and there is a
model of size $\aleph_{n}$, it follows that the pinned cardinal of
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1}))$ is
$\aleph_{n}^{+}$, in a model where $|\mathbb{R}|\geq\aleph_{n}$. It follows
from Lemma 22 that
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n+1}))$ is not
Borel reducible to
$E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{n}))$.

Both $E_{\mathrm{inj}}(\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2}))$ and
$=^{+}\restriction Z$ have the same pinned cardinal $\aleph_{1}^{+}$. 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 $\sigma$ 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
$\mathrm{Str}(\sigma,\mathbb{N})$ has algebraic dimension less than or equal
to $1$. By Theorem 18 and the observation in Example 10 that
$\mathrm{Aut}(\mathcal{N}_{2}^{\mathrm{BKL}})$ is $2$-free we have that
$E\big{(}\mathrm{Aut}(\mathcal{N}^{\mathrm{BKL}}_{2})\big{)}$ is generically
ergodic with respect to
$\big{(}\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\sigma,\mathbb{N}),\simeq_{\mathrm{iso}}\big{)}$.
Therefore it suffices to show that $=^{+}\restriction Z$ is Borel reducible to
$\big{(}\mathrm{Str}^{\mathrm{inj}}_{\mathbb{R}}(\sigma,\mathbb{N}),\simeq_{\mathrm{iso}}\big{)}$.
This is done by sending $x\in Z$ to the pair $(x,\mathcal{N})$ where
$\mathcal{N}$ is the structure on $\mathbb{N}$ defined by $h_{i}(n)=m\iff
f_{i}(x(n))=x(m)$ (where $f_{i}$ are the functions from Example 27). ∎

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