This question is meant to be viewed under moderate large cardinal hypotheses, e.g., enough to ensure $\aleph_1^{L[x]}<\aleph_1$ for all reals $x$.

In analogy with the (well-developed) theory of countable Borel equivalence relations, what can be said about countable $\mathbf\Sigma^1_2$ (or $\mathbf\Delta^1_3$, whatever make the most sense) equivalence relations?

A natural example of such an equivalence relation is relative constructability, i.e., $x\equiv_c y$ if and only if $L[x]=L[y]$ for reals $x$ and $y$.

Specifically, I'm looking for:

  • An analogue of the Feldman-Moore theorem on countable equivalence relations being induced by appropriately-measurable actions of countable groups.
  • Analogues of the Silver and Glimm-Effros dichotomies.
  • A theory of hyperfiniteness (i.e., equivalence relations which are a increasing union of $\mathbf\Sigma^1_2$ finite equivalence relations), and its relationship to appropriately-measurable $\mathbb{Z}$-actions.

Are there any references which address these issues?

I suspect both that many arguments in the Borel case can be adapted (assuming appropriate uniformization results under large cardinals), but also that I might be blind to possible pitfalls...

  • 2
    $\begingroup$ Concerning Silver's theorem on $\Sigma^1_2$-eq relations, it was proved by Kechris in the paper ``On transfinite sequences of projective sets with an application to $\Sigma^1_2$-equivalence relation". $\endgroup$
    – 喻 良
    Mar 19 '18 at 3:25

We will show the generalization of the Feldman-Moore theorem on countable equivalence relations to the context of thin $\kappa$-Suslin equivalence relations where $\kappa$ is any infinite cardinal. Under further hypotheses, that is determinacy axioms, then every $\Sigma^1_{2n+2}$ set of reals is $\delta^1_{2n+1}$-Suslin and every thin equivalence relation will be a countable equivalence relation by the perfect set theorem for all sets under $AD$. Just in $ZFC$, the pointclass $\Sigma^1_2$ is $\delta^1_1$-Suslin where $\delta^1_1=\aleph_1$ but the perfect set theorem at this level requires a bit more.

We will thus show the following: let $X$ be a Hausdorff space and assume $E\subseteq X\times X$ is a thin $\kappa$-Suslin equivalence relation (in the non $AD$ context we may just say $E$ is an equivalence relation of size $\leq \kappa$). Call a group thin if it does not contain a perfect set of elements. Then there exists a $\kappa$-Suslin action of a thin group $F$ on $X$ such that $E$ is obtained as an orbit equivalence relation, that is we have that $$E=\{(x,y)\in X\times X: Fx=Fy\}.$$ Notice that the second set is always $\kappa$-Suslin by definition. Furthermore the group $F$ is generated by the set $$\{f_k:k\in \omega\}$$ such that $f_k$ has order 2, $(f_k)^2=e$ for all $k\in \omega$ and $$E(x,y) \text{ holds if and only if either } x=y, \text{ or } f_k(x)=y$$ for some $k\in \omega$.

We now start the proof. By Caicedo, Clemens, Conley and Miller we have the generalization of the Luzin-Novikov uniformization theorem to the $\kappa$-Suslin sets. More specifically: If $\kappa$ is an infinite cardinal, if $X$ is a Hausdorff space and if $A\subseteq X^{\omega}$ is $\kappa$-Suslin then one of the following holds:

1) The set $A$ is the union of $\kappa$-many graph intersecting sets which are $\kappa^+$-Borel when considered as subsets of $A$,

2) The set $A$ has a pairwise disjoint perfect subset.

Two sequences $x$ and $y$ are graph disjoint if $x(n)=y(n)$ for all $n\in \omega$. A set $A$ is called graph intersecting if no two sequences in $A$ are graph disjoint. The Luzin-Novikov uniformization theorem follows as a special case of the Caicedo-Clemens-Conley-Miller theorem by considering constant sequences.

Next, since the equivalence relation $E\subseteq X\times X$ has $\leq \kappa$-size sections then by the Caicedo-Clemens-Conley-Miller theorem we have $E=\bigcup_{\alpha\in \kappa} Y_{\alpha}$ where for every $\alpha\in \kappa$, $Y_{\alpha}\subseteq X\times X$ is the graph of a $\kappa^+$-Borel function. We define $$Y_{\alpha}^{-1}=\{(x,y)\in X\times X: (y,x)\in Y_{\alpha}\}$$ By symmetry of $E$ we obtain that $$E=\bigcup_{\alpha<\kappa} Y_{\alpha}$$ Define next the following sets: $$Y_{\alpha,\beta}=Y_{\alpha}\cap Y^{-1}_{\beta}$$ We also must have that $$E=\bigcup_{\alpha,\beta<\kappa} Y_{\alpha,\beta}$$ We may assume for the remainder of the proof that $X$ is the unit interval $I$.

Let $J,K\subseteq I$ be two disjoint intervals with rational endpoints. Since $J$ and $K$ are disjoint we must have $$J,K\subseteq (I\times I)\setminus\{(x,x):x\in I\}$$ For all $\alpha,\beta<\kappa$, define also the set $$A_{\alpha,\beta}(J,K)$$ by $$A_{\alpha,\beta}(J,K)=proj(\{(x,y)\in Y_{\alpha,\beta}: x\in J \wedge y\in K\})$$ where we take the projection onto the first coordinate. Associated to each such set $A_{\alpha\beta}(J,K)$ we have a map $$f_{\alpha,\beta}(J,K):A_{\alpha,\beta}(J,K) \to X$$ such that the graph $$\Gamma_f=\{(x,y)\in Y_{\alpha,\beta}: x\in Z\}$$ By definition of the set $A$ we must have $$f(A)\cap A=\emptyset$$ and $$E(x,f(x))$$ since $Y_{\alpha,\beta}\subseteq E$ and $(x,f(x)\in Y_{\alpha,\beta}$ for all $x\in A$. In addition notice that $f$ is an injective function.

We thus define a $\kappa$-Suslin bijection $$f': f(A)\cup A\to f(A)\cup A$$ by letting $f(f(x))=x$ and finally define an automorphism of the space $X$ fixing everything outside of $A\cup f(A)$, that is $f'(x)=x$ where $x\in X\setminus (A\cup f(A))$.

By definition of the intervals $J$ and $K$ there are only $\aleph_0$ many such automorphisms $f$ of $X$. Let $F$ be the group generated by the set $$\{f_k:k\in \omega\}$$ $F$ satisfies the theorem and the $\kappa$-Suslin equivalence relation $E$ comes as an orbit equivalence relation.

There are probably more results that can be generalized along these lines and more natural equivalence relations that could be studied by looking at inner models with Woodin cardinals and the $\mathcal{Q}$-degrees. Here's a reference of the Caicedo-Clemens-Clinton-Miller article I'll stop here, this is already too long.


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