%\documentclass[options]{asl} \documentclass[options,bibay2]{asl} \usepackage{amssymb} % optional \usepackage{oldlfont} \usepackage{psfig} \usepackage[all]{xy} \title[Group actions] {Group actions and\\ and countable models} \author{Greg Hjorth} \revauthor{Hjorth, G.} \address{Department of Mathematics\\ UCLA\\ Los Angeles CA90095-1555, USA} \email{greg@math.ucla.edu} % These will be typeset in italics \newtheorem{Theorem}{Theorem}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{lemma}[Theorem]{Lemma} \newtheorem{corollary}[Theorem]{Corollary} % These will be typeset in Roman \theoremstyle{definition} \newtheorem{Definition}[Theorem]{Definition} \newtheorem{definition}[Theorem]{Definition} \newtheorem{examples}[Theorem]{Examples} \newtheorem{example}[Theorem]{Example} \newtheorem{Fact}[Theorem]{Fact} \newtheorem{Conjecture}[Theorem]{Conjecture} \newtheorem{conjecture}[Theorem]{Conjecture} \newtheorem{question}[Theorem]{Question} \newtheorem{notation}[Theorem]{Notation} \begin{document} \def\R{{\mathbb R}} \def\N{{\mathbb N}} \def\Q{{\mathbb Q}} \def\Z{{\mathbb Z}} \def\C{{\mathbb C}} \def\P{{\mathbb P}} \def\no{\noindent} \def\Box{\square} \def\forces{\Vdash} %\def\Ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\sim$}}}{}} %\def\ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\scriptscriptstyle\sim$}}}{}} \def\Ubf{\tboldsymbol} \def\ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\scriptscriptstyle\sim$}}}{}} \maketitle \tableofcontents \section{Introduction} The field of Polish group actions has emerged in the last couple of years as a kind of sub-sub-discipline in its own right. To some extent one can think of the study of countable models as being a special case of a Polish group action -- since the infinite symmetric group, consisting of all permutations of the natural numbers, is a Polish group whose action on countable models gives rise to the isomorphism relation as its orbit equivalence relation. In these talks I will try to emphasize the connections between recent work on Polish groups and some basic concepts one might see in a first course in logic or model theory. I will begin the discussion at a very general level, which hopefully might make some sense to almost any kind of logician. Of course this means that the initial pace will seem tedious to an expert. On the other hand as the paper progresses I want to increasingly mention connections to other areas and recent results and include stick figure sketches of their proofs. Perhaps the later parts of the paper will only be completely clear for someone who already knows much of this material or is willing to spend hours in the library digging up references. Thus the criticism may be raised that the level of exposition is inconsistent; and by the end perhaps no one will be completely happy. But short of turning this paper into a book that problem strikes me as an inevitable consequence of its goals. My own outlook on the history of this sub-area would really be to place {\bf The descriptive set theory of Polish group actions}, Becker-Kechris[1996], as being central. This book represented the intersection of two seemingly unrelated schools of thought. On the one hand was a sequence of papers treating topics in what one might call ``invariant descriptive set theory." Perhaps the most well known papers here are those by Vaught, in which the development of the notion of Vaught transform led to unified proofs for results by Ryll-Nardzewski (every orbit is Borel) and Lopez-Escobar (invariant Borel sets in the space of countable structures are described by ${\cal L}_{\omega_1, \omega}$ sentences). For this Vaught was awarded the first Karp prize. A decade later, in work that was completed in the 1980's but not published until 1994, Sami used Vaught's work to obtain results on changing topologies for the ``logic action" of $S_\infty$ and showed that the topological Vaught conjecture holds for abelian Polish groups. (More on this below.) The second sequence of papers comes from researchers in operator algebras and the study of infinite dimensional group representations. Although the topic of Polish group actions for its own sake was somewhat incidental to the main direction of their research, the group responsible for conjugacy on infinite dimensional representations is the infinite dimensional unitary group and thus Polish but not locally compact. By the time of Effros[1982] one has a completely general result for Polish groups acting on Polish spaces, largely divorced from the original context of Effros[1965], Glimm[1961], and Mackey[1957]. %\newpage %\begin{figure}[ht] %{\psfig{figure=people.ps}} %\end{figure} \begin{figure} $$\xymatrix@=10pt{ & & *+[F]{\txt{ Mackey}} \ar[d] & \\ & & *+[F]{\txt{ (Glimm) }} \ar[dd] & \\ *+[F]{\txt{(Ryll-Nardzewski)[1964]}} \ar[dd] & & & \\ & & *+[F]{\txt{Effros[1965, 1982]}} \ar[ddddl] & \\ *+[F]{\txt{Vaught[1974, 1975]}}\ar[d] & & & \\ *+[F]{\txt{Miller[1977, 1978]}}\ar[d] & & & \\ *+[F]{\txt{Sami[1994]}}\ar[dr] & & & \\ & *+[F]{\txt{Becker-Kechris[1996]}} \ar[d] & & \\ & *+[F]{\txt{Lots of people (Becker,\\ Camerlo, Dougherty,\\ Gao, Farah, Kechris,\\ Louveau, Sami,\\ Solecki, and others) }} & }$$ \end{figure} Both these traditions were very much in the mind of Becker and Kechris by the time of {\bf The descriptive set theory of Polish group actions}. I have drawn up a small flow chart to try to illustrate how these papers might be viewed as interrelating, but it goes without saying that this interpretation is personal and subjective. Certainly there are papers off to one side of this chart (such as Friedman-Stanley[1989] and Burgess[1978]) which have proved extremely influential, but do not easily fall into the cartoon like summary of the history I have tried to give above. \section{Dynamic changes in topologies} The main issue here is to discuss some of the tricks that enable us to change topologies and remain in the category of Polish spaces. \subsection{Polish spaces and Polish groups [definitions; the statement of the topological Vaught conjecture]} \begin{Definition} A topological space is said to be {\it Polish} if it is separable and there is some complete metric which generates its topology. \end{Definition} \no{\bf Examples} (i) $\R$, the reals equipped with the topology generated by the open intervals. (The usual euclidean distance gives a complete metric.) (ii) $C([0, 1])$, the continuous functions from the unit interval to $\R$ with the topology generated by the sup norm, $d(f_1, f_2)=$ sup$_{0\leq x\leq 1}|f_1(x)-f_2(x)|.$ (iii) $\N$ in the discrete topology. As a metric, \leftskip 0.6in \noindent $d(x_1, x_2)=1$ if $x_1\neq x_2$, \noindent $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=0$ if $x_1=x_2$. \leftskip 0in (iv) $\N^{\N}$ in the product topology. (v) More generally if $(X_i)_{i\in\N}$ is a sequence of Polish spaces, then their infinite product \[\prod_\N X_i\] is a Polish space in the product topology. \begin{proof} After possibly replacing each metric $d_i$ with \[\hat{d}_i(x, y)=\frac{d_i(x, y)}{d_i(x, y)+1}\] we may assume that each $d_i$ is bounded by 1. Then we can obtain a compatible complete metric on the product given by \[d(\vec x, \vec y)=\sum_{i\in \N}2^{-i}d_i(x_i, y_i)\] where \[\vec x=(x_0, x_1, x_2, \ldots, x_i, \ldots)\] \[\vec y= (y_0, y_1, y_2, \ldots, y_i, \ldots)\] are both in $\prod_\N X_i$. \end{proof} \begin{lemma} \label{G_d} Let $X$ be a Polish space and $A\subset X$ a $G_{\delta}$ subset -- that is to say, defined by a countable intersection of open sets. Then $A$ in the subspace topology is again a Polish space. \end{lemma} \begin{proof} First for open ${\cal O}\subset X$. Let $d$ be complete metric for $X$, and find continuous \[f: {\cal O}\rightarrow \R\] with $f(x)\rightarrow \infty$ as $x\rightarrow X\setminus {\cal O}$. (For instance, $f(x)=$ inf$\{\frac{1}{d(x, c)}: c\notin {\cal O} \}$.) Then define $d_{\cal O}$ on ${\cal O}$ by \[d_{\cal O}(x, y)=d(x, y)+|f(x)-f(y)|.\] Suppose $(x_i)$ is a sequence in ${\cal O}$ that is $d_{\cal O}$-Cauchy. From the definition of $d_{\cal O}$ it must be $d$-Cauchy, and hence by assumption that $d$ witnesses $X$ Polish, there is a limit $x_\infty$. But we must also have that $(f(x_i))_{i\in\N}$ will be Cauchy in $\R$, and thus $x_\infty\in {\cal O}$. For the general case \[A=\bigcap_{i\in\N}{\cal O}_i\] we can take \[f_n: {\cal O}_n\rightarrow\R\] as above and let \[d_A(x, y)=d(x, y)+\sum_{n\in\N} 2^{-n}|f_n(x)-f_n(y)|.\] \end{proof} \begin{Definition} A topological group is said to be {\it Polish} if it is Polish as a space. \end{Definition} \begin{examples} (i) $(\R, +)$, the reals equipped with its usual additive structure. (ii) Any discrete group. Countable products of discrete groups, and more generally the class of Polish groups is closed under countable products. (iii) $S_\infty$ the group of all permutations of $\N$, equipped with the topology of pointwise convergence. So as a basis we may take all sets of the form \[\{\pi\in S_\infty: \pi(\ell_1)=k_1, ..., \pi(\ell_n)=k_n\}.\] (iv) $U_\infty$ the unitary group of $\ell^2$, infinite dimensional (separable) Hilbert space. Thus $U_\infty$ is the group of all bijective linear operators \[T:\ell^2\rightarrow \ell^2\] such that for all $\xi, \zeta\in \ell^2$ \[\langle \xi, \zeta\rangle=\langle T(\xi), T(\zeta)\rangle.\] We give this the topology generated by the subbasis consisting of sets of the form \[\{T\in U_\infty: |\langle T(\xi), \zeta\rangle - \langle T_0(\xi), \zeta\rangle|<\epsilon\}.\] \end{examples} \begin{definition} Let $G$ be a Polish group. We say that a Polish space $X$ is a {\it Polish $G$-space} if it is a Polish space which is equipped with a continuous action by $G$. We then let $E_G$ denote the orbit equivalence relation: $x_1 E_G x_2$ if \[\exists g\in G(g\cdot x_1=x_2).\] For $x\in X$ let $[x]_G$ indicate the orbit of $x$: \[\{y\in X: \exists g\in G(g\cdot x=y)\}.\] $X/G$ indicates the set of orbits. \end{definition} \begin{examples} (i) Let $\Z$ act on the complex unit circle $=\{e^{ix}: x\in \R\}$ in the following manner: For $e^{ix}$ and $k\in\Z$ we let \[k\cdot e^{ix}=e^{i(x+2\pi k\surd 2)}.\] In other words the action is induced by a rotation of $\surd 2$ radians. In this way, among many others, we could view the complex unit circle as a Polish $\Z$-space. (ii) Given a countable set $X$ we can consider the space $(U_\infty)^X$ of function from $X$ to the unitary group of Hilbert space; this space is Polish since it is naturally isomorphic to a countable product of $U_\infty$ with itself. We can let $U_\infty$ act on this space by point-wise conjugation: \[(T\cdot f)(x)=T\circ f(x)\circ T^{-1}\] for any $x\in X$, $f\in (U_\infty)^X$. $U_\infty$ is a Polish group and under this action $(U_\infty)^X$ is a Polish $U_\infty$-space. (iii) Perhaps the most natural example for a logician of a Polish group acting on a Polish space is the action of $S_\infty$ on the space of countable models. There are a few fiddley points here related to precisely which topology we would want to place on this space, so I will postpone the details to section \ref{2.2}. \end{examples} \begin{conjecture}{\bf Topological Vaught Conjecture}\footnote{ This conjecture is very much still open.}: Whenever $G$ is a Polish group and $X$ a Polish $G$-space, then either \begin{enumerate} \item $X/G\leq \aleph_0$ (countably many orbits\footnote{In general I use ``countably many" to mean either finitely many or exactly $\aleph_0$ many.}), or \item $X/G=2^{\aleph_0}$ (continuum many orbits). \end{enumerate} \end{conjecture} This conjecture should be compared with its counterparts for first order logic and countably infinitary logic. \begin{conjecture} {\bf Original Vaught Conjecture} Let $T$ be a first order theory. Then either $T$ has countably many countable models up to isomorphism or it has continuum many countable models up to isomorphism. \end{conjecture} \begin{definition} For ${\cal L}$ a language we let ${\cal L}_{\omega_1, \omega}$ be the collection of formulas obtained by closing under countable disjunctions and conjunctions, as well as the usual first operations of existential and universal quantification, substitutions, and negation. \end{definition} \begin{conjecture} {\bf Vaught Conjecture for ${\cal L}_{\omega_1, \omega}$} Let ${\cal L}$ be a countable language and $\sigma$ a sentence in ${\cal L}_{\omega_1, \omega}$. Then either $\sigma$ has countably many countable models up to isomorphism or it has continuum many countable models up to isomorphism. \end{conjecture} As stated here, the topological Vaught conjecture is trivially true under CH (the continuum hypothesis). Frequently people use a reformulation of this conjecture that is equivalent under the negation of CH by Burgess' theorem \ref{burgess} below. And in fact this equivalence is local, action by action, not just global: For each Polish group $G$ and Polish $G$-space $X$, if CH fails then there are perfectly many orbits if and only if there are continuum many. In fact I will find it convenient to refer to the following as {\it the} topological Vaught conjecture. \begin{conjecture}{\bf Topological Vaught Conjecture} (Modern form): Whenever $G$ is a Polish group and $X$ a Polish $G$-space, then either \begin{enumerate} \item $X/G\leq \aleph_0$ (countably many orbits), or \item there is a perfect set $P\subset X$ (that is to say, a set which is closed, non-empty, and has no isolated points) such that any two points in $P$ are orbit inequivalent under the action of $G$. \end{enumerate} \end{conjecture} Since any perfect set necessarily has size $2^{\aleph_0}$ it is easily seen that this alternative form of the topological Vaught conjecture implies the earlier. In fact it turns out that these two conjectures are equiprovable over ZFC -- one can prove the first using only the axioms of ZFC if and only if one can prove the second using only the axioms of ZFC. This equiprovability follows by a short absoluteness argument. If the stronger form of the topological Vaught conjecture failed, then we can add lots of reals to blow up the size of the continuum; in the resulting model there must still be uncountably many orbits, by Shoenfield absoluteness; but equally there cannot be a perfect set of orbit inequivalent points, again by Shoenfield absoluteness; and thus we arrive at a contradiction with the local equivalence of the two forms of the topological Vaught conjecture under $\neg$CH. We sketch this argument due to Sami at \ref{borel} below. The proof of this equivalence for the model theoretic versions of the Vaught conjecture has also been written up precisely in Steel[1978]. One way to think of this alternate form of the topological Vaught conjecture is this: It is the conjecture that there cannot be ``exactly $\aleph_1$ many orbits", where we understand ``exactly" to mean that there are $\aleph_1$ many orbits but not perfectly many orbits. It might be helpful before continuing with the discussion of the painfully open topological Vaught conjecture to look at some parallel problems. One can pose similar kinds of questions for other classes of equivalence relations, and in almost every case the answer is known. I will need a couple more definitions from descriptive set theory. If you have not seen these before they may appear technical, but the motivation is a natural one: We wish to describe certain classes of equivalence relations which are not intractably complicated and compare them with the kind that arise for Polish group actions. \begin{definition} A subset $B$ of a Polish space $X$ is said to be {\it Borel} if it appears in the smallest collection ${\cal B}$ of subsets of $X$ such that \begin{enumerate} \item ${\cal B}$ includes the open sets in $X$; \item ${\cal B}$ is an algebra -- that is to say it is closed under complementation in $X$ and finite unions and intersections; \item moreover ${\cal B}$ is a $\sigma$-algebra -- that is to say, an algebra which is also closed under countable intersections. \end{enumerate} (Thus the collection of {\it the Borel sets} is the smallest $\sigma$-algebra containing the open sets.) An equivalence relation $E\subset X\times X$ is {\it Borel} if it is Borel in the product topology. A subset $A$ of a Polish space $X$ is said to be {\it analytic} or $\Ubf {\Sigma}^1_1$ if there is a Polish space $Y$ and a Borel set $B\subset X\times Y$ (that is to say, Borel in the product topology) with \[A=\{x\in X| \exists y\in Y((x, y)\in B)\}.\] $P\subset X$ is $\Ubf {\Pi}^1_1$ or {\it co-analytic} if its complement is analytic. \end{definition} Clearly every Borel set is analytic, and then since the Borel sets are closed under complementation we equally have that every Borel set is co-analytic. It is not hard to see that if a Polish group $G$ acts continuously on a Polish space $X$ then the set \[\{(x_1, g, x_2)| g\cdot x_1=x_2\}\] is closed in $X\times G\times X$, and thus the resulting orbit equivalence relation $E_G$ is analytic. In the context of model theory Borel sets have an intuitive meaning. They correspond roughly to sets which can be defined by sentences in ${\cal L}_{\omega_1, \omega}$. (See corollary \ref{lopez} below.) Analytic sets can be thought of as like a generalization of the set of models in a language which admit an expansion to a model of some given theory in a larger language. \begin{theorem} (Silver[1980]) Let $E$ be a Borel (or even $\Ubf {\Pi}^1_1$) equivalence relation on a Polish space $X$. Then either \begin{enumerate} \item $X/E\leq \aleph_0$, or \item $X/E=2^{\aleph_0}$ (continuum many equivalence classes), and in fact in this case there is a perfect set of $E$-inequivalent points in $X$. \end{enumerate} \end{theorem} This would be very nice for those of us who slave away trying to prove the topological Vaught conjecture if it were the case that the equivalence relations arising from Polish group actions were always Borel. Unfortunately while they must be analytic they can in general be non-Borel. \begin{theorem} (Burgess[1978]) \label{burgess} Let $E$ be a $\Ubf {\Sigma}^1_1$ equivalence relation on a Polish space $X$. \begin{enumerate} \item $X/E\leq \aleph_1$, or \item $X/E=2^{\aleph_0}$, and in fact in this case there is a perfect set of $E$-inequivalent points in $X$. \end{enumerate} \end{theorem} Burgess' theorem is optimal. If one considers the space $2^\N$ and sets $x_1Ex_2$ if $\omega_1^{{\rm ck}(x_1)}=\omega_1^{{\rm ck}(x_2)}$ (the supremum of the lengths of the $x_1$-recursive well orders of $\N$ equals the supremum of the lengths of the $x_2$-recursive well orders of $\N$) then we obtain an equivalence relation which has exactly $\aleph_1$ many classes and is in the complexity class $\Ubf {\Sigma}^1_1$. By Cohen we know that $2^{\aleph_0}$ may be bigger than $\aleph_1$, and thus we cannot hope to extend Burgess' result in ZFC. Thus any proof of the topological Vaught conjecture must use more than the simple fact that orbit equivalence relations being considered are analytic. It is known from Ryll-Nardzewski[1964] that the orbit equivalence relations arising from continuous actions of Polish groups on Polish spaces are a special kind of $\Ubf {\Sigma}^1_1$ equivalence relation in which every orbit is equivalence class is Borel, but unfortunately the counterexample above with $\aleph_1$ many equivalence classes is also one in which every class is Borel. It does not seem as if any easily stated fact about the kinds of equivalence relations arising from Polish group actions would allow some general descriptive set theoretic proof that there cannot be exactly $\aleph_1$ many orbits. The proof, if the conjecture is indeed true, would need to use very specific facts about the context of a Polish group action. It is unlikely to be soft. \subsection{The space of countable models [isomorphism of countable models viewed as a kind of Polish group action; different Polish topologies we may place on spaces of countable models]} \label{2.2} \begin{definition} Let ${\cal L}$ be a countable language, and for notational simplicity assume that ${\cal L}$ is generated by a single binary relation $R$. We then let Mod$({\cal L})$ be the space of all ${\cal L}$-structures whose underlying set is $\N$, equipped with the topology generated by sets of the form \[\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models \varphi(k_1, \ldots, k_n)\}\] where $\varphi$ is quantifier free and $k_1, \dots, k_n$ are in $\N$. There is then a natural homeomorphism \[{\rm Mod}({\cal L})\rightarrow \{0, 1\}^{\N\times \N}\] \[{\cal M}\mapsto \chi_{R^{\cal M}}\] associating to each ${\cal M}$ the characteristic function of its interpretation of $R$. Thus Mod$({\cal L})$ is a Polish space. We let $S_\infty$ act on Mod$({\cal L})$ by \[\pi\cdot {\cal M}\models R(k_1, k_2)\] if and only if \[{\cal M}\models R(\pi^{-1}(k_1), \pi^{-1}(k_2)).\] Then Mod$({\cal L})$ is a Polish $S_\infty$ space and the orbit equivalence relation is just the usual isomorphism relation. \end{definition} Our next goal is to see that the topological Vaught conjecture really does generalize the model theoretic versions of the Vaught conjecture. To do this we need to consider alternative Polish topologies we may place on the space of countable models. \begin{definition} (i) Let $\tau_{\rm FO}$ be the topology on ${\rm Mod}({\cal L})$ whose basic open sets have the form \[\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models \varphi(k_1, \ldots, k_n)\}\] where $\varphi$ is first order and $k_1, \dots, k_n$ are in $\N$. (ii) For $F\subset {\cal L}_{\omega_1, \omega}$ a countable fragment (closed under subformulas, instantiations, and first order operations), let $\tau_F$ be the topology on ${\rm Mod}({\cal L})$ whose basic open sets have the form \[\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models \varphi(k_1, \ldots, k_n)\}\] where $\varphi\in F$. \end{definition} All these provide Polish topologies on ${\rm Mod}({\cal L})$. (Proof for (i) below). They are all finer that the topology of quantifier free formulas. They all give rise to the same Borel structure as the topology of quantifier free formulas, which I will continue to use as my default choice of topology on the space. Thus for any theory $T$ the set \[\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models T\}\] is a closed $S_\infty$-invariant subspace of $({\rm Mod}({\cal L}), \tau_{\rm FO})$, and hence a Polish $S_\infty$ space in its own right. Sets of the form $\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models T\}$ correspond exactly to the closed invariant subspaces; the subspace is minimal iff $T$ complete. A key step in the proof of Becker's theorem (which is coming up in section \ref{3.3}) is to pass from a not so good topology, like the quantifier free topology, to a better topology, like $\tau_{\rm FO}$. \begin{lemma} (Gregorczyk, Mostowski, Ryll-Nardzewski[1960]) $({\rm Mod}({\cal L}), \tau_{\rm FO})$ is a Polish space. \end{lemma} \begin{proof} Let $B$ be the set of substitutions of the form $\varphi(k_1, \ldots, k_n)$ where $\varphi$ is a first order ${\cal L}$ formula. Let $X=\{0, 1\}^B$, the space of all functions from $B$ to $\{0, 1\}$. We may naturally identify $({\rm Mod}({\cal L}), \tau_{\rm FO})$ with the functions in $f\in X$ which preserve the ``inductive unraveling of truth" -- for instance, $f(\varphi_1(k)\wedge \varphi_2(k))=1$ if and only if $f(\varphi_1(k))=f(\varphi_2(k))=1$. Most of these requirements correspond to closed conditions in $X$, except for the rule governing existential quantifiers, \[f(\exists x\varphi(k, x))=1\Leftrightarrow\exists l\in \N f(\varphi(k, l))=1{\rm ,}\] which is still no worse than $G_\delta$. Thus $({\rm Mod}({\cal L}), \tau_{\rm FO})$ is homeomorphic to a $G_\delta$ subspace of a Polish space and therefore Polish. \end{proof} In many respects $\tau_{\rm FO}$ would seem to be a nicer topology than the topology generated by quantifier free formulas. This becomes more apparent after we introduce the notion of {\it Vaught transform} below and note that $\tau_{\rm FO}$ has a basis that is closed under the Vaught transforms. The next subsection discusses an important theorem due to Becker and Kechris which states that among other things that for any Polish $G$-space there are ``cofinally" many Polish topologies in which that action is still continuous and that are closed under the Vaught transforms. This was arguably the main result of Becker-Kechris[1996]. Sami[1994] had previously shown the same thing for a general class of continuous actions of $S_\infty$, including its natural action on $({\rm Mod}({\cal L}), \tau_{\rm FO})$. \subsection{Becker-Kechris theorem [Vaught transforms; adding to the topology of a Polish $G$-space and keeping the action continuous]} \begin{definition} Let $G$ be a Polish group and $X$ a Polish $G$-space. For $V\subset G$ open and $B\subset X$ (preferably Borel), we let \[B^{\Delta V}\] be the set of $x\in X$ for which there is a non-meager\footnote{A set is meager if it is included in a countable union of closed sets each of which contain no non-empty open subset.} set of $g\in V$ for which $g\cdot x\in B$. We symmetrically define $B^{* V}$ to be the set of $x\in X$ for which there is a relatively comeager set of $g\in V$ with $g\cdot x\in B$. \end{definition} \begin{lemma} (Vaught[1973]) For $\varphi\in{\cal L}_{\omega_1, \omega}$, $B$ the set of ${\cal M}\in {\rm Mod}({\cal L})$ with $ {\cal M}\models \varphi(k_1, \ldots, k_n, a_1, \ldots, a_m)\}{\rm ,}$ and $V=\{\pi\in S_\infty: \pi(a_1)=b_1,\ldots, \pi(a_m)=b_m\}$, \[B^{\Delta V}=\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models \exists x_1\ldots\exists x_n \varphi(\vec x,\vec b)\}{\rm ,}\] \[B^{* V}=\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models \forall x_1\ldots \forall x_n \varphi(\vec x, \vec b)\}{\rm .}\] \end{lemma} Using this one can show by an induction on the Borel complexity of $B$ that: \begin{corollary} (Lopez-Escobar[1965]) \label{lopez} If $B\subset {\rm Mod}({\cal L})$ is Borel and invariant then there is a sentence $\varphi\in {\cal L}_{\omega_1, \omega}$ with \[B=\{{\cal M}\in {\rm Mod}({\cal L}): {\cal M}\models \varphi\}.\] \end{corollary} \begin{theorem} (Becker-Kechris[1996]) \label{change} Let $G$ be a Polish group and $X$ a Polish $G$-space. Let $B\subset X$ be Borel. Let ${\cal B}$ be a countable basis for $G$. Then there is a Polish topology $\tau$ on $X$ such that: \begin{enumerate} \item $\tau$ is richer than the original topology on $X$; \item for any $V\in {\cal B}$ \[B^{\Delta V}\in \tau{\rm ;}\] \item $\tau$ has a countable basis ${\cal C}$ such that for all $U_1, U_2\in {\cal C}$ and $V_1, V_2\in {\cal B}$, \[(U_1\cap (X\setminus U_2)^{*V_1})^{\Delta V_2}\in {\cal C}{\rm ;}\] {\rm (This rather ominous looking condition plays a central role in the proof of Becker's theorem \ref{beckertheorem} below; it represents an approximation to the basis having the kind of closure conditions presented by the natural basis for $\tau_{\rm FO}$ on Mod$({\cal L})$.)} \item the action of $G$ on $X$ is still continuous with respect to $\tau$ (and thus $(X, \tau)$ is still a Polish $G$-space). \end{enumerate} \end{theorem} In the case that $B$ is invariant under the action of $G$ we obtain that $B^{\Delta G}=B^{*G}=B$, and thus $B$ is an invariant clopen subset of $(X,\tau)$. In particular, since for any sentence $\sigma\in {\cal L}_{\omega_1, \omega}$ we have that \[\{{\cal M}: {\cal M}\models \sigma \}\] is Borel and invariant, Vaught's conjecture for ${\cal L}_{\omega_1, \omega}$ is implied by the topological Vaught conjecture for $S_\infty$. It would take us too far afield to go into the details of their proof. Instead I will present a lemma, which by a transfinite induction can be used to prove the Becker Kechris theorem. \begin{lemma} $X$ a Polish $G$-space, $V\subset G$ open and $C\subset X$ closed. Then there is a richer Polish topology on $X$ for which the action is still continuous and $C^{\Delta V}$ is now open. \end{lemma} \begin{proof} By a theorem of Birkhoff and Kakutani (see 1.1.1 Becker-Kechris[1996]) we may find a compatible right invariant metric $d$ on $G$ that is bounded by 1. Let ${\cal L}(G)$ be the space of $f: G\rightarrow [0, 1]$ such that for all $g_1, g_2 \in G$ \[|f(g_1)-f(g_2)|\leq d(g_1, g_2){\rm .}\] Then ${\cal L}(G)$ is a Polish $G$-space under the natural action $(g\cdot f)(h)= f(hg).$ For $x\in X$ let $f_x\in {\cal L}(G)$ be given by \[f_x(g)={\rm \: inf}\{d(g, h): h\cdot x\notin C\}{\rm .}\] $X_C=\{(x, f_x): x\in X\}$ is an invariant subset of $X\times {\cal L}(G)$ and $\pi_C:X\rightarrow X_C$ \[x\mapsto (x, f_x)\] is a bijection which respects the $G$-action. One then verifies, as in Hjorth[1999], that $X_C$ is a $G_\delta$ subset of $X\times {\cal L}(G)$ and thus a Polish $G$-space. The pullback of the topology on $X_C$ to $X$ is as required. \end{proof} \section{Smoothness and the Glimm-Effros dichotomy} \subsection{Smoothness [Effros lemma; $G_\delta$ orbits as the analogue of atomic models]} \begin{definition} An orbit equivalence relation $E_G$ on Polish space $X$ is said to be {\it smooth} if there is a Borel function\footnote{That is to say, for any open $O\subset \R$ the set $f^{-1}(O)$ appears in the $\sigma$-algebra on $X$ generated by the open sets --i.e., $f^{-1}[O]$ is Borel.} $f: X\rightarrow \R$ such that for all $x_1, x_2\in X$ \[x_1 E_G x_2 \Leftrightarrow f(x_1)=f(x_2).\] \end{definition} Burgess[1979] showed that for $E_G$ given by the continuous action of a Polish group action we have $E_G$ smooth if and only if there is a Borel set meeting every orbit in exactly one point. \begin{example} For ${\cal M}, {\cal N}\in {\rm Mod}({\cal L})$ set \[{\cal M}\equiv {\cal N}\] if they have the same (first order) theory. Then $\equiv$ is smooth. To see this, let $(\varphi_n)_{n\in\N}$ enumerate the first order sentences of ${\cal L}$. For ${\cal M}\in {\rm Mod}({\cal L})$ associate the real $x_{\cal M}\in (0, 1)$ in whose decimal expansion we have a 5 at the $n^{\rm th}$ position if ${\cal M}\models \varphi_n$ and 6 if it does not. ${\cal M}\mapsto x_{\cal M}$ is a Borel function witnessing smoothness. \end{example} \begin{example} Let $\Z_2^{<\N}$ be the infinite binary sequences which are eventually constantly 0. This is a countable set, and we give it the discrete topological structure. We further introduce the group structure obtained by pointwise addition mod 2; thus for $\vec g, \vec h\in \Z_2^{<\N}$ and $n\in\N$ we have \leftskip 0.5in \no $(\vec g +\vec h)(n)=0$ if $\vec h(n)= \vec g(n)$ and \no $\:\:\:\:\:\:\:\: \:\: \:\: \:\:\:\:\:=1$ otherwise. \leftskip 0in We then let it act on $2^\N$ in the natural manner: For $\vec x\in 2^\N=_{df}{\{0, 1\}}^\N$, $\vec g\in \Z_2^{<\N}$, $n\in\N$, we again have $(\vec g\cdot \vec x)(n)=0$ if and only if $\vec x(n)=\vec g(n)$. \end{example} \begin{lemma} \label{non-smooth} The above orbit equivalence relation $E_{\Z_2^{<\N}}$ is non-smooth. \end{lemma} \begin{proof} Suppose instead (by Burgess[1979]) $B$ is a Borel selector. Let $\mu$ be the product (``coin flipping") measure on $2^\N$. Note that $(2^\N, \mu)$ is a probability space and ${\Z_2^{<\N}}$ acts by measure preserving transformations on $(2^\N, \mu)$. Since countably many translates of $B$ cover $2^\N$ we must have $\mu(B)>0$. But then there are $\aleph_0$ many {\it disjoint} translates of $B$, each having the same measure, and hence a contradiction to $\mu(2^\N)=1$. \end{proof} There are many theorems giving necessary conditions for non-smoothness in terms of being able to {\it embed} this orbit equivalence relation $E_{\Z_2^{<\N}}$. For a wide variety of groups (but not $S_\infty$!) one can show that $E_G$ is smooth if and only if we cannot so embed this canonical example of a non-smooth equivalence relation. For instance: \begin{theorem} (Becker-Kechris) \label{E_0} Let $G$ be a Polish group and $X$ a Polish $G$-space, $x\in X$. Suppose that no orbit $[y]_G$ with the same closure as $[x]_G$ is comeager in \[\overline{[x]_G}{\rm ,}\] the closure of the orbit. Then \begin{enumerate} \item $E_G$ is not smooth and \item there is a continuous injection \[\rho: 2^\N\rightarrow X\] such that for all $\vec x, \vec y\in 2^\N$ \[\vec x E_{\Z_2^{<\N}} \vec y\Leftrightarrow (\rho(\vec x) E_G \rho(\vec y)).\] \empty \end{enumerate} \end{theorem} In fact 2. implies 1. in the statement of the theorem. After this Becker-Kechris theorem one might conclude that it would be nice to understand when an orbit is comeager in its own closure. Here a theorem was proved some 30 years earlier by Effros: \begin{theorem} (Effros[1965]) Let $G$ be a Polish group and $X$ a Polish $G$-space. Then for $x\in X$ the following are equivalent: \begin{enumerate} \item $[x]_G$ is comeager in $\overline{[x]_G}$; \item $[x]_G$ is $G_\delta$; \item the map \[G\rightarrow [x]_G\] \[g\mapsto g\cdot x\] is open. \end{enumerate} \end{theorem} In the case of the logic action of $S_{\infty}$ on $({\rm Mod}({\cal L}), \tau_{\rm FO})$ -- the ${\cal L}$-structures on $\N$ with the topology generated by first order logic -- one has that the $G_\delta$ orbits correspond to atomic models, see Miller[1978]. (The omitting types theorem shows that the realization of any non-principal type is meager.) \subsection{Examples [algebraic characterization of discrete groups with smooth dual (Thoma); Bernoulli shifts up to isomorphism are smooth (Ornstein); but not general mpt's (Feldman)]} \begin{example} As before let $U_\infty$ be the infinite dimensional unitary group, consisting of all isomorphisms of Hilbert space. For $H$ a countable (discrete) group we define Rep$(H)$ to be the space of all homomorphism \[\rho: H\rightarrow U_\infty.\] This may naturally be identified with a closed subspace of the $H$-fold product of $U_\infty$, \[\prod_H U_\infty{\rm ,}\] and thus is a Polish space. We let $U_\infty$ act on Rep$(H)$ by ``pointwise conjugation", \[(T\cdot \rho)(h)=T\circ\rho (h) \circ T^{-1}.\] We then let Irr$(H)$ be the {\it irreducible} representations in Rep$(H)$ -- that is to say those $\rho$ for which there are no non-trivial closed subspaces of Hilbert space closed under the operators $\{\rho(h): h\in H\}$. It turns out that Irr$(H)$ is a $G_\delta$ subset of Rep$(H)$, and thus a Polish $U_\infty$-space in its own right. \end{example} \begin{theorem} (Thoma[1964]) \label{thoma} The orbit equivalence $E_{U_\infty}$ on {\rm Irr}$(H)$ is smooth if and only if $H$ is abelian-by-finite. \end{theorem} One can extend the definition of Rep$(H)$ and Irr$(H)$ to a general locally compact group. Here one demands that the representations all be continuous, and we give these spaces the ``compact-open" topology, generated by sets of the form \[\{\rho: \rho[C]\subset O\}{\rm ,}\] where $C\subset H$ is compact and $O\subset U_\infty$ is open. Again we obtain that Rep$(H)$ and Irr$(H)$ are $U_\infty$ Polish spaces. \begin{theorem} (Harish-Chandra[1953], more or less\footnote{I fib terribly. In fact he proved that the representations of semi-simple connected Lie groups are all ``type I". While at the time it was apparently felt that this should turn into a proof of the smoothness of $E_{U_\infty}|_{{\rm Irr}(H)}$, this does not seem to have been confirmed until the proof of the ``Mackey conjecture" a few years later by Glimm}) If $H$ is a semi-simple connected Lie group, then the orbit equivalence $E_{U_\infty}$ on {\rm Irr}$(H)$ is smooth. \end{theorem} \begin{example} Let $M_\infty$ be the group of measure preserving transformations of the unit interval, subject to identification of transformations agreeing almost everywhere. We give this the topology generated by sets of the form \[\{\pi\in M_\infty: \mu(\pi(A)\Delta \pi_0(A))<\epsilon\}{\rm ,}\] for $\pi_0\in M_\infty$, $A\subset [0,1]$ measurable, $\epsilon>0$. In this topology and the operation of composition it becomes a Polish group. It seems natural to think of measure preserving transformations as really being ``isomorphic" if they are the same up to some isomorphism of the underlying standard Borel probability space. Thus we are led to the orbit equivalence relation of $M_\infty$ acting on itself by conjugation: \[\pi\cdot \rho=\pi\circ \rho\circ \pi^{-1}.\] \end{example} \begin{theorem} (Feldman\footnote{Actually something stronger and rather more specific is shown in Feldman[1974]. He obtains non-smoothness even for the property K transformations. See a Foreman[to appear] for a thorough discussion of this and related results.}) The orbit equivalence relation $E_{M_\infty}$ obtained above is not smooth. \end{theorem} For some special classes of measure preserving transformations one obtains smoothness of the restriction of $E_{M_\infty}$. For instance Ornstein showed that the isomorphism relation on {\it Bernoulli shifts} is smooth and that {\it entropy} provides a complete invariant. \subsection{Becker's theorem [Glimm-Effros dichotomy, and hence Vaught's conjecture, for solvable groups]} \label{3.3} \begin{definition} (Becker) A Polish group $G$ satisfies the {\it Glimm-Effros dichotomy} if whenever $G$ acts continuously on a Polish space $X$ either: \begin{enumerate} \item $E_G$ is smooth; or \item there is a continuous injection \[\rho: 2^\N\rightarrow X\] such that for all $\vec x, \vec y\in 2^\N$ \[\vec x E_{\Z_2^{<\N}} \vec y\Leftrightarrow (\rho(\vec x) E_G \rho(\vec y)).\] \end{enumerate} \end{definition} It is not hard to see that the Glimm-Effros property for a group $G$ implies that it does not provide a counterexample for the topological Vaught conjecture. \begin{lemma} Let $G$ be a Polish group and $X$ a Polish $G$-space. If $G$ satisfies the Glimm-Effros dichotomy then $X$ does not have exactly $\aleph_1$ many orbits. \end{lemma} \begin{proof} Let $X$ be a Polish $G$-space. We wish to show that either there are perfectly many orbits or (at most) countably many orbits. Apply the definition of the Glimm-Effros dichotomy. Suppose first we are in case 2, and so we have a continuous embedding $\rho$ of $E_{\Z_2^{<\N}}$ into $E_G$. We can consider the compact set \[C=\{\vec x| \forall n(n \: {\rm not\: a\: prime\: power \:}\Rightarrow x(n)=0) \] \[\wedge \forall p \: {\rm prime}\: \forall k>0 (x(p)=x(p^k))\}.\] Any two elements of $C$ are $ E_{\Z_2^{<\N}}$-inequivalent. $\rho[C]$ is again compact and without isolated points. And thus $\rho[C]$ provides a perfect set of orbit inequivalent points. Alternatively suppose we are in case 1. Then there is Borel $G$-invariant \[f: X\rightarrow \R{\rm ,}\] such that for all $x_1, x_2 \in X$ \[x_1E_G x_2 \Leftrightarrow f(x_1)=f(x_2).\] A routine calculation shows that $E_G$ must consequently be Borel, and we can finish by applying Silver's theorem. \end{proof} Ever expanding classes of groups have been shown to satisfy the Glimm-Effros dichotomy: \begin{enumerate} \item Locally compact Polish groups (Effros, after Glimm). \item Abelian Polish groups (Solecki). \item Nilpotent (Hjorth). \item Solvable (Becker). \end{enumerate} Notably $S_\infty$ fails this dichotomy. \begin{theorem} (Becker) Solvable Polish groups satisfy the Glimm-Effros dichotomy. \label{beckertheorem} \end{theorem} \begin{proof} Let $G$ be a solvable Polish group and $X$ a Polish $G$-space. Assume that 2. fails, that is to say, we are unable to continuously inject $E_{\Z_2^{<\N}}$ into $E_G$. Since $G$ is solvable it has a {\it complete} compatible right invariant metric (Gao[1998], Hjorth-Solecki[1999]), call it $d_G$. Following theorem \ref{change} we may assume that we have a basis ${\cal C}$ for the topology $\tau$ on $X$ and ${\cal B}$ on $G$ such that: \begin{enumerate} \item for all $U_1, U_2\in {\cal C}$ and $V_1, V_2\in {\cal B}$, \[(U_1\cap (X\setminus U_2)^{*V_1})^{\Delta V_2}\in {\cal C}{\rm ;}\] \item the action of $G$ on $X$ is still continuous with respect to $\tau$ (and thus $(X, \tau)$ is still a Polish $G$-space). \end{enumerate} It is not hard to show that there is a Borel function \[\theta: X\rightarrow \R\] such that for all $x_1, x_2 \in X$ \[\theta(x_1)=\theta(x_2)\Leftrightarrow \overline{[x_1]_G}= \overline{[x_2]_G}.\] (We may code the closure of an orbit by a real number, and the coding can be completed in a Borel manner.) Thus it suffices to show that $\overline{[x]_G}$ is a complete invariant of $[x]_G$; that is to say, $\overline{[x_1]_G}= \overline{[x_2]_G}$ if and only if $[x_1]_G =[x_2]_G$. Fix some $x\in X$. Now by theorem \ref{E_0}, the Becker-Kechris theorem on embedding $E_{\Z_2^{<\N}}$, we know that there must an $[y]_G$ with the same closure as $[x]_G$ which is comeager in $\overline{[x]_G}=\overline{[y]_G}$. Using the Effros theorem, we can obtain for each open \[V_n=\{g\in G: d_G(1_G, g)<2^{-n}\}\] some open $U_n\in{\cal C}$ containing $y$ with diameter less than $2^{-n}$ such that \[V_n\cdot y\supset U_n\cap [y]_G.\] Then using the closure assumptions on the the basis ${\cal C}$ for the topology and a calculation which we do not present here, it can be argued that whenever $U\in {\cal C}$ and \[U\cap U_n\cap [y]_G\neq\emptyset\] \[\hat{x}\in U_n\cap [x]_G\] then there are $g$ arbitrarily close to the identity with \[g\cdot \hat{x}\in U^{\Delta V_n^2},\] and thus in particular we may find $g,h\in V_n^2$ with $hg\cdot \hat{x}\in U$. Repeating this observation infinitely often we may find a sequence $(h_n)_{n\in\N}$ in $G$ and $(x_n)_{n\in\N}$ in $[x]_G$ with: \begin{enumerate} \item $x_n\in U_n$; \item $h_n\in V_n^4$, and thus $d_G(h_n, 1_G)<2^{-n+2}$; \item $x_{n+1}=h_n\cdot x_n$. \end{enumerate} We find $h_{n+1}$ by applying the preceding observation with $U_{n+1}$ taking the place of the set $U$. Thus if we let $g_n=h_nh_{n-1}\ldots h_1h_0$ then we have $x_n=g_n\cdot x_0\in [x]_G$ and by 1. we have \[x_n\rightarrow y.\] On the other hand 2. and right invariance of the metric gives \[d_G(g_n, g_{n-1})< 2^{-n+2}\] and thus $(g_n)$ is Cauchy. But for $g_\infty$ the limit we obtain from continuity of the action, \[g_\infty\cdot x_0=y.\] Thus we have established that $[x]_G=[y]_G$ is comeager in $\overline{[x]_G}$. Generalizing we therefore have that whenever $\overline{[x_1]_G}= \overline{[x_2]_G}$ we must have both $[x_1]_G$ and $[x_2]_G$ are comeager in this common closure. Hence the orbits overlap. Hence they are equal. \end{proof} \begin{example} $S_\infty$ does not satisfy the Glimm-Effros dichotomy. Let ${\cal L}$ be generated by binary function $+$, unary function $-(\:)$, and constant symbol $0$. Let AG be the subset of Mod$({\cal L})$ consisting of ${\cal G}$ in which $(+^{\cal G}, -(\:)^{\cal G}, 0^{\cal G})$ defines an abelian group structure on $\N$. Certainly this corresponds to a first order definable property on ${\cal G}$, and so AG is a closed invariant subspace of Mod$({\cal L})$. Then for $p$ a prime let TAG$_p$ be the set of all $p$-groups -- that is to say, ${\cal G}$ such that for all $g\in {\cal G}$ there exists $n>0$ with \[{\cal G}\models p^n\cdot g=0.\] TAG$_p$ is an invariant $G_\delta$ set, and hence a Polish $S_\infty$-space in its own right. The Ulm analysis of abelian $p$-groups shows that we may find a {\it reasonably nice} (e.g. absolutely $\Delta^1_2$ in the codes, in the sense of Hjorth[2000]) function $U:{\rm TAG}_p\rightarrow 2^{<\omega_1}$ assigning bounded subsets of $\aleph_1$ as complete invariants. {\bf Claim:} ${\rm TAG}_p/S_\infty$ is not smooth. Roughly this follows because in general there can be no {\it reasonably simply definable} (e.g. absolutely $\Delta^1_2$!) $\omega_1$ sequence of reals. {\bf Claim:} We cannot embed $E_{\Z_2^{<\N}}$ into ${\rm TAG}_p/S_\infty$. Otherwise by postcomposing with $U$ we would obtain an assignment $\hat{U}$ of bounded subsets of $\aleph_1$ as complete invariants for $2^\N/{\Z_2^{<\N}}$. Then we would obtain some fixed $\alpha<\omega_1$ and $\mu$-measure\footnote{$\mu$ the ``coin flipping" measure from before} one set on which the reduction $\hat{U}$ takes values inside $2^\alpha$. (To see this, we can take $(M; \in)$ some countable transitive model containing the real parameter used in the embedding of $E_{\Z_2^{<\N}}$; then there will be a $\mu$-measure 1 set of $x\in 2^\N$ that are ``random" generic for $M$ -- and for any such $x$ there will be some $\alpha< \omega_1^M=\omega_1^{M[x]}<\omega_1^V$ with $\hat{U}(x)\in 2^\alpha$). But then since $2^\alpha$ is naturally isomorphic to $2^\N$, and hence Borel isomorphic to $\R$, we obtain a contradiction to our proof before that $2^\N/{\Z_2^{<\N}}$ is non-smooth on any $\mu$-measure one set. (Here we have skipped over technicalities relating to the fact that $\hat{U}$ is not necessarily Borel. The issue here however is that it can be chosen to be universally measurable in the codes; hence on a measure one set it will equal a Borel function, or alternatively the proof of \ref{non-smooth} can be seen to work in the more general context of $\mu$-measurable functions.) \end{example} \section{Vaught's conjecture on analytic sets} \subsection{Variations on the conjecture} \begin{notation} For $H$ a Polish group let TVC$(H)$ be the statement that whenever $X$ is a Polish $H$-space then either there are only countably many orbits or there is a perfect set $P\subset X$ of orbit inequivalent points: \[\forall x_1, x_2 \in P(x_1\neq x_2 \Rightarrow [x_1]_H\neq [x_2]_H).\] Let WVC$(H)$ be the statement that when $X$ is a Polish $H$-space then either there are only countably many orbits or there are $2^{\aleph_0}$ many orbits. \end{notation} \begin{lemma} (Sami) \label{borel} If $H$ is a Polish group and $X$ is a Polish $H$-space witnessing the failure of TVC$(H)$, then $X$ continues to witness the failure of TVC$(H)$ through all generic extensions. \end{lemma} \begin{proof} {\bf Claim:} In no generic extension can $X$ contain a perfect set of orbits. Proof of Claim: The statement that there are perfectly many orbits may be expressed as: \[\exists P\subset X(P {\rm \: perfect\:} \wedge\] \[\forall x_1, x_2 \in P\forall g\in G(x_1\neq x_2\Rightarrow g\cdot x_1\neq x_2)).\] This is $\Ubf{\Sigma}^1_2$, and hence absolute by Shoenfield. \hfill (Claim$\square$)\footnote{This proof also works to show that if a $\Ubf{\Sigma}^1_1$ $A$ does not contain a perfect set of orbits then in no generic extension it will contain a perfect set of orbits. The point here is that if $P$ is a perfect subset of $A$ then by say an appeal to the Baire category theorem and the Jankov-von Neumann uniformization we may find a perfect $P_0\subset P$ and a continuous function $f$ on $P_0$ such that for all $x\in P_0$ we that $f(x)$ witnesses $x\in A$.} {\bf Claim:} In every generic extension $X/H$ is uncountable. Proof of Claim: For conceptual simplicity let us do the case that \[X=\{{\cal M}\in {\rm Mod}({\cal L}):{\cal M}\models T\}\] and $H=S_\infty$. Then for any ${\cal M}\in$ Mod$({\cal L})$ we have that $[{\cal M}]_{S_\infty}$ is uniformly Borel in (any code for) the Scott sentence of ${\cal M}$. Moreover, in some natural parameter space $Y$ we have that \[{\cal S}=\{({\cal M}, y): y {\rm \: codes \: the \: Scott \: sentence \: of \: }{\cal M}\}\] is $\Ubf{\Pi}^1_1.$ Then the statement that the models of $T$ has uncountably many isomorphism types among its countable models becomes the statement that for all sequences $({\cal M}_i), (y_i)$, either: \begin{enumerate} \item there exists $i$ with $({\cal M}_i, y_i)\notin {\cal S}$; or \item there exists ${\cal N}\models T$ with ${\cal N}$ not satisfying any of the sentences coded by the $\{y_i: i\in\N\}$. \end{enumerate} Thus it is $\Ubf{\Pi}^1_2$ and subject to Shoenfield absoluteness. The general case for arbitrary Polish group actions follows by working not with the Scott sentence, but the {\it stabilizer} of the points in question; again $[x]_H$ is uniformly Borel in (any code for) the stabilizer of $x$ (see the proof of 1.2.4, 7.1.2 Becker-Kechris[1996]). \hfill (Claim$\square$) \end{proof} \begin{corollary} It is provable in ZFC that every Polish group $H$ satisfies WVC$(H)$ (i.e. the original topological Vaught conjecture) if and only it is provable that every Polish group satisfies TVC$(H)$. \end{corollary} \begin{proof} $\Leftarrow$ since any perfect set has size $2^{\aleph_0}$. $\Rightarrow$ since if we have a group $H$ and a Polish $H$-space $X$ showing the failure of TVC$(H)$ then we may go to a forcing extension in which $2^{\aleph_0}=\aleph_2$. In the forcing extension Sami's lemma and Burgess' theorem from \ref{burgess} on analytic equivalence relations imply that we must have $|X/H|=\aleph_1$. \end{proof} \begin{notation} For $H$ a Polish group let TVC$(H, \Ubf{\Sigma}^1_1)$ be the statement that whenever $X$ is a Polish $H$-space and $A\subset X$ is $\Ubf{\Sigma}^1_1$ then either \[A/H\leq \aleph_0\] or there is perfect $P\subset A$ such that for all $x_1, x_2\in P$ \[x_1\neq x_2\Rightarrow [x_1]_H\neq [x_2]_H.\] \end{notation} Sami's argument also shows absoluteness of TVC$(H, \Ubf{\Sigma}^1_1)$. \begin{lemma} (H. Friedman[1973]) Up to isomorphism there are exactly $\aleph_1$ many possible order types for the ordinals in countable $\omega$-models of Kripke-Platek set theory. \end{lemma} \begin{corollary} TVC$(S_\infty, \Ubf{\Sigma}^1_1)$ fails. \end{corollary} \subsection{Some sketches around the proof for characterizing Vaught's conjecture on analytic sets} %\begin{definition} Let us say that $S_\infty$ {\it divides} a %Polish group $H$ if there is a closed subgroup $K