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\title{\Huge{Two applications of model theory to the study of topological groups}}   
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\author{\Huge Greg Hjorth \footnote{Research partially supported by NSF grant DMS 96-22977}}      
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\date{\LARGE{1997}}          % Enter your date or \today between curly braces
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{\bf Definition.} $S_{\infty}$ is the symmetric group consisting of all permutations of 
$\N$. Equipped with the topology generated by basic open sets of the form 
$\{g\in S_{\infty}:g(0)=k_0,...,g(n)=k_n\}$ it becomes a completely metrizable 
separable topological group -- in other words, a {\it Polish group}.\\

The following theorem suggests it may be helpful to use model theory when trying to 
build closed subgroups of 
$S_{\infty}$.\\

{\bf Theorem} (folklore/Reyes). A subgroup $G<S_{\infty}$ is closed if and only if 
there is a strucure ${\cal M}$ whose underlying set is 
$\N$ with $G=$Aut$({\cal M})$ -- the group of all automorphisms of ${\cal M}$. 

\newpage 

{\bf Question} (Cameron). Is every closed {\it cofinitary} (every element fixing infinitely many 
natural numbers is the identity) subgroup of $S_{\infty}$ locally compact?\\

\bigskip

\bigskip

\bigskip

\bigskip

{\bf Question} (Becker). Are the Polish groups with a {\it complete left invariant metric} the only 
ones satisfying the {\it topological Vaught conjecture}?\\

\newpage 

{\bf Cofinitary groups.}\\ 

{\bf Definition.} A subgroup $G<S_{\infty}$ is {\it cofinitary} if for all $g\in S_{\infty}$ 
\[\exists ^{\infty}n\in\N (g(n)=n)\Leftrightarrow g= 1.\]

Every countable discrete group can be realized as a closed cofinitary subgroup of $S_{\infty}$, 
as can every compact subgroup of $S_{\infty}$. It is not obvious that there are any more 
examples -- and indeed all abelian cofinitary groups fall into one of these two categories. 
Cameron presents an uncountable locally compact non-compact closed cofinitary group in 
\cite{cameron}, and goes on to conjecture that all closed cofinitary groups are locally compact. 
This would imply that all closed continuous homomorphic images of closed cofinitary groups 
are locally compact. However:\\

{\bf Theorem.} There is a closed cofinitary $G<S_{\infty}$ that is not locally compact.\\

In fact\\

{\bf Theorem.} Every closed subgroup of $S_{\infty}$ is the continuous homomorphic image 
of a closed cofinitary subgroup of $S_{\infty}$. 

\newpage 

{\bf The idea of the proof of the first theorem.} Build $G$ as Aut$({\cal M})$ where the 
language of ${\cal M}$ includes infinitely many equivalence relations, $E_0, E_1, ..., E_n,..., $ 
where each $E_{n+1}$ refines each $E_n$ equivalence class into exactly two equivalence 
classes, each $E_n$ having exactly $2^n$ many equivalence classes. We also require 
that any two distinct elements in ${\cal M}$ are $E_n$-inequivalent 
for some $n$. For instance, this much   
can be achieved by letting ${\cal M}\subset 2^{\N}$ and setting $xE_ny$ if $x|_n=y|_n$. 

\bigskip

\bigskip

We then demand that from any sequence of $n+1$ many elements in ${\cal M}$ we may define a 
linear ordering of the $E_n$-equivalence classes. This will suffice to show that 
any element $g\in G=_{df}$Aut$({\cal M})$ is either the identity or fixes only finitely many elements 
of ${\cal M}$ -- in other words, $G$ is cofinitary. 

\bigskip

\bigskip

Finally, we do all the above as {\it homogeneously as possible}, so that ${\cal M}$ has no real 
structure except as explicitly defined. This serves to guarantee that Aut$({\cal M})$ is not 
locally compact. 

\newpage

{\bf Complete left invariant groups}\\

{\bf Definition.} If $G$ is a Polish group, then the {\it topological Vaught conjecture for $G$} is the 
assertion that whenever $G$ acts continuously on a Polish space $X$, either $|X/G|\leq\aleph_0$ or 
$|X/G|=2^{\aleph_0}$. A Polish group is said to have a {\it complete left invariant metric} -- or just be 
{\it cli} for short -- if there is a complete compatible metric $d$ on $G$ so that for all $g_0, g_1, h\in G$, 
$d(g_0, g_1)=d(hg_0, hg_1)$.\\

The class of cli groups includes all abelian, or even just solvable, Polish groups, all locally compact 
Polish groups, and is closed under countable products. In \cite{becker} Howard Becker shows that every 
group in this class satisfies the topological Vaught conjecture and raises the question of whether this 
conclusion characterizes the class.\\

{\bf Theorem} (Knight). There is a countable model ${\cal M}$ whose Scott sentence has a model of 
size $\aleph_1$, but no higher.\\

We have that $G$=Aut$({\cal M})$ is a non-cli group satsfying the topological Vaught 
conjecture by the following disparate facts. 

\newpage

{\bf (I).} Aut$({\cal M})$ is not a cli group by work of Su Gao; indeed
in general for ${\cal N}$ a countable model Aut$({\cal N})$ is cli if and only if 
the Scott sentence of ${\cal N}$ has no uncountable model.\\

{\bf (II).} (Becker-Kechris) If $G$=Aut$({\cal M})$ acts continuously on a Polish space $X$ then there is 
a sentence $\sigma\in{\cal L}'_{\omega_1\omega}$, ${\cal L}'$ some language including the language of 
${\cal M}$, such that: (a) every countable model of $\sigma$ is (isomorphic to) an expansion of 
${\cal M}$; (b) the $G$-orbits in $X$ have a natural correspondence with the countable models of 
$\sigma$ considered up to isomorphism.\\

{\bf (III).} Knight's construction actually gives that if ${\cal N}$ is an expansion of ${\cal M}$, 
and $\varphi_{\cal N}$ is its Scott sentence, then in any inner model of ZFC containing $\varphi_{\cal N}$  
we have $|\varphi_{\cal N}|\leq\aleph_1$.\\

{\bf (IV).} (Folklore) Assume $\neg$CH. If $\sigma\in{\cal L}'_{\omega_1\omega}$, ${\cal L}'$ 
some countable language, 
provides a counterexample to Vaught's conjecture for ${\cal L}'_{\omega_1\omega}$ then in ${\Bbb V}$ 
there is a transfinite sequence $(\varphi_{\alpha})_{\alpha\in Ord}$ of distinct infinitary 
sentences such that for each $\alpha$, $G\subset$Coll$(\omega, \alpha)$ ${\Bbb V}$-generic,\\

\leftskip 0.5in

${\Bbb V}[G]\models \varphi_{\alpha}$ is the Scott sentence of some model of $\sigma$. 

\leftskip 0in

\newpage

\begin{thebibliography}{99}
\bibitem{becker}H. Becker, {\it Vaught's conjecture for complete left invariant 
groups}, handwritten notes, University of North Carolina at Columbia, 1996. 

\bibitem{beckerkechris} H. Becker and A.S. Kechris, {\bf The descriptive set theory of 
Polish group actions}, to appear in the London Mathematical Society Lecture Notes Series. 

\bibitem{cameron} P.J. Cameron, {\it Cofinitary permutation groups,} 
{\bf Bulletin of the London Mathematical Society,} vol. 28(1996), 
pp. 113-140.

%\bibitem{devries} J. de Vries, {\it Universal topological transformation groups,} 
%{\bf General topology and its applications,} vol. 5(1975), pp. 107-122. 

%\bibitem{effros} E. Effros, {\it Transformation groups and $C^*$-algebras,} 
%{\bf Annals of Mathematics,} ser 2, vol. 81(1975), pp. 38-55. 

%\bibitem{fehamo} J. Feldman, P. Hahn, C.C. Moore, {\it Orbit structure and countable sections 
%for actions of countable groups,} {\bf Advances in Mathematics,} vol. 28(1978), pp. 186-230. 

\bibitem{gao} S. Gao, {\it Automorphisms of countable structures,} 
to appear in the {\bf Journal of Symbolic Logic.} 

%\bibitem{hewittross} E. Hewitt, K.A. Ross, {\bf Abstract harmonic analysis,} vol. I, 
%Springer-Verlag, Berlin and New-York, 1979. 

\bibitem{hodges} W. Hodges, {\bf Model theory,} Cambridge University Press, Cambridge, 1993. 

%\bibitem{kechris2} A.S. Kechris, {\bf Classical descriptive set theory}, 
%Springer-Verlag Graduate Texts in Mathematics, Berlin, 1995. 

\bibitem{knight} J. Knight, {\it A complete $L_{\omega_1\omega}$ sentence 
characterizing $\aleph_1$}, {\bf Journal of Symbolic Logic,} vol. 42(1977), 
pp. 59-62. 

%\bibitem{meg} M.G. Megrelishvili, {\it A Tikhonov $G$-space that does not have compact 
%$G$-extensions and $G$-linearization,} {\bf Uspekhi Matematicheskikh Nauk,} 
%vol. 43(1988), pp. 145-6. 

\bibitem{reyes} G.E. Reyes {\it Local definability theory}, {\bf Annals of Mathematical 
Logic,} vol. 1(1970), pp. 95-137. 

%\bibitem{zimmer} R. Zimmer, {\it Ergodic theory and semi-simple groups}, Birkhauser, Basel, 1984.

\end{thebibliography}

%6363 MSB

%Mathematics

%UCLA

%CA90095-1555

greg@math.ucla.edu


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