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\title{\huge Bi-Borel reducibility of 
essentially countable Borel equivalence relations\footnote{Key words and 
phrases: Equivalence relation, Borel reducibility, near hyperbolic group, 
rigidity. 2000 Mathematics Subject Classifications: Primary 03E15. 
Secondary 20F67, 37A20.}}         
\author{Greg Hjorth\footnote{
Partially supported by 
NSF grant DMS  0140503}}        % Enter your name between curly braces
%\date{\today}          % Enter your date or \today between curly braces
\maketitle



%\section{Introduction} 

This note 
answers a questions from 
\cite{jacksonkechrislouveau} by showing that considered up to Borel reducibility, there 
are more essentially countable Borel equivalence relations than countable Borel 
equivalence relations. 
Namely: 

\begin{theorem} There is an essentially countable Borel equivalence relation 
$E$ such that for {\rm no} countable Borel equivalence relation $F$ (on a 
standard Borel space) do we have 
\[E\sim_B F.\] 
\end{theorem} 

%Here $E\sim_B F$ is an abbreviation for $E\leq_B F \leq_B E$. 

The proof of the result is short. It does however require an extensive 
rear guard campaign to extract from the techniques of 
\cite{hjorthkechris} the following 

\begin{mess} There are countable Borel equivalence relations $(E_x)_{x\in 2^\N}$ 
such that:- 

\leftskip 0.4in 

\noindent (i) each $E_x$ is defined on a standard Borel probability space 
$(X_x, \mu_x)$; each $E_x$ is $\mu_x$-invariant and $\mu_x$-ergodic; 

\noindent (ii) for $x_1\neq x_2$ and $A$ $\mu_{x_1}$-conull, we have $E_{x_1}|_A$ 
not Borel reducible to $E_{x_2}$; 

\noindent (iii) if $f: X_x\rightarrow X_x$ is a measurable reduction of $E_x$ to itself, 
then $\mu_x({\rm im}(f))>0$; 

\noindent (iv) 
\[\bigcup_{x\in 2^\N}\{x\}\times X_x\] 
is a standard Borel space on which the projection function 
\[(x, z)\mapsto x\] 
is Borel and the equivalence relation $\hat{E}$ given by 
\[(x, z) \hat{E} (x', z')\] 
if and only if $x=x'$ and $zE_x z'$ is Borel; 

\noindent (v) 
\[2^\N\rightarrow M(\bigcup_{x\in 2^\N}\{x\}\times X_x)\]
\[x\mapsto \delta_x\times \mu_x\] 
is Borel. 

\label{messy}

\end{mess} 

We first prove the theorem granted this messy fact. We then prove the fact. 

(iv) and (v) are messy and unpleasant to 
state precisely, but are intended to express the idea that we have an effective 
parameterization of countable Borel equivalence relations by points in a standard 
Borel space. Examples along these lines appear already in the Adams-Kechris 
constructions; the new feature is (iii). 

Simon Thomas has pointed out to me that in light of 
theorem 4.4 \cite{thomas} the 
Gefter-Golodets 
examples of section 5 \cite{thomas} also 
satisfy the conclusion of \ref{messy}. 
None the less I think it is worth including the 
construction below, since it ultimately relies on the 
far more elementary techniques of 
\cite{hjorthkechris}. 

\medskip 

\noindent{\bf Acknowledgments:} I am 
exceedingly grateful 
to an anonymous referee for a thorough and critical   
reading of the original version  
of this paper and for making many helpful comments. 

\medskip 


\section{Introduction} 

We include a few of the main definitions, 
some notational conventions, and facts about measure 
theory in the context of descriptive set theory. 
A more complete introduction to the theory of 
Borel equivalence relations can be found in 
\cite{hjorthkechris}, \cite{jacksonkechrislouveau}, or 
\cite{thomas}. 

\begin{definition} For equivalence relations 
$E, F$ defined on Polish spaces $X, Y$ we write 
\[E\leq_B F\] 
if there is a Borel function $f:X\rightarrow Y$ 
with 
\[x_1 E x_2\Leftrightarrow f(x_1) F f(x_2).\] 
We write $E\sim_B F$ if $E\leq_B F$ and $F\leq_B E$. 
We say that $E$ is {\it countable} if every 
equivalence class is countable and {\it Borel} 
if it is Borel as a subset of 
$X\times X$. We say that it is 
{\it essentially countable} if there is 
some countable, Borel $F$ with $E\leq_B F$. 
\end{definition} 

\def\m{{\mathcal M}}

\begin{definition} For 
$X$ a Polish space we denote by 
$\m(X)$ the collection of 
Borel probability measures on 
$X$. This is given the $\sigma$-algebra 
generated by sets of the form 
\[\{\mu:a< \int fd\mu <b\}\] 
for $f$ a Borel function on $X$ and $a, b\in \R$. 
Following 
$\S$17\cite{kechris} $\m(X)$ becomes a 
standard Borel space.  

For $x\in X$ we use $\delta_x$ to denote the 
{\it Dirac measure} with $\delta_x(\{x\})=1$ 
and $\delta_x(X\setminus\{x\})=0$. 

For $\varphi(\cdot)$ some 
formula, possibly with parameters, 
and $\mu\in \m(X)$ we write 
\[\forall^\mu x (\varphi(x))\]
if the set of $x\in X$ with $\varphi(x)$ is 
co-null and 
\[\exists^\mu x (\varphi(x))\]
if the set of $x\in X$ with $\varphi(x)$ is
non-null.  
\end{definition} 

\begin{proposition} 
\label{measurelemma} 
For $Y$ Polish and $A\subset X\times Y$ Borel, 
the set 
\[\{(y, \mu)\in Y\times \m(X): \exists^\mu x\in X
((x, y)\in A)\}\] 
is Borel. 
\end{proposition} 

\begin{proof} It suffices to see that for 
$a>0$ the sets  
\[\{(y, \mu)\in Y\times \m(X): \mu(\{x: (x, y)\in A\})\}>a\]
\[\{(y, \mu)\in Y\times \m(X): \mu(\{x: (x, y)\notin A\})\}>a\]
are both $\Ubf{\Sigma}^1_1$, which follows 
from 29.26 \cite{kechris}. 
\end{proof} 

\begin{definition} For a group $G$ 
acting on a set $X$ and $F\subset X$ we use 
stab$_F$ to denote the {\it set wise stabilizer} -- 
that is to say, the collection of $g\in G$ 
such that $\{g\cdot x: x\in F\}=F$. 
Thus in particular, for $x\in X$, stab$_{\{x\}}$ 
denotes the group elements leaving $x$ fixed. 

In the case that $G$ acts on a set $X$ we will use 
$[X]^n$ to denote the size $n$ subsets of $X$ equipped with the 
natural action 
\[g\cdot \{x_1,..., x_n\}=\{g\cdot x_1, ..., g\cdot x_n\}.\] 
We use $[X]^{<\infty}$ to denote the union of the $[X]^n$'s. 
\end{definition} 



\begin{lemma} The action of $G$ on its finite subsets, 
$[G]^{<\infty}$, is free if and only if $G$ is torsion 
free. 
\end{lemma} 

\begin{proof} If $g$ has infinite order, then for any 
$h\in G$, $\{g^n\cdot h: n\in \N\}$ 
is infinite, and thus there is no $g$-invariant 
finite subset of $G$. 
Conversely, if $H$ is a finite subgroup of 
$G$ then it is a fixed point in $[G]^{<\infty}$ for any 
$h\in H$. 
\end{proof} 

\begin{lemma} If $G$ acts freely on $X$, then there is a 
$G$-map from $X$ to $G$. 
\end{lemma} 

\begin{proof} 
There is no requirement that this $G$-map be Borel, and 
thus we can select from each orbit $[x]_G$ a unique 
point $x_0\in [x]_G$. From this we 
obtain a well defined map by 
letting $\theta(g\cdot x_0)=g$. 
\end{proof} 


We will at some later point need the following corollary. 

\begin{corollary} \label{map} 
If $G$ is a torsion free group 
then there is a $G$-map from $[G]^{<\infty}$ to $G$. 
\end{corollary}


\begin{definition} For $X$ a standard Borel space, we let 
$\m_{\leq n}(X)$ be the probability 
measures which concentrate on 
at most $n$ points and $\m_{\geq n}(\m)$ the measures which 
do {\it not} concentrate on at most $n-1$ points. Thus 
$\m_{\geq n}(X)=\m(X)\setminus \m_{\leq n-1}(X)$. 
\end{definition} 

The next lemma is a routine generalization of the argument 
given at C2.3 \cite{hjorthkechris} along with credit to Lyons. 

\begin{lemma} 
\label{lyons} 
Suppose that $G$ is a group, $X$ is a standard Borel 
$G$-space, $Y$ is a countable $G$-space, 
and there is a Borel $G$-map from 
$[X]^n$ to $Y$. 
Then there is a Borel $G$-map from $\m_{\geq n}(X)$ to 
$[Y]^{<\infty}$. 
\end{lemma} 

\begin{proof} Let $\phi$ be the Borel $G$-map from 
$[X]^n$ to $Y$. 
Let $\{*\}$ be some single point not in 
$Y$ and form the space $Y^*=Y\dot{\bigcup} \{*\}$.  
Define the Borel map 
\[\eta: X^n\rightarrow Y^*\] 
by $\eta(x_1, x_2,..., x_n)=\phi(\{x_1, x_2,..., x_n\})$ if the 
points are distinct, and equal to the default value $*$ otherwise. 
For $\mu\in \m_{\geq n}(X)$ we let $\mu^*$ be the measure 
on $Y^*$ which has at each $y_0\in Y^*$ 
\[\mu^*(\{y_0\})=\mu^n(\{(x_1,...x_n): \eta(x_1,...x_n)=y_0\}).\] 
The assumption that $\mu\in \m_{\geq n}(X)$ gives 
that $\mu^*$ does not concentrate entirely on the new point 
$*$ of $Y^*$, and thus we can select 
the {\it finitely} many 
points in $Y$ for which $\mu^*(\{y_0\})$ realizes the maximum 
value. 

It is routinely seen that this assignment is Borel. 
\end{proof} 





\section{Proof of theorem granted messy fact} 

We begin with a collection of $(E_x, X_x, \mu_x)$ as described there. 
Let $A\subset 2^\N$ be $\Ubf{\Sigma}^1_1$ non-Borel. Let $C\subset 
\N^\N\times 2^\N$ be closed with the canonical projection function 
\[p_2: \N^\N\times 2^\N\rightarrow 2^\N\]
\[(z, x)\mapsto x\]
having 
\[p_2[C]=A;\] 
so $C$ is our witness to $A\in\Ubf{\Sigma}^1_1$. 

We let $\hat{X}$ be the set of $(z, x, w)$ which have $(z, x)\in C$ and 
$w\in X_x$. We give $\hat{X}$ the standard Borel structure it inherits 
as a Borel subset of 
\[\N^\N\times \bigcup_{x\in 2^\N}\{x\}\times X_x.\] 
For $\vec v_1=(z_1, x_1, w_1), \vec v_2=(z_2, x_2, w_2)\in \hat{X}$, we set 
\[\vec v_1 E \vec v_2\] 
if and only if $x_1=x_2$ and $w_1 E_{x_1} w_2$. 
This is clearly an essentially countable equivalence relation. 

For a contradiction suppose $F$ is a countable Borel equivalence relation 
on a standard Borel space $Y$ and 
\[\theta: \hat{X}\rightarrow Y\]
\[\rho: Y\rightarrow \hat{X}\] 
witness 
\[E\sim_B F.\] 
Let $\bar{\rho}: Y\rightarrow \bigcup_{x\in 2^\N}\{x\}\times X_x$ be obtained by 
setting $\bar{\rho}(y)=(p_2(\rho(y)), p_3(\rho(y)))$ -- 
so if $\rho(y)=(z, x, w)$ then $\bar{\rho}(y)=(x, w)$. 

\begin{claim} \label{claim1} If $w_0\in X_x$ then 
$\{y: \bar{\rho}(y)=(x, w_0)\}$ is countable. 
\end{claim} 

\begin{proof} 
For any $y_0$ with $\bar{\rho}(y_0)=(x, w_0)$ 
we have that for all other $y$ with 
\[\bar{\rho}(y)=(x, w_0)\] 
that $y F y_0$; thus the claim follows from $F$ being 
countable. 
\end{proof} 

\begin{claim} \label{claim2} 
If $x\in A$ then 
\[\{x\}\times X_x\cap {\rm im}(\bar{\rho})\] 
is non-null in 
\[(\{x\}\times X_x, \delta_x\times \mu_x).\] 
\end{claim} 

\begin{proof} 
We let $z$ be such that $(z, x)\in C$. We define 
\[\gamma_{z, x}: X_x\rightarrow Y\] 
by $\gamma_{z, x}(w)= \theta(z, x, w)$; the assumptions on 
$\theta$ entail that $\gamma_{z, x}$ witnesses $E_x\leq_B F$. 
By composition we have 
\[\rho\circ \gamma_{z, x}: X_x\rightarrow \hat{X}\] 
witnessing $E_x\leq_B E$. And then by $\mu_x$-ergodicity of 
$E_x$ we have some single $x_0$ such that 
\[\forall^{\mu_x}w \in X_x(\bar{\rho}(\gamma_{z, x}(w))\in \{x_0\} \times X_{x_0}).\] 
Postcomposing with the projection to the second coordinate we 
obtain a reduction 
\[\bar{\gamma}_{z, x}=p_2\circ \rho\circ \gamma_{z, x}\]
\[X_x\rightarrow X_{x_0}\] 
on a conull subset of $E_x$ to $E_{x_0}$, and 
hence $x=x_0$ by part (ii) of our messy fact. And then by part (iii) 
$\mu_x({\rm im }(\bar{\gamma}_{z, x}))>0$, and, after unraveling the 
definitions, im$(\bar{\rho})$ non-null in $\{x\}\times X_x$. 
\end{proof} 

Putting together these claims we have 
\[\begin{array}{ll}x\in A& \Leftrightarrow \exists ^{\mu_x}w\in X_x( (x, w)\in {\rm im}(\bar{\rho}))\\
 & \Leftrightarrow  \exists ^{\delta_x\times\mu_x}(x', w')
\in \bigcup_{y\in 2^\N}\{y\}\times X_{y}
( (x', w')\in {\rm im}(\bar{\rho}))
\end{array}.\] 
$\bar{\rho}$ is countable to one, 
and so it has Borel image by Lusin-Novikov (see 18.10 
\cite{kechris}). 
Applying lemma \ref{measurelemma} we have $A$ is Borel with a 
contradiction. 




\def\gp{{\Gamma_{\vec p}}}


\section{Messy fact} 

\begin{notation} 
For $\vec p=(p_1, p_2,...)$ a sequence of distinct primes, let $\Q_{\vec p}$ be the subgroup of 
$\Q$ generated by 
\[\{(p_i)^{-n}: i, n \in \N\}\] 
(so that we have infinite divisibility in $\Q_{\vec p}$ by each prime on the list). 
Then let 
\[\Gamma_{\vec p}=H_{\vec p} * G_{\vec p}\] 
where $H_{\vec p}$ and $G_{\vec p}$ are disjoint copies of 
$\Q_{\vec p}$. 

Following $\S$C3 of \cite{hjorthkechris} we let $[S_{\gp}]$ be the collection of 
all infinite sequences 
\[(a_0, a_1, a_2,...a_n,...)\] 
such that at each $i$ we have $a_i\in H_{\vec p}$ iff $a_{i+1}\in G_{\vec p}$ 
iff $a_{i}\notin G_{\vec p}$; we let $\gp$ act in the natural way on 
$[S_\gp]$ -- concatenate and reduce. 

\end{notation} 




\begin{lemma} 
\label{2.1} 
(a) If $w, a\in \Gamma_{\vec p}$ with $g^2 = w^\smallfrown 
a^\smallfrown w^{-1}$ then $g$ is in the cyclic subgroup 
generated by $wc w^{-1}$ for any $c$ with $c^2 = a$. 

(b) If a non-identity element $h$ of $\Gamma_{\vec p}$ is infinitely 
divisible by a prime $q$, then $q$ appears in the sequence $\vec p$ 
and $h$ has the form 
\[h=w a w^{-1}\] 
for some $a\in G_{\vec p}\cup H_{\vec p}$. 


\end{lemma} 

\begin{proof} (a)
Suppose $h, g\in \gp$ 
and $g^2 =h$. In general we will measure the {\it length} 
of a word in $\Gamma_{\vec p}$ by looking at the number of 
times it alternates between $G_{\vec p}$ and $H_{\vec p}$. 

We write $h$ as a reduced word in the form 
\[h=w^\smallfrown (a^k)^\smallfrown w^{-1}\] 
where $w$ is as long as possible and $a$ is as short as 
possible.   
We write $g$ as a reduced word 
\[g=v^\smallfrown b^\smallfrown v^{-1}\] 
where again $v$ is as long as possible. Then it follows that 
\[g^2 = v^\smallfrown (b^2)^\smallfrown v^{-1},\] 
and hence $v=w$, $b^2 =a^k$. 

Now there are two cases. 

In one case we have $a\notin G_{\vec p}\cup H_{\vec p}$, 
when indeed if follows that $b$ appears as an initial 
string in $a^k$, and hence, by minimization of $a$, that 
$2$ divides $k$, and we have $g=w^\smallfrown 
(a^{k/2})^\smallfrown w^{-1}$ as required. 

The second case is $a\in G_{\vec p}\cup H_{\vec p}$. 
Assume $a\in G_{\vec p}$. Then since $G_{\vec p}$ is torsion 
free we have again that $b=a^{k/2}$ is the unique element 
with $b^2=a^k$. 

(b) If we have a prime $q$ and 
$h, g\in \gp$ with $g^q=h$ then as above
either 


\leftskip 0.4in

\noindent (i) for some $\bar{h}, 
\bar{g}$ either both in $G_{\vec p}$ or both
in $H_{\vec p}$ and some $w\in \gp$ we have
\[h = w^\smallfrown \bar{h} ^\smallfrown w^{-1},\]
\[g = w^\smallfrown \bar{g} ^\smallfrown w^{-1},\]
as reduced words and
\[\bar{g} +\bar{g}+...(q \: {\rm times}) ...+ \bar{g}=\bar{h}\]
in the respective copy of $\Q_{\vec p}$; or

\noindent (ii) for some $b, w\in \gp, b\notin 
G_{\vec p}\cup H_{\vec p}$ with 
$w^\smallfrown b^\smallfrown w^{-1}$ reduced
we have
\[h=w^\smallfrown (b^q)^\smallfrown w^{-1},\]
\[g= w^\smallfrown b ^\smallfrown w,\]
and as a reduced word
\[b^q=b^\smallfrown b^\smallfrown...(q\: {\rm times}) ...^\smallfrown b.\]

\leftskip 0in

The only possibility for infinite divisibility by $q$ is the 
first case, when we indeed have to have $q$ on the $\vec p$ 
sequence.   
\end{proof}  



\def\hv{{H_{\vec p}}}
\def\gv{{G_{\vec p}}}





\begin{lemma} \label{2.2} 
If $e\in [S_\gp]$, $g\in \gp$, 
$g\cdot e =e$ (i.e. $g\in {\rm stab}\{e\}$), $g$ not the 
identity, then for some $w, a\in \gp$, $i\in\Z$, $i\neq 0$,  
\[e=w^\smallfrown a^\smallfrown a^\smallfrown...\] 
\[g=w^\smallfrown (a^i)^\smallfrown w^{-1}.\] 

\end{lemma} 

\begin{proof} 
It is easily seen that $e$ is eventually periodic, and so 
for some $a, w$ 
\[e= w^\smallfrown a^\smallfrown a^\smallfrown...\]
Note then that 
\[g\cdot w^\smallfrown a^\smallfrown a^\smallfrown...=
w^\smallfrown a^\smallfrown a^\smallfrown...\] 
if and only if 
\[w^{-1}gw\cdot a^\smallfrown a^\smallfrown...
=a^\smallfrown a^\smallfrown...\] 
Thus we may as well assume that $e$ is outright periodic and 
suppose that $e$ has period $n$ 
-- where the {\it period} is the least $n$ for which it is 
it is $n$-periodic. We may also 
assume without loss that there are $g_i$'s in $G_{\vec p}$ 
and $h_i$'s in $H_{\vec p}$ with 
\[a=g_1^\smallfrown h_2^\smallfrown g_3^\smallfrown...^\smallfrown h_n.\] 
We suppose $g\cdot e = e$ with 
\[g=k_1^\smallfrown k_2^\smallfrown ...^\smallfrown k_\ell,\] 
where, as usual, $k_i\in H_{\vec p}$ iff $k_{i+1}\in G_{\vec p}$ 
iff $k_{i}\notin G_{\vec p}$. 

Then by considering the fact that $g\cdot e =e $ 
still has period $n$  either we obtain  $k_\ell\in \hv$ 
and $k_\ell=h_n$, and then $k_{\ell -1}=g_{n-1}$, $k_{\ell -2}=h_{n-2}$, and so on, until 
in the final end we have $g=a^i$ some $i>0$, or we obtain 
$k_\ell\in \gv$ and $k_{\ell} = g_1^{-1}$, and then $k_{\ell-1}=h_2^{-1}$, and it 
all unravels to the end with $g=a^{-i}$ some $i>0$. 
\end{proof} 

\begin{corollary} 
\label{2.3} 
If $e_1, e_2\in [S_\gp]$, then {\rm stab}$\{e_1, e_2\}$ is abelian. 
\end{corollary} 

\begin{proof} 
Whenever $g\in$ stab$\{e_1, e_2\}$ we have 
$g^2$ is in stab$\{e_1\}$ and thus we may appeal to 
\ref{2.2} to write  $e_1$ as 
\[e_1=w^\smallfrown a^\smallfrown a^\smallfrown...\] 
where $lh(a)=$ period of $e_1$ and $g^2$ as 
\[g^2=w^\smallfrown (a^i) ^\smallfrown w^{-1}.\] 
Then by \ref{2.1} we have that any such $g$ will be in the cyclic subgroup 
generated $w^\smallfrown a^\smallfrown w^{-1}$. 
\end{proof} 



\begin{lemma} 
\label{2.4} 
There is a Borel $\gp$ map 
\[\varphi: [S_\gp]^3\rightarrow \gp.\] 

\end{lemma} 


\begin{proof} 
Since $\hv$, $\gv$ are torsion free there are, 
by \ref{map}, respective $\hv$- and $\gv$-maps 
\[\theta_H: [\hv]^3\rightarrow \hv,\] 
\[\theta_G:[\gv]^3\rightarrow \gv.\] 
Now for any $\{f_1, f_2, f_3\}\in [S_\gp]^3$ we follow C3.2 \cite{hjorthkechris} and consider a 
split in cases. 

\medskip 

{\bf Case(a)} There is a reduced word 
$v$ such that any two of the ends agree exactly up to 
$v$. 

\medskip 

Thus we may write these as 
\[f_1=v^\smallfrown k_1^\smallfrown k_1'^\smallfrown...\]
\[f_2=v^\smallfrown k_2^\smallfrown k_2'^\smallfrown...\]
\[f_3=v^\smallfrown k_3^\smallfrown k_3'^\smallfrown...\]
where at each $i$ we have 
 $k_i\in H_{\vec p}$ iff $k_{i}'\in G_{\vec p}$ 
iff $k_{i}\notin G_{\vec p}$. We then let 
\[\varphi(\{f_1, f_2, f_3\})=v\theta(\{k_1, k_2, k_3\})\] 
where 
$\theta=\theta_G$ or $\theta=\theta_H$ depending on which group the 
$k_i$'s exist in. 

\medskip 

{\bf Case(b)} Two of the ends agree with one another more than either does with the third. 

\medskip 

So we may write $\{e_1, e_2, e_3\}=\{f_1, f_2, f_3\}$ with 
\[e_1=u^\smallfrown w^\smallfrown k_1...\]
\[e_2=u^\smallfrown w^\smallfrown k_2...\]
\[e_3=u^\smallfrown \hat{w}^\smallfrown ...\] 
as reduced words and $w$, $\hat{w}$ having no initial segment in common. 

We then let 
\[\varphi(\{f_1, f_2, f_3\})=(u^\smallfrown w)\theta(\{k_1, k_2, 0\})\]
where $\theta$ is again either $\theta_G, \theta_H$ and $0$ is the zero 
(abelian group identity) in the respective copy of $\Q_{\vec p}$. 

\medskip 

Note that case(b) can be converted into case(a) by acting on the triple by any 
reduced group element of the form 
\[g=v^\smallfrown k^\smallfrown (w^{-1})^\smallfrown u^{-1},\] 
where $k$ is in the same copy of $\Q_{\vec p}$ as $k_1, k_2$ but is not 
equal to either $k_1^{-1}, k_2^{-1}$. 
Thus to verify that $\varphi$ is indeed a $\gp$-map it suffices to 
check the invariance property for $g$ in this form. 

But here one has 
\[\varphi(\{g\cdot f_1, g\cdot f_2, g\cdot f_3\}) =\varphi(v^\smallfrown (k k_1)^\smallfrown..., 
v^\smallfrown (k k_2)^\smallfrown..., v^\smallfrown k ^\smallfrown (w^{-1})^\smallfrown \hat{w}^\smallfrown...)\] 
\[=v\cdot \theta(\{kk_1, k k_2, k \}) = v^\smallfrown k \cdot \theta(\{k_1, k_2, 0\})\]
\[=v^\smallfrown k\cdot((w^{-1})^\smallfrown u^{-1}\cdot(u^\smallfrown w\cdot \varphi(\{k_1, k _2, 0\}))) 
= v^\smallfrown k ^\smallfrown (w^{-1})^\smallfrown u^{-1} \cdot \theta(\{e_1, e_2, e_3\}),\] 
as required. 

\end{proof} 



\begin{definition} 
Let $H^\infty_{\vec p}$ and $G^\infty_{\vec p}$ be the respective one point compactifications 
of $H_{\vec p}$ and $G_{\vec p}$. 


So $H^\infty_{\vec p}$ consists of $H_{\vec p}$ along 
with a new point $\infty_{H}$, and the open sets are all sets which do {\it not} contain the 
point $\infty_H$ along with all cofinite sets which do contain $\infty_H$. Similarly 
$G_{\vec p}^\infty$ contains the new point $\infty_G$. 

We let $H_{\vec p}$ and $G_{\vec p}$ act on their one point compactifications so that 
the action extends the usual action on themselves by left translation and the new point 
at infinity becomes a fixed point -- so 
\[h\cdot a= h+a\] 
for $h, a\in H_{\vec p}$, whilst 
\[h\cdot \infty_H = \infty_H\] 
for $h\in H_{\vec p}$, and similarly on the $G_{\vec p}$ side. 


\end{definition} 

It is easily checked that these actions are continuous. 


\begin{definition} 
We let $\gp^*$ consist of all finite sequences 
\[a_0^\smallfrown a_1 ^\smallfrown...a_n\] 
where the $a_i$'s, $i<n$, 
alternate between $H_{\vec p}$ and $G_{\vec p}$, as in the definition 
of $[S_{\gp}]$, but the final point $a_n$ is either $\infty_H$ or 
$\infty_G$, depending on whether $a_{n-1}$ is in $G_{\vec p}$ or 
$H_{\vec p}$. 

Note that there is a natural analog of ``concatenate and reduce" which 
defines an action of $H_{\vec p}$, or $G_{\vec p}$, on $\gp^*$: Given 
$h\in H_{\vec p}$ and $w=a_0^\smallfrown a_1^\smallfrown a_2...^\smallfrown a_n$ 
we define $h\cdot w$ by cases: 

\leftskip 0.4in 

\noindent (i) if $a_0\in G^\infty_{\vec p}$, then 
\[h\cdot w=(h^\smallfrown a_0^\smallfrown a_1...a_n);\] 

\noindent (ii) if $n=0$ and $a_n =\infty_H$ then $h\cdot w$ just equals $w$ again; 

\noindent (iii) if $n>0$, $a_0\in H_{\vec p}$, $a_0\neq -h$, then 
\[h\cdot w=((h+a_0)^\smallfrown a_1^\smallfrown...a_n);\] 

\noindent (iv) if $n>0$ and $a_0=-h$, then 
\[h\cdot w=(a_1^\smallfrown a_2^\smallfrown...a_n).\] 

\leftskip 0in 

\noindent The action by $G_{\vec p}$ is the obvious analog, and we then obtain 
an action by the free product $\gp=G_{\vec p}* H_{\vec p}$. 

\end{definition} 

\begin{lemma} 
\label{further3.3} 
The stabilizers of points in the action of $\gp$ on 
$\gp^*$ are abelian. 
\end{lemma} 


\begin{proof} 
Consider some point $w=a_0^\smallfrown a_1^\smallfrown...a_n\in \gp^*$. 
We 
let $u=a_0^\smallfrown a_1...a_{n-1}$, then the stabilizer of $w$ in the 
action of $\gp$ on $\gp^*$ equals $uH_{\vec p} u^{-1}$ or $uG_{\vec p} u^{-1}$ 
depending on whether $a_n$ equals $\infty_H$ or $\infty_G$. 
\end{proof} 

\def\ss{[S_{\gp}]^{*\infty}} 

\begin{definition} 
We let $\ss$ be the union of $[S_{\Gamma_{\vec p}}]$ and 
$\Gamma_{\vec p}^*$; thus it consists in all finite sequences 
alternating between $G_{\vec p}^\infty, H_{\vec p}^\infty$ and 
only taking the respective $\infty_G$ or $\infty_H$ at the final point 
along with all infinite sequences that alternate between 
$G_{\vec p}$ and $H_{\vec p}$. 

We endow it with a topology 
as follows. Given a finite sequence 
$\vec a=a_0^\smallfrown a_1^\smallfrown...^\smallfrown a_n$ in ${\gp}$ 
and $V$ an open subset of 
either $H_{\vec p}^\infty$ or $G_{\vec p}^\infty$, depending on 
whether $a_n$ is in 
$G_{\vec p}$ or $H_{\vec p}$ respectively, we let 
\[{\cal O}(V, \vec a)\] 
be the set of all $\vec b\in [S_{\gp}]\cup \gp^*$ where 
\[b_i = a_i\] 
all $i\leq n$, and 
\[b_{n+1}\in V.\]  
We then equip $\ss$ with the topology generated by taking the collection of 
all such ${\cal O}(V, \vec a)$ as our basis. 
\end{definition} 


\begin{lemma} $\ss$ is a compact metric space. 
\end{lemma} 


\begin{proof} It is an easy exercise to check sequential 
compactness of the space 
directly by hand, which then suffices by 
second countability of the space; 
alternatively we may observe that $\ss$ is the continuous 
image of a closed subset of the compact space 
\[(H^\infty_{\vec p}\cup G^\infty_{\vec p})^\N.\] 
This second argument also serves to show that the space is separable. 

Since $\ss$ is zero dimensional and compact, any two disjoint closed sets 
can be separated by a clopen sets. As the space 
is obviously Hausdorff, we may then use Urysohn's criterion to 
determine that the space 
is metrizable. 
\end{proof} 









\begin{lemma} $\gp$ is near-hyperbolic (in the sense of \cite{hjorthkechris}). 
\end{lemma} 


\def\m{{\cal M}}

\begin{proof} 
Following C3.3 \cite{hjorthkechris} we consider the action on the compact 
metric space 
$\ss$. 
The equivalence relation induced on $[S_\gp]$ is hyperfinite since it is included in 
the tail equivalence relation, and there are only 
two orbits in the action of $\gp$ on $\gp^*$.  
Thus certainly the induced orbit equivalence relation on 
the whole space $\ss$ is hyperfinite and hence amenable, as is the induced orbit 
equivalence relation on $\m_{\leq 2}(\ss)$.  

From \ref{2.3} and \ref{further3.3} 
we have that the stabilizers of points in $\m_{\leq 2}(\ss)$ 
are abelian, and so certainly amenable. From \ref{2.4} and \ref{lyons} we 
obtain a Borel $\Gamma_{\vec p}$-map  
\[\eta:\m_{\geq 3}(\ss)\rightarrow [\Gamma_{\vec p}]^{<\infty}.\]  
Since $\Gamma_{\vec p}$ is torsion free 
we have from \ref{map} that that the stabilizers of the 
action on $[\Gamma_{\vec p}]^{<\infty}$ and hence $\m_{\geq 3}(\ss)$ 
are trivial. It also follows that the induced orbit 
equivalence relation 
$E_{\gp}^{\m_{\geq 3}(\ss)}$ is smooth, since 
$\{\eta^{-1}[F]: F\in [\Gamma_{\vec p}]^{<\infty}\}$ partitions the 
space into countably many Borel pieces, 
each of which meets a given orbit 
at most once. 
\end{proof} 

\def\Gp{{K_{\vec p}}}
\def\Gq{{K_{\vec q}}}



\begin{lemma} 
\label{2.6} 
Suppose $\vec p, \vec q$ are sequences of primes, 
$\gp, \Gamma_{\vec q}$ are constructed as 
above, $X_{\vec p}$, $X_{\vec q}$ are standard Borel spaces, 
and that 
$K_{\vec p}=\gp\times\gp$, $K_{\vec q}=\Gamma_{\vec q}\times\Gamma_{\vec q}$
act freely and in a mixing and measure preserving manner 
on $X_{\vec p}$, $X_{\vec q}$, with 
the action of any non-amenable subgroup being $E_0$-ergodic. 
Then $E_{\Gp}\leq_B E_\Gq$ implies $\vec p$ is included in $\vec q$. 
\end{lemma} 

\begin{proof} 
We consider the induced 
\[E_{\gp\times\Z}\subset E_{\Gp}\] 
obtained by finding a copy of $\Z$ inside the right hand copy of $\Gp$. 
The reduction of $E_\Gp$ to $E_\Gq$ induces a countable to one homomorphism 
$f$ of $E_{\gp\times\Z}$ to $E_\Gq=E_{\Gamma_{\vec q}\times\Gamma_{\vec q}}$ and 
we obtain induced cocycles 
\[\alpha_1: (\gp\times \Z)\times X_{\vec p}\rightarrow \Gamma_{\vec q},\]
\[\alpha_2: (\gp\times \Z)\times X_{\vec p}\rightarrow \Gamma_{\vec q},\]
by considering the two copies of $\Gamma_{\vec q}$ which underpin $E_{\Gq}$. 
Note that we cannot have 
both of these cocycles equivalent to cocycles into 
amenable subgroups of $\Gamma_{\vec q}$ since $E_{\Gamma_{\vec p}\times \Z}$ is 
$E_0$-ergodic. 

The action of 
$K_{\vec p}=\Gamma_{\vec p}\times \Gamma_{\vec p}$ 
is mixing, and therefore every infinite subgroup acts 
ergodically. In particular for $\Z$ in the subgroup 
$\Gamma_{\vec p}\times \Z$ we have that $\Z$ acts in an ergodic 
fashion.  
\[ [f(x)]_{\Gamma_{\vec q}\times \Gamma_{\vec q}}\] 
is not a.e. constant. Thus 
2.2 \cite{hjorthkechris} applied to 
$\Gamma=\Gamma_{\vec p}
\times \Z$, $\Delta=\Z \triangleleft \Gamma$ 
yields that one of $\alpha_1, \alpha_2$ 
is 
equivalent to a cocycle 
induced by a homomorphism 
$\pi: \Gamma_{\vec p}\times \Z\rightarrow \Gamma_{\vec q}
\times \Gamma_{\vec q}$  
with non-amenable image. 
Since the image is in particular non-trivial it follows that 
there is some $g\neq 0$ in $G_{\vec p}\cup H_{\vec q}$ 
with $\pi(g)\neq 0$. 

However $g$ is divisible by every power of every prime on the 
$\vec p$ sequence, and hence so is $\pi(g)$.  
Thus the lemma follows by \ref{2.1}. 
\end{proof} 

In the following lemma 
\[p_1, p_2: \gp\times \gp\rightarrow \gp\] 
are the respective projections to the first and 
second copies of $\gp$. 

\begin{lemma} 
\label{2.7} 
Suppose $\Gp=\gp\times\gp$ acts freely and in a measure preserving and 
mixing manner on 
the standard Borel probability space 
$(X, \mu)$ and the action of each non-amenable subgroup is $E_0$-ergodic. 
Let 
\[\alpha:\Gp\times X\rightarrow 
\Gamma_{\vec p}\times \Gamma_{\vec p} (=\Gp)\] 
be a Borel cocycle such that its restriction to any non-amenable 
subgroup is not 
equivalent to a cocycle taking values in an amenable subgroup of $\Gp$. 

Then there is an infinite cyclic subgroup $\Delta$ of $K_{\vec p}$ 
such that the \
restricted cocycle 
\[\alpha:\Delta\times X\rightarrow \Gamma_{\vec p}\times\Gamma_{\vec p}\] 
is equivalent to a cocycle 
\[\hat{\alpha}:\Delta\times X\rightarrow 
\Gamma_{\vec p}\times\Gamma_{\vec p}\] 
such that 

\leftskip 0.4in 

\noindent (i) $p_1\circ \hat{\alpha}:\Delta\times 
X\rightarrow \Gamma_{\vec p}$ 
given by a non-trivial homomorphism of 
$\Delta$ into $\Gamma_{\vec p}$ a.e; 

\noindent (ii) $p_2\circ \hat{\alpha}:
\Delta\times X\rightarrow \Gamma_{\vec p}$ 
is trivial a.e. (that is to say, $p_2\circ\hat{\alpha}(\delta, x)$ is the 
identity $\mu$ a.e. $x$). 

\leftskip 0in 

\end{lemma} 

\begin{proof} 
We find some copy $\F_2=\langle a, b\rangle$ of the free group inside $\gp$ and 
consider the induced cocycles 
\[\alpha_{1,j}=p_j\circ \alpha|_{(\F_2\times\langle a\rangle)\times X}: 
(\F_2\times \langle a\rangle )\times X\rightarrow \Gamma_{\vec p},\] 
\[\alpha_{2,j}=p_j\circ \alpha|_{(\langle a\rangle\times \F_2)\times X}: 
(\langle a\rangle\times \F_2 )\times X\rightarrow \Gamma_{\vec p},\] 
into the $j^{\rm th}$ copy of $\Gamma_{\vec p}$, for $j=1, 2$. 
We let 
\[\beta_{1,j}= \alpha_{1, j}|_{(\{1\}\times\langle a\rangle)\times X}: 
(\{1\}\times \langle a\rangle )\times X\rightarrow \Gamma_{\vec p},\] 
\[\beta_{2,j}=p_j\circ \alpha_{2, j}|_{(\langle a\rangle\times \{1\})\times X}: 
(\langle a\rangle\times  \{1\} )\times X\rightarrow \Gamma_{\vec p},\] 
be the respective restrictions. 

\medskip 

\noindent {\bf Claim(1):} For either $j=1$ or $j=2$ we have that the cocycle 
\[\beta_{1,j}:(\{1\}\times \langle a\rangle )
\times X\rightarrow \Gamma_{\vec p}\] 
is equivalent to a trivial cocycle (a.e.). 

\medskip 


\noindent {\bf Proof of claim:} 
Or else 2.2 \cite{hjorthkechris} gives that the induced cocycles 
\[\alpha_{1,j}|_{(\F_2\times \{1\})\times X}: 
(\F_2\times \{1\} )\times X\rightarrow \gp,\] 
$(j=1, 2)$,  
are both equivalent a.e. to cocycles into amenable subgroups of $\gp$ and hence the 
restricted cocycle 
\[\alpha|_{(\F_2\times \{1\})\times X}: 
(\F_2\times \{1\} )\times X\rightarrow \Gp,\] 
is equivalent to a cocycle a.e. into an amenable group, in contradiction to assumptions 
of lemma. \hfill ($\square$Claim) 

\medskip 

\noindent{\bf Claim(2):} For either $\ell=1$ or $\ell=2$ 
\[\alpha_{2,\ell}: (\langle a\rangle \times \F_2)\times X\rightarrow \gp\] 
is equivalent to a homomorphism with  non-amenable image in $\gp$. 

\medskip 

\noindent {\bf Proof of claim:} 
Again by 2.2 \cite{hjorthkechris}. 
\hfill ($\square$Claim) 

\medskip 

%Note then by 3.10 \cite{hjorthkechris} we have $\ell\neq j$ and the homomorphism from 
%$\langle a\rangle\times\F_2$ will be injective on $\{1\}\times \langle a\rangle$ since 
%$\gp$ is torsion free. 

\noindent{\bf Claim(3):} Given $j$ as in claim(1) and $\ell$ as 
in claim(2), $\ell\neq j$. 

\medskip 

\noindent{\bf Proof of Claim:} Suppose instead 
$\ell =j$. Then $\beta_{1,\ell}$ is equivalent to the 
trivial cocycle, which always returns the identity element 
of the group $\Gamma_{\vec p}$. 
On the other hand,  the assumption on $j=\ell$ gives that 
\[\beta_{1,\ell}=\alpha_{2, \ell}
|_{(\{1\}\times \langle a\rangle)\times X}\] 
is equivalent to a cocycle by a non-trivial homomorphism, which 
will then necessarily be injective since $\Gamma_{\vec p}$ 
is torsion free. 

Thus we have established that 
$\beta_{1,\ell}$ is simultaneously equivalent to the 
trivial cocycle and a cocycle given by an injective homomorphism, 
and this directly contradicts 3.5 \cite{hjorthkechris}. 
\hfill ($\square$Claim) 

Thus we get an induced cocycle 
\[\alpha|_{(\{1\}\times \langle a\rangle)\times X} =(\beta_{1,1}, \beta_{1, 2}): 
(\{1\}\times \langle a\rangle)\times X\rightarrow \gp\times\gp\] 
as required. 
\end{proof} 

We these lemmas in place we can complete the proof of the messy fact. 

\begin{lemma} For any infinite string of primes $\vec p$ there is a 
corresponding standard Borel probability space space 
\[(X_{\vec p}, \mu_{\vec p})\] 
on which $K_{\vec p}$ acts in a mixing, measure preserving 
manner, with every non-amenable subgroup having an $E_0$-ergodic 
action and every infinite cyclic subgroup acting in a 
uniquely ergodic manner. 
This can be done so that 
\[\bigcup_{\vec p}\{\vec p\}\times X_{\vec p}\] 
is a standard Borel space, the projection function 
\[(\vec p, x)\mapsto \vec p\] 
is Borel, and the assignment of measures 
\[\vec p\mapsto \delta_{\vec p}\times \mu_{\vec p}\] 
is Borel as a function from infinite sequences of 
primes to $\m(\bigcup_{\vec p}\{\vec p\}\times X_{\vec p})$. 
\end{lemma} 

\begin{proof} 
From A6.1, A4.1, and 3.7 of \cite{hjorthkechris}, but we do need to 
see that this assignment of $X_{\vec p}$ to $\vec p$ is 
effective.  
 

For each sequence of primes $\vec p$ let $X_{\vec p}\subset 2^{\Gp}$ be the 
collection of 
all 
\[f:\Gp\rightarrow \{0,1\}\] 
such that for all infinite cyclic groups 
$\langle\sigma\rangle\subset \Gp$ and 
clopen $U\subset 2^\Gp$ we have 
\[{\rm lim}_{n\rightarrow\infty}\frac{1}{2n+1}|\{i: -n\leq i \leq n , \sigma^i\cdot f\in U\}| 
=\mu_{\vec p}(U),\] 
where $\mu_{\vec p}$ is the usual Bernoulli ``coin flipping" measure on 
$2^\Gp$. Thus by the ergodic theorem $(X_{\vec p}, \mu_{\vec p})$ is uniquely 
ergodic on all infinite subgroups. By A4.1 \cite{hjorthkechris} 
every non-amenable subgroup gives rise to an 
$E_0$-ergodic equivalence relation. 
By A6.1 \cite{hjorthkechris} the action is mixing. 

The various conditions about the assignment of the space and the 
measure being suitably Borel in $\vec p$, as stated above in the 
lemma, are routinely verified since $X_{\vec p}$ is a Borel 
subset of $2^{K_{\vec p}}$ uniformly in $\vec p$. 
\end{proof}  

For $x\in 2^\N$ we use almost disjoint 
coding to assign continuously an infinite 
sequence ${\vec p}(x)$ of distinct primes such that for $x\neq y$ we have 
${\vec p}(x)$ and ${\vec p}(y)$ eventually disjoint. We let 
\[(X_x, \mu_x, E_x)=
(X_{\vec p(x)}, \mu_{{\vec p}(x)}, E_{{K}_{{\vec p}(x)}});\] 
we need to verify conditions (i)-(v) of the messy fact. 

(i), (iv), and (v) are all immediate from the construction. 
(ii) follows from 
\ref{2.6}. For (iii), we suppose $f:X_x\rightarrow X_x$ is a homomorphism of 
$E_x$ to $E_x$. Then \ref{2.7} gives some infinite cyclic 
$\langle \sigma\rangle, 
\langle \tau\rangle\subset K_{{\vec p}(x)}$ and measurable 
\[\theta: X_x\rightarrow K_{{\vec p}(x)}\] 
with 
\[\theta(\sigma\cdot x)\cdot f(\sigma\cdot x)= 
\tau\cdot\theta(x)\cdot f(x).\] 
Thus if we define 
\[\hat{f}: X_x\rightarrow X_x,\]
\[x\mapsto \theta(x)\cdot f(x)\] 
then this gives a measurable map which conjugates the action of 
$\sigma$ with that of $\tau$, and hence, by our unique ergodicity 
assumption, we have 
\[\mu_x({\rm im}(\hat{f}))>0,\]
\[\therefore \mu_x({\rm im}(f))>0.\] 

\section{Is the notion of near hyperbolic interesting?} 

The definition of {\it near hyperbolic} was formulated in 
\cite{hjorthkechris} as a purely ad hoc measure to help 
organize the proofs, and the example discussed there 
were in fact {\it hyperbolic}.  
In the present paper the examples 
given are not hyperbolic, and yet they 
would seem to natural 
instances of near hyperbolic groups. 
%The method from the last section along with 
%\cite{hjorthkechris} 
%can adapted to obtain a relatively broad class of near 
%hyperbolic groups. 
%\begin{lemma} Let $A$ and $B$ be amenable groups. Then the 
%free product 
%\[A*B\] 
%is near hyperbolic. 
%\end{lemma} 
%It is also not true that every 
%group is near hyperbolic. It follows from the 
%rigidity theorems of \cite{hjorthkechris} 
%that products of near hyperbolic groups give 
%rise to actions which are very different to those 
%produced by near hyperbolic groups themselves, and 
%in particular, ${\mathbb F}_2\times {\mathbb F}_2$ 
%is not near hyperbolic. 

Given that the methods of \cite{hjorthkechris} 
work exactly for the class of groups which are 
near hyperbolic, it might be interesting to determine 
which groups lie inside that class, just as it might 
be interesting to determine which groups allow the 
kinds of rigidity theorems found in either 
\cite{hjorthkechris} or \cite{monodshalom}. 
In particular, we do not know what relationship exists 
between the class of near hyperbolic groups and the 
classes ${\mathcal C}$ and ${\mathcal C}_{\rm reg}$ of 
Monod and Shalom in \cite{monodshalom}. They prove that 
every group embeds into one in ${\mathcal C}_{\rm reg}$ 
However while the notion of near hyperbolic goes down 
to subgroups, groups such as ${\mathbb F}_2\times 
{\mathbb F}_2$ can not be near hyperbolic since, 
for instance, they do not satisfy the kinds of 
rigidity theorems shown in 
\cite{hjorthkechris} for this class, and thus, by 
Monod and Shalom, not every 
${\mathcal C}_{\rm reg}$ group 
is near hyperbolic.  
This leaves open the reverse inclusion, 
just as they leave open whether ${\mathcal C}_{\rm reg}$ is strictly 
included in ${\mathcal C}$, and indeed the real task at this 
stage would seem to be to 
characterize the  groups having varying levels of rigidity for their 
actions.

Here it is tempting to ask the following 
audacious question: 

\begin{question} Let $\Gamma_1$ and $\Gamma_2$ be 
torsion free infinite groups, and let $\Gamma_1\times \Z$ and 
$\Gamma_2\times \Z$ act in a mixing and measure preserving 
manner on standard Borel probability spaces 
$X_1$ and $X_2$. If $E_{\Gamma_1}$ is orbit equivalent to 
$E_{\Gamma_1}$, must there exist a homomorphism from 
$\Gamma_1$ to $\Gamma_2$ with amenable kernel? 
\end{question}  


This, along with a great deal more, certainly holds when 
$\Gamma_1, \Gamma_2$ are in the class of near hyperbolic 
by \cite{hjorthkechris} or when they are in ${\mathcal C}$ 
by 2.23 \cite{monodshalom}. However it is hard to imagine that the 
answer to the above question could possibly be affirmative, 
since the methods used in the known cases are so specific. 




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%\bibitem{kechris} A.S. Kechris, {\it Lectures on cost of equivalence relations and 
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\end{thebibliography}







\end{document}



