# Topological generators and full groups

The number of topological generators for full groups of ergodic equivalence relations

François Le Maître, UMPA-ENSL

This paper relates two important invariants associated to orbit equivalence relations.  So let’s define what types of equivalence relations we’re talking about (and a related group), what the two invariants are, and then we’ll say what the main result of the paper actually is.

Suppose we have a countable group $\Gamma$ acting on a probability space $(X,\mu)$.  Then the action is measure-preserving, well, if it preserves the measure.  So $\mu(g\cdot B)=\mu(B)$ for any $g\in\Gamma$ and measurable $B\subseteq X$.  The orbit equivalence relation $E_\Gamma$ on $X$ is defined by

$x E_\Gamma y \Leftrightarrow \exists g\in\Gamma (g\cdot x = y)$

We call the equivalence relation probability measure-preserving if it arises from a measure-preserving action.  We call it ergodic if every measurable union of $E_\Gamma$-classes is either measure 0 or measure 1.  This means, for instance, that if $C\subseteq X$ has positive measure and we look at $O=\cup_{g\in\Gamma} g\cdot C$ (i.e. the union of the $E_\Gamma$-classes which meet $C$), then $\mu(O)=1$.  So in some sense, group actions which give rise to ergodic equivalence relations (unsurprisingly called ergodic group actions) must move things around a lot, since sets of positive measure will cover the whole space (except for a null set) when you act on them by the group.  Of course, there are stronger mixing properties out there, but that doesn’t concern us presently.

Given a probability measure preserving equivalence relation $E$, one can talk about all measure-preserving automorphisms $T$ of $(X,\mu)$ for which $T(x) E x$ for $\mu-a.e.$  $x\in X$.  This forms a group $[E]$, called the full group of $E$.  Note that if $E=E_\Gamma$ for $\Gamma$ acting on $X$ in a measure-preserving way, then $\Gamma\leq [E]$.

The full group of $E$ can be turned into a topological group without too much trouble (the metric is the obvious one, where the distance between two functions is the measure of the elements on which they differ), and then we can bring topological group theory to bear when we talk about it.  The structure of $[E]$ is related to properties of $E$ in ways that we’re still exploring.  One fact is that $[E]$ is topologically simple (i.e. it has no nontrivial closed normal subgroups) iff $E$ is ergodic.

The invariants in question

Suppose we have a measure-preserving equivalence relation $E$.  We extend our viewpoint beyond even $[E]$ to look at partial Borel automorphisms of $(X,\mu, E)$.  By this I mean Borel bijections of the form $f\colon A\rightarrow B$, where $A,B\subseteq X$ are Borel, $f(x) E x$ for $\mu-a.e.$ $x\in X$, and $f$ preserves the measures induced on $A$ and $B$.  We write this set as $[[E]]$.

So suppose we have a collection $F=\{f_1,f_2,\ldots\}\subseteq [[E]]$.  We can define a graph $G_F$ by $x G_F y \Leftrightarrow \exists f_i\in F (f^{\pm 1}(x)=y)$.  If the connected components of $G_F$ are the equivalence classes of $E$ we call $F$ a graphing of $E$.  Then the cost of $F$ is

$\sum_i \mu(\text{dom}(f_i))$,

which one can check is the same as the average degree of a vertex in $G_F$.  The cost of $E$ is the infimum of the costs of its graphings.  We’ll write it $C_\mu(E)$.  (The standard reference for this stuff, by the way, is Topics in Orbit Equivalence, by Kechris and Miller.)

The other invariant we’ll look at is the number of topological generators of $[E]$.  We say $\{g_1, g_2,\ldots\}\subseteq [E]$ is a set of topological generators of $[E]$ if the group generated by $\{g_1,g_2,\ldots\}$ is dense in $[E]$.  We write $t([E])$ for the least number of elements in a set of topological generators for $[E]$.

The main theorem

If $E$ is a probability measure-preserving ergodic equivalence relation, then $t([E])=\lfloor C_\mu(E) \rfloor + 1$.

The proof itself is fairly short, although it does rely on some earlier results on cost and full groups.  Really, this is as nice a relationship as one could hope for.  The author mentions that they are currently working on a paper in the case of non-ergodic equivalence relations, and have some similar results there.