## Notes of François Le Maître’s lecture

Full groups in the locally compact measure preserving setting

1. Orbit equivalence

Let ${G}$ be a locally compact secound countable group, ${(X,\mu)}$ a standard probability measure space, on which ${G}$ acts preserving ${\mu}$. Let ${R_G}$ denote the corresponding equivalence relation.

1.1. A general lemma

Lemma 1 Let ${G}$ and ${H}$ act on ${X}$. The following are equivalent.

1. There is a full measure subset on which the equivalence relations ${R_G}$ and ${R_H}$ restrict to the same equivalence relation.
2. For all ${g\in G}$,

$\displaystyle \begin{array}{rcl} \mu(\{x\in X\,;\,gx\in Hx\})=1. \end{array}$

Proof: set ${A=\{x\in X\,;\, \textrm{for a.e. }g\in G,\,gx\in Hx\}}$.

1.2. Full groups

Definition 2 Say two actions ${G}$, ${H}$ on ${(X,\mu)}$ are orbit equivalent if there exists a subset ${A}$ of full measure and an automorphism (measure preserving bijection mod null sets) of ${(X,\mu)}$ that maps ${R_G}$ to ${R_G}$ in restriction to ${A}$.

Definition 3 Given an action of ${G}$ on ${(X,\mu)}$, the associated full group ${[R_G]}$ is defined by

$\displaystyle \begin{array}{rcl} [R_G]=\{T\in Aut(X,\mu)\,;\,\forall x\in X,\,T(x)\in Gx\}. \end{array}$

Orbit equivalence boils down to conjugacy of full groups. This follows from the Lemma above.

Proposition 4 Let ${G}$ and ${H}$ act on ${(X,\mu)}$. Let ${S\in Aut(X,\mu)}$. The following are equivalent.

1. ${S}$ is an orbit equivalence between the actions of ${G}$ and ${H}$.
2. ${S[R_G]S^{-1}=[R_H]}$.

2. From isomorphism to conjugacy

2.1. Dye’s reconstruction theorem

In 1959, Dye defined full groups a sfollows: a group of automorphisms of ${X,\mu)}$ is a full group if it is stable under cutting and pasting. This amounts, given a countable partition of ${X}$ and elements of ${G}$ that move them around in order to obtain again a partition of ${X}$, to put these elements together to define a new automorphism.

Theorem 5 (Dye’s reconstruction theorem, 1963) Let ${G_1}$, ${G_2}$ be ergodic full groups. Suppose ${\psi:G_1\rightarrow G_2}$ is an abstract isomorphism. Then ${\psi}$ coincides with conjugation by some automorphism.

Corollary 6 ${Aut(X,\mu)}$ itself has no outer automorphisms.

Theorem 7 (Carderi-Le Maitre) Let ${G}$ and ${H}$ act ergodically on ${(X,\mu)}$. Then ${[R_G]}$ and ${[R_H]}$ are abstractly isomorphic iff the actions are orbit equivalent.

3. More properties of full groups

3.1. The countable discrete case

If ${G}$ is countable discrete, ${[R_G]}$ has a topology (the uniform metric) that turns it into a Polish space (separable, metrizable). Here are a few known fact.

Theorem 8 (Giordano-Pestov 2005) Let ${G}$ be countable discrete and act freely on ${(X,\mu)}$. The following are equivalent.

1. ${G}$ is amenable.
2. ${[R_G]}$ is amenable.

Definition 9 Let ${G}$ be a topological group. Its topological rank ${t(G)}$ is the minimum number of elements needed to generate a dense subgroup.

Example. ${t({\mathbb R}^n)=n+1}$.

Theorem 10 Let ${G}$ be countable discrete and act ergodically. Then

$\displaystyle \begin{array}{rcl} t([R_G])=\lfloor Cost(R_G)\rfloor +1. \end{array}$

For instance, for a free action of the free group ${\mathbb{F}_n}$, ${t([R_{\mathbb{F}_n}])=n+1}$.

3.2. The locally compact second countable case

We use the topology of convergence in measure. If ${Y}$ is a Polish space, pick a metric on ${Y}$ and equip the space of measurable maps ${X\rightarrow Y}$ with the distance ${d(f,g)=\int_X d(f(x),g(x))\,d\mu(x)}$. The resulting topology does not depend on the choice of metric on ${Y}$.

Theorem 11 (Carderi-Le Maitre) With this topology, ${[R_G]}$ is a Polish group.

Giordano-Pestov’s result generalizes.

Theorem 12 (Carderi-Le Maitre) Let ${G}$ be locally compact, second countable and unimodular. Assume that ${G}$ acts freely on ${(X,\mu)}$. The following are equivalent.

1. ${G}$ is amenable.
2. ${[R_G]}$ is amenable.

Theorem 13 (Carderi-Le Maitre) Let ${G}$ be non discrete and act ergodically. Then

$\displaystyle \begin{array}{rcl} t([R_G])=2. \end{array}$

In fact, a dense ${G_\delta}$ of pairs of elements generate a dense subgroup.

Proof is inspired by the work of Kyed-Vaas on ${\ell^2}$ Betti numbers of locally compact groups. Especially, their notion of discrete section is useful, it more or less allows to reduce to the case ${G=S^1\times\Gamma}$, ${X=S^1\times Y}$, where ${\Gamma}$ is countable discrete acting on ${Y}$ ergodically, and ${S^1}$ acts on itself by translation. Cost 2 comes from the ${S^1}$ factor.