** Full groups in the locally compact measure preserving setting **

**1. Orbit equivalence **

Let be a locally compact secound countable group, a standard probability measure space, on which acts preserving . Let denote the corresponding equivalence relation.

** 1.1. A general lemma **

Lemma 1Let and act on . The following are equivalent.

- There is a full measure subset on which the equivalence relations and restrict to the same equivalence relation.
- For all ,

Proof: set .

** 1.2. Full groups **

Definition 2Say two actions , on are orbit equivalent if there exists a subset of full measure and an automorphism (measure preserving bijection mod null sets) of that maps to in restriction to .

Definition 3Given an action of on , the associated full group is defined by

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

Proposition 4Let and act on . Let . The following are equivalent.

- is an orbit equivalence between the actions of and .
- .

**2. From isomorphism to conjugacy **

** 2.1. Dye’s reconstruction theorem **

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

Theorem 5 (Dye’s reconstruction theorem, 1963)Let , be ergodic full groups. Suppose is an abstract isomorphism. Then coincides with conjugation by some automorphism.

Corollary 6itself has no outer automorphisms.

Theorem 7 (Carderi-Le Maitre)Let and act ergodically on . Then and are abstractly isomorphic iff the actions are orbit equivalent.

**3. More properties of full groups **

** 3.1. The countable discrete case **

If is countable discrete, 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 be countable discrete and act freely on . The following are equivalent.

- is amenable.
- is amenable.

Definition 9Let be a topological group. Its topological rank is the minimum number of elements needed to generate a dense subgroup.

**Example**. .

Theorem 10Let be countable discrete and act ergodically. Then

For instance, for a free action of the free group , .

** 3.2. The locally compact second countable case **

We use the topology of convergence in measure. If is a Polish space, pick a metric on and equip the space of measurable maps with the distance . The resulting topology does not depend on the choice of metric on .

Theorem 11 (Carderi-Le Maitre)With this topology, is a Polish group.

Giordano-Pestov’s result generalizes.

Theorem 12 (Carderi-Le Maitre)Let be locally compact, second countable and unimodular. Assume that acts freely on . The following are equivalent.

- is amenable.
- is amenable.

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

In fact, a dense of pairs of elements generate a dense subgroup.

Proof is inspired by the work of Kyed-Vaas on Betti numbers of locally compact groups. Especially, their notion of discrete section is useful, it more or less allows to reduce to the case , , where is countable discrete acting on ergodically, and acts on itself by translation. Cost 2 comes from the factor.