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.


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