Download Full Groups of Equivalence Relations

Full Groups of Equivalence Relations
Todor Tsankov
California Institute of Technology
First European Set Theory Meeting, Bȩdlewo 2007
joint work with John Kittrell
Countable Borel equivalence relations
Definition
An equivalence relation E on a standard Borel space X is called
countable if every equivalence class is countable.
Theorem (Feldman–Moore)
Every countable, Borel equivalence relation is the orbit equivalence
relation of a Borel action of a countable group.
Hence, the study of countable Borel equivalence relations reduces to
the study of group actions.
Invariant measures
Some of the main tools in the theory of Borel equivalence relations
come from ergodic theory and require the presence of a measure.
A distinct disadvantage of those methods is that they yield results
which hold only almost everywhere, while interesting features of the
equivalence relations are often concentrated on null sets.
(X, µ) is a standard probability space.
An equivalence relation E on X is called measure-preserving if it
is generated by a measure-preserving group action. It is ergodic
if every E-invariant set is either null or conull.
All equivalence relations from now on are assumed countable,
Borel, and measure-preserving.
Orbit equivalence
Definition
Two measure-preserving equivalence relation E and F on spaces
(X, µ) and (Y, ν), respectively are isomorphic if there are invariant
Borel sets A ⊆ X and B ⊆ Y of full measure and a measure-preserving
map f∶ A → B such that
x1 E x2 ⇐⇒ f(x1 ) F f(x2 ).
Two measure-preserving actions Γ ↷ (X, µ) and ∆ ↷ (Y, ν) are
called orbit equivalent if their orbit equivalence relations EΓX and E∆Y
are isomorphic.
Orbit equivalence has become an important meeting point of ergodic
theory, Borel equivalence relations, and von Neumann algebras.
Amenable groups
Theorem (Dye, 1963)
All ergodic actions of Z are orbit equivalent.
An equivalence relation generated by a Z action is called hyperfinite.
Theorem (Ornstein–Weiss, 1980)
If Γ ↷ X is measure-preserving and Γ is amenable, then EΓX is
hyperfinite a.e.
Corollary
All ergodic actions of amenable groups are orbit equivalent.
However, for non-amenable groups, there are many non-orbit
equivalent actions.
Full groups
Aut(X, µ) denotes the group of all measure-preserving
automorphisms of (X, µ) (where two automorphisms are identified if
they are equal a.e.).
Definition
Let E be a countable, measure-preserving equivalence relation. The
full group of E, denoted by [E], is the group of all automorphisms of
(X, µ) preserving E, i.e.,
[E] = {T ∈ Aut(X, µ) ∶ Tx E x
for a.e. x ∈ X}.
The uniform topology
Definition
The uniform topology of Aut(X, µ) is given by the metric
d(T, S) = µ({T ≠ S}).
The metric d is complete but the topology on Aut(X, µ) is not
separable.
However, [E] is a closed subgroup of Aut(X, µ) which is separable
and therefore Polish. It also has a two-sided invariant metric.
Dye’s theorem
Theorem (Dye, 1963)
Let E and F be two countable, measure-preserving, ergodic
equivalence relations on the standard probability space (X, µ). Then
the following are equivalent:
E and F are isomorphic;
[E] and [F] are isomorphic (algebraically);
there exists f ∈ Aut(X, µ) such that f[E] f−1 = [F].
Moreover, every algebraic isomorphism between [E] and [F] is
realized by a conjugacy.
In short, full groups are complete invariants for the equivalence
relations.
A motivating example
We look for topological group properties which distinguish
equivalence relations. The only known result of this kind to date is the
following.
Definition
A topological group is called extremely amenable if every time it acts
continuously on a compact space, the action has a fixed point.
Theorem (Giordano–Pestov, 2002)
Let E be an ergodic equivalence relation on (X, µ). Then the following
are equivalent:
E is hyperfinite;
[E] is extremely amenable.
A first attempt
Question: Can one distinguish equivalence relations by looking
at the topology of the full groups alone, forgetting the group
structure?
Answer: No.
Theorem
Let E be a measure-preserving equivalence relation on (X, µ) which is
not equality a.e. Then [E] is homeomorphic to ℓ2 .
Other groups homeomorphic to ℓ2
Topological groups known to be homeomorphic to ℓ2 include:
infinite-dimensional separable Fréchet spaces (Anderson, Kadec,
1960’s);
MALG µ (Bessaga–Pełczyński, 1972);
Homeo+ ([0, 1]) (Anderson, 197?);
Aut(X, µ) (with the weak topology) (Nhu, 1990);
the unitary group of ℓ2 (with the strong topology);
the isometry group of the Urysohn metric space (Melleray, 2006);
. . . and many others.
Topological generators
Another possible invariant (suggested by Kechris) one could look at is
the number of topological generators.
Definition
Let G be a separable topological group. The number of topological
generators of G, denoted by t(G), is the minimal number n such that
there is an n-generated dense subgroup of G.
t(G) can be 1, 2, . . . , ℵ0 .
If a subgroup Γ ≤ [E] is dense, then Γ generates E.
Gaboriau’s theory of cost implies that if E is generated by a free
action of Fn , then t([E]) ≥ n.
So as long as t([E]) is finite for some E, t([E]) is a non-trivial
invariant.
The hyperfinite case
Theorem
Let E be ergodic, hyperfinite. Then t([E]) ≤ 3.
Since [E] is non-abelian, clearly t([E]) ≥ 2.
Question
Let E be ergodic, hyperfinite. Is t([E]) equal to 2 or 3?
An upper bound for t([E])
Theorem
Let E1 , E2 be measure-preserving equivalence relations. Then
⟨[E1 ] ∪ [E2 ]⟩ is dense in [E1 ∨ E2 ].
Lemma
If F ⊆ E are equivalence relations, F is finite and E ergodic, then there
exists a hyperfinite, ergodic equivalence relation E′ such that
F ⊆ E′ ⊆ E.
Lemma
If E is hyperfinite, there exist finite equivalence relations F1 , F2 such
that E = F1 ∨ F2 .
Corollary
If E is generated by an ergodic action of Fn , then t([E]) ≤ 6n.
Finitely many generators
Theorem
Let E be an ergodic equivalence relation on (X, µ). Then the following
are equivalent:
1
E can be generated by a finitely generated group;
2
t([E]) < ∞;
3
cost E < ∞.
Proposition
If E is generated by a free, ergodic action of Fn , then
n + 1 ≤ t([E]) ≤ 6n.
Question
If E and F are generated by free, ergodic actions of Fn , is it the case
that t([E]) = t([F])? If yes, what is the value?
Automorphisms are continuous
Dye’s theorem implies that for ergodic E, every algebraic
automorphism of [E] is automatically continuous. In a certain
sense, the algebra of the group determines the topology.
Ergodicity is essential: if T is a non-trivial involution, then
[T] ≅ (MALG µ , △) and MALG µ , being a vector space over
Z/2Z, has many non-continuous automorphisms.
The phenomenon of automatic continuity
Classical results:
Every Baire measurable homomorphism between Polish groups
is continuous (Pettis).
Every measurable homomorphism between locally compact
groups is continuous (Kleppner).
A recent discovery: if the source group is sufficiently complicated,
then every homomorphism into a separable group is continuous.
This can be thought of as the inability of the axiom of choice to
produce pathological homomorphisms.
Automatic continuity examples
Groups which enjoy this strong automatic continuity property include:
groups with ample generics (Kechris–Rosendal) (e.g., S∞ the
automorphism group of the random graph, and the group of
measure-preserving homeomorphisms of 2N );
Homeo(2N ), Aut(Q), Homeo+ (R), Homeo+ (S1 )
(Rosendal–Solecki);
homeomorphism groups of compact 2-manifolds (Rosendal).
The Steinhaus property
Definition (Rosendal–Solecki)
A topological group G is said to have the Steinhaus property (with
exponent k) if for every set W ⊆ G such that countably many left
translates of it cover G, W k contains an open neighborhood of the
identity.
Proposition (Rosendal–Solecki)
If G and H are topological groups, f∶ G → H is a homomorphism, G
has the Steinhaus property, and H is separable, then f is continuous.
Automatic continuity for full groups
Theorem
Let E be ergodic. Then [E] is Steinhaus with exponent 38.
Corollary
Let E be ergodic. Then every homomorphism from [E] to a separable
group is continuous.
Corollary
Let E be ergodic, hyperfinite. Then every action of [E] on a compact
metrizable space is continuous. Hence (by Giordano–Pestov), it has a
fixed point.
Note that “metrizable” is essential. By a theorem of Veech, every
discrete group admits a free action on a compact space.
Topological full groups
Our example for a 3-generated dense subgroup of [E] (E hyperfinite)
comes from topological dynamics.
Let ϕ be an aperiodic homeomorphism of the Cantor space X. For
every homeomorphism γ of X preserving the orbits of ϕ, define its
associated cocycle nγ ∶ X → Z by
nγ (x) = n ⇐⇒ γ(x) = ϕn (x).
Definition
The topological full group of ϕ is defined by:
[[ϕ]] = {γ ∈ Homeo(X) ∶ ∀x ∃n ∈ Z γ(x) = ϕn (x)
and nγ is continuous}.
Topological full groups (cont.)
Proposition
Let ϕ be a minimal homeomorphism of the Cantor space X and µ be a
ϕ-invariant measure. Let E be the equivalence relation induced by ϕ.
Then the countable group [[ϕ]] is dense in [E].
Theorem (Matui, 2006)
If ϕ is minimal, the following are equivalent:
[[ϕ]] is finitely generated;
ϕ is conjugate to a minimal subshift and K0 (X, ϕ)/2K0 (X, ϕ) is a
finite group.
K0 (X, ϕ) = C(X, Z)/{ f − f ○ ϕ−1 }
Matui also computed explicit examples of 3-generated topological full
groups.
The cost
Definition
Let E be an equivalence relation. A Borel graph on X (a symmetric,
irreflexive relation) is called a graphing of E if its connected
components are the E-equivalence classes.
Definition
The cost of a graphing G is defined by:
cost G =
1
∫ degG (x) dµ(x).
2 X
The cost of E is defined by:
cost E = inf{cost G ∶ G is a graphing of E}.
The cost (cont.)
If E is generated by a group action Γ ↷ X and S is a generating set for
Γ, then G defined by
xG y ⇐⇒ ∃γ ∈ S
γ ⋅ x = y or γ ⋅ y = x
is a graphing of E. In particular,
cost E ≤ ∣S∣.
If the action Γ ↷ X is free, G gives the structure of a Cayley graph of Γ
to every E-equivalence class.
Theorem (Gaboriau)
If E is generated by a free action of Fn , cost E = n.