Outer actions of finite groups

Chapter III: Lifts of Borel actions on quotient spaces

3.5 Outer actions of finite groups

Definition 3.4.9. A countable group𝐺 is treeableif it admits a free pmp Borel action whose induced equivalence relation is treeable.

Example 3.4.10. There are many examples of groups which are not treeable (see [KM04, p. 30], [Kec22, p. 10.8]):

• Infinite property (T) groups.

• 𝐺×𝐻, where𝐺 is infinite and𝐻 is non-amenable.

• More generally, lattices in products of locally compact Polish groups𝐺×𝐻, where𝐺is non-compact and 𝐻is non-amenable.

The proof of the next result is motivated by [CJ85, Theorem 5] and the remark following the proof of [FSZ89, Theorem 3.4].

Proposition 3.4.11. Suppose that every outer action of𝐺 lifts. Then𝐺 is treeable.

Proof. We can assume that 𝐺 = 𝐹_∞/𝑁 for some 𝑁 ⊳ 𝐹_∞, where 𝐹_∞ is the free group on infinitely many generators. Fix a free pmp Borel action𝐹_∞ ↷^𝐵 (𝑋 , 𝜇) (for instance, the Bernoulli shift on 2^𝐹∞), and consider the induced free outer action 𝐺 →Out𝐵(𝐸^𝑋

𝑁)(seeExample 3.4.1). By assumption, there is a lift𝐺 →Aut𝐵(𝐸^𝑋

𝑁), which is also a free action. Then 𝐸^𝑋

𝐺 is treeable and preserves𝜇, since𝐸^𝑋

𝐹_∞ satisfies these properties and contains𝐸^𝑋

𝐺. □

Note that we have no control over the treeable CBER in the proof ofProposition 3.4.11.

In particular, the following is open:

Problem 3.4.12. Does every outer action on𝑋/𝐸₀lift?

Let𝐺 ≤ Aut𝐵(𝐸)be a countable subgroup such that𝐹 =𝐸^∨𝐺. For every𝑥 ∈ 𝑋\𝑌, let 𝑔_𝑥 ∈ 𝐺 be least (in some enumeration of 𝐺) such that 𝑔_𝑥 ·𝑥 ∈ 𝑌; this exists by Φ-maximality of 𝑅. Then the equivalence relation generated by 𝑅 ↾ 𝑌 and {(𝑥 , 𝑔_𝑥·𝑥) :𝑥 ∈ 𝑋 \𝑌} is an(𝐸 , 𝐹)–link. □ Corollary 3.5.2. Every outer action of a finite group has a class-bijective lift.

Proof. Follows fromProposition 3.3.4andTheorem 3.5.1. □ The following is a special case ofCorollary 3.6.14, whose proof is much harder.

Corollary 3.5.3. Every outer action ofZhas a class-bijective lift.

Proof. On the finiteZ-orbits, applyCorollary 3.5.2. On the infiniteZ-orbits of𝑋/𝐸,

just lift uniquely. □

We next introduce lifts of morphisms:

Definition 3.5.4. Let𝐻 → 𝐺be a morphism of countable groups. Then𝐻 → 𝐺has theclass-bijective lifting propertyif for any CBER 𝐸 and any diagram of the form

𝐻 Aut𝐵(𝐸)

𝐺 Out𝐵(𝐸)

𝑝_𝐸

with𝐻 →Aut𝐵(𝐸)class-bijective, there is a class-bijective lift𝐺 → Aut𝐵(𝐸).

Proposition 3.5.5. Let𝐻 be a countable group, let(𝐺_𝑛)𝑛be a countable family of countable groups, let𝐻 →𝐺_𝑛 be morphisms, and let𝐺 be the amalgamated free product of the𝐺_𝑛over𝐻. If every outer action of𝐻 has a class-bijective lift, and each𝐻 →𝐺_𝑛has the class-bijective lifting property, then every outer action of𝐺 lifts.

Proof. Let 𝐸 be a CBER, and fix 𝐺 → Out𝐵(𝐸). By assumption, there is a class-bijective lift of𝐻 →Out𝐵(𝐸). Then for each𝑛, there is a class-bijective lift 𝐺_𝑛 →Aut𝐵(𝐸) such that the following diagram commutes:

𝐻 Aut𝐵(𝐸)

𝐺_𝑛 Out𝐵(𝐸)

𝑝𝐸

Thus by the universal property of amalgamated products, there is a lift 𝐺 →

Aut𝐵(𝐸). □

Theorem 3.5.6. Let 𝐺 be a countable group and let 𝑁 ⊳ 𝐺 be a finite normal subgroup such that every outer action of𝐻 =𝐺/𝑁has a class-bijective lift.

(1) The inclusion𝑁 ↩→𝐺 has the class-bijective lifting property.

(2) Every outer action of𝐺 has a class-bijective lift.

Proof. (1)implies(2)byCorollary 3.5.2, so it suffices to show(1).

Let𝐸 be a CBER on𝑋, and suppose we have 𝑁 Aut𝐵(𝐸)

𝐺 Out𝐵(𝐸)

𝑝_𝐸

with 𝑁 → Aut𝐵(𝐸) class-bijective, and let 𝐹 = 𝐸^∨𝑁. Note that 𝐿 = 𝐸^𝑋

𝑁 is an (𝐸 , 𝐹)–link. There is an induced outer action𝐻 →Out𝐵(𝐹). We can assume that [𝐹 : 𝐸] =𝑛 < ∞. Let𝑆be a transversal for 𝐿, and fix a Borel actionZ/𝑛Z ↷ 𝑋 generating 𝐿.

Define an injection Aut𝐵(𝐹 ↾𝑆) ↩→Aut𝐵(𝐹) as follows: given𝑇 ∈Aut𝐵(𝐹 ↾𝑆), let𝑇^′∈Aut𝐵(𝐹) be the unique morphism satisfying𝑇^′(𝑘·𝑥) =𝑘 ·𝑇(𝑥) for every 𝑥 ∈ 𝑆 and 𝑘 ∈ Z/𝑛Z. This descends to an injection Out𝐵(𝐹 ↾ 𝑆) ↩→ Out𝐵(𝐹) satisfying the following commutative diagram:

Out𝐵(𝐹 ↾𝑆) Out𝐵(𝐹)

Sym𝐵(𝐹 ↾𝑆) Sym𝐵(𝐹)

𝑖_𝐹↾𝑆 𝑖𝐹

We claim that this injection is a bijection. To see this, let 𝑇 ∈ Aut𝐵(𝐹). Since 𝑋 = Ã

𝑘∈Z/𝑛Z𝑘 · 𝑆, we have 𝑛𝑆e= 𝑋e in the cardinal algebra K (𝐹 × 𝐼

N). Thus 𝑛𝑇(𝑆) =𝑇(𝑋) = 𝑋e, so by the cancellation law, we havee𝑆=𝑇(𝑆), i.e., there is some 𝑇^′ ∈ Inn𝐵(𝐹) with𝑇^′(𝑇(𝑆)) = 𝑆. Then (𝑇^′𝑇) ↾ 𝑆 ∈ Aut𝐵(𝐹 ↾ 𝑆) is the desired map.

Thus we obtain an outer action 𝐻 → Out𝐵(𝐹 ↾ 𝑆) and by assumption, there is an (𝐹 ↾ 𝑆, 𝐸^∨𝐺 ↾𝑆)–link𝐿^′. Then the equivalence relation generated by𝐿 and 𝐿^′is

an(𝐸 , 𝐹^′)–link. □

We will prove next a generalization ofCorollary 3.5.2to morphisms. For that, we need the following result.

Proposition 3.5.7. Let𝐸 ⊆ 𝐹 be a bounded index extension of CBERs. Then the following are equivalent:

(1) 𝐸 ⊳ 𝐹.

(2) There is a finite subgroup𝐺 ≤ Out𝐵(𝐸)such that𝐹 =𝐸^∨𝐺. Proof. (2) =⇒ (1)Immediate.

(1) =⇒ (2)Let 𝐻 =(ℎ_𝑛)𝑛 ≤ Aut𝐵(𝐸) be a countable subgroup such that𝐹 =𝐸^∨𝐻. We define inductively a sequence(𝑔_𝑛)𝑛 ⊆ Inn𝐵(𝐹) ∩Aut𝐵(𝐸)as follows: for every

𝐹-class𝐶, if there is𝑖 such that 𝑝_𝐸

↾𝐶(ℎ_𝑖 ↾𝐶) ≠ 𝑝_𝐸

↾𝐶(𝑔_𝑗 ↾𝐶) for all 𝑗 < 𝑛, then for the least𝑖with this property, set𝑔_𝑛↾𝐶 =ℎ_𝑖 ↾𝐶; otherwise set𝑔_𝑛 ↾𝐶 =id ↾𝐶. Note that the sequence(𝑔_𝑛)𝑛is eventually equal to id𝑋, since𝐸 is of bounded index in𝐹. Thus the group ˜𝐺 =⟨𝑔_𝑛⟩_𝑛<∞ ≤ Inn𝐵(𝐹) ∩Aut𝐵(𝐸)is finitely generated. Note also that 𝐹 = 𝐸^∨𝐺˜. Now the image of Inn𝐵(𝐹) ∩Aut𝐵(𝐸) in Out𝐵(𝐸) is locally finite, since it is a subgroup of (𝑆_𝑛)^𝑋/𝐹 for some finite symmetric group𝑆_𝑛. So the image𝐺of ˜𝐺in Out𝐵(𝐸)is finite, and we are done. □ We have a generalization ofTheorem 3.5.1:

Theorem 3.5.8. Let𝐸 ⊆ 𝐹 ⊆ 𝐹^′be CBERs such that𝐸 has finite index in𝐹^′and 𝐸 ⊳ 𝐹^′. Then every(𝐸 , 𝐹)–link is contained in an (𝐸 , 𝐹^′)–link.

Proof. By partitioning the underlying standard Borel space𝑋, we can assume that there is some 𝑛 < ∞ such that every 𝐹^′-class contains at most 𝑛 𝐹-classes. We proceed by induction on𝑛. The case𝑛=1 is trivial.

Let 𝐿 be an (𝐸 , 𝐹)–link and let 𝑆 be a transversal for 𝐿. Let Φ be the set of 𝐴 ∈ [𝐹^′↾𝑆]^<∞which are a transversal for𝐹 ↾𝐶 for some𝐹^′-class𝐶. By [KM04, Lemma 7.3], there is aΦ-maximal fsr 𝑅. Let𝑌 ⊆ 𝑋 be the set of𝑥 ∈ 𝑋 such that [𝑥]𝐹 ⊆ [dom(𝑅)]𝐿 and let𝑍 =𝑋\𝑌. We can assume that no𝐹^′-class is contained in𝑌, since the equivalence relation generated by 𝑅 and 𝐿 is an (𝐸 , 𝐹^′)–link on such a class. ByΦ-maximality of𝑅, no 𝐹^′-class is contained in 𝑍 either. By(2) of Proposition 3.4.3, we have 𝐸 ↾ 𝑌 ⊳ 𝐹^′ ↾ 𝑌, so by the induction hypothesis,

there is an (𝐸 ↾ 𝑌 , 𝐹^′ ↾ 𝑌)–link 𝐿_𝑌 containing 𝐿 ↾ 𝑌. Similarly, there is an (𝐸 ↾𝑍 , 𝐹^′↾𝑍)–link𝐿_𝑍 containing𝐿 ↾𝑍.

Let𝑆_𝑌 and𝑆_𝑍 be transversals for𝐿_𝑌 and𝐿_𝑍respectively. It suffices to show that there is some𝑇 ∈ Inn𝐵(𝐹^′) such that𝑇(𝑆_𝑌) = 𝑆_𝑍, since then the smallest equivalence relation containing 𝐿_𝑌 and 𝐿_𝑍 and {(𝑥 , 𝑇(𝑥)) : 𝑥 ∈ 𝑆_𝑌} is an (𝐸 , 𝐹^′)–link. In other words, we need to show thatf𝑆_𝑌 =f𝑆_𝑍 in the cardinal algebraK (𝐹^′×𝐼

N). By Proposition 3.5.7, there is a finite subgroup𝐺 ≤ Out𝐵(𝐸) such that𝐹^′=𝐸^∨𝐺. By partitioning𝑋, we can assume that[𝐹^′↾𝑌 : 𝐸 ↾𝑌] =𝑛_𝑌 and[𝐹^′↾ 𝑍 : 𝐸 ↾ 𝑍] =𝑛_𝑍 for some𝑛_𝑌, 𝑛_𝑍 < ∞. Then𝑌e=𝑛_𝑌f𝑆_𝑌 and𝑍e=𝑛_𝑍f𝑆_𝑍. Let𝑘 = ^|𝐺|

𝑛𝑌+𝑛𝑍. Then for every 𝑥 ∈ 𝑋, we have

|{𝑔 ∈𝐺 : [𝑥]𝐸 ⊆ 𝑔·𝑌}| = ∑︁

[𝑦]𝐸⊆𝑌

|{𝑔 ∈𝐺 : [𝑥]𝐸 =𝑔· [𝑦]𝐸}| =𝑘 𝑛_𝑌, and thus|𝐺|e𝑌 =𝑘 𝑛_𝑌𝑋e. Similarly,|𝐺|e𝑍 =𝑘 𝑛_𝑍𝑋e. Thus

|𝐺|𝑛_𝑌𝑛_𝑍f𝑆_𝑌 =|𝐺|𝑛_𝑍𝑌e= 𝑘 𝑛_𝑌𝑛_𝑍𝑋e= |𝐺|𝑛_𝑌e𝑍 = |𝐺|𝑛_𝑌𝑛_𝑍𝑆f_𝑍,

which yieldsf𝑆_𝑌 =𝑆f_𝑍 by the cancellation law. □

Corollary 3.5.9. Every morphism of finite groups has the class-bijective lifting property.

Proof. Suppose we have

𝐻 Aut𝐵(𝐸)

𝐺 Out𝐵(𝐸)

𝑝𝐸

with𝐻 and𝐺finite, and 𝐻 → Aut𝐵(𝐸) class-bijective. Then𝐸_𝐻 is an(𝐸 , 𝐸^∨𝐻)–

link, so byTheorem 3.5.8, there is an(𝐸 , 𝐸^∨𝐺)–link𝐿_𝐺 containing𝐸_𝐻. This lets us define an action of𝐺by setting𝑔·𝑥to be the unique element in both [𝑥]𝐿_𝐺 and

𝑔· [𝑥]𝐸. □

Corollary 3.5.10. Every outer action of an amalgamated free product of finite groups has a lift.

Proof. Let 𝐻 be a finite group, let (𝐺_𝑛)𝑛<∞ be finite groups, let 𝐻 → 𝐺_𝑛 be morphisms, and let 𝐺 be the amalgamated free product of the 𝐺_𝑛 over 𝐻. By Corollary 3.5.2, every outer action of𝐻has a class-bijective lift. ByCorollary 3.5.9, the morphisms𝐻→ 𝐺_𝑛have the class-bijective lifting property. Thus byProposi-

tion 3.5.5, every outer action of𝐺lifts. □

Given CBERs 𝐸 ⊆ 𝐹, we say that 𝐹/𝐸 is hyperfinite if there is an increasing sequence(𝐹_𝑛)𝑛of finite index extensions of𝐸 such that𝐹 =Ð

𝑛𝐹_𝑛.

Corollary 3.5.11. Let 𝐸 ⊳ 𝐹 be CBERs with 𝐹/𝐸 hyperfinite. Then there is an (𝐸 , 𝐹)–link.

Proof. ApplyTheorem 3.5.8countably many times. □

Corollary 3.5.12. Every outer action of a locally finite group has a class-bijective lift.

Proof. Immediate fromCorollary 3.5.11. □