Descriptive set theory and dynamics of countable groups

Introduction

Descriptive set theory

Borel equivalence relations

Countable Borel equivalence relations

Hyperfinite Borel equivalence relations

Lifts of Borel actions

Dichotomies for Polish modules

Quotients by countable normal subgroups are hyperfinite

Introduction

The theory of countable Borel equivalence relations has been applied in numerous areas of mathematics. In fact, they are equivalent to the universal countable Borel equivalence relation𝐸∞, which is the most difficult countable Borel equivalence relation.

Preliminaries and examples

Proofs

Then there is a morphism𝐺 →Out𝐵(𝐹), induced by the action of𝐺 on𝐺/𝐻, so we get a lift𝐺 →Aut𝐵(𝐹). Then for every𝑛 there is a class-bijective rise 𝐺𝑛 →Aut𝐵(𝐸) such that the following diagram commutes:.

Lifts of Borel actions on quotient spaces

Introduction

Using this result and a variant of [KM04, Corollary 13.3], we show, in Corollary 3.3.11, that the full lifting problem has a positive solution generically for an arbitrary periodic (ie with all its classes infinite) CBER𝐸. A Borel action of a countable group 𝐺 on a standard Borel space 𝑋 is pmp if it has an invariant Borel probability measure.

Preliminaries

𝐺 The CBER caused by this action, i.e. an equivalence relation whose classes are exactly the orbits of this action. The Feldman-Moore theorem (see e.g. [Kec22, Theorem 3.3]) states that for every CBER𝐸 on𝑋 there exists a countable group𝐺 and a Borel action𝐺 on𝑋 such that 𝐸 =𝐸𝑋.

Borel actions on quotient spaces

So, if 𝐸 is incompressible and satisfies 𝐸 𝐵 𝐸 ⊕ (𝐸×𝐼 . N), then every non-trivial countable group admits a non-external action on 𝑋/𝐸. It is shown in [RM21] that there exists a countable basis of pairs 𝐸 ⊆ 𝐹 CBER such that there is no Borel action 𝐺 ↷𝐵 𝑋/𝐸with𝐹 =𝐸∨𝐺 (see Section 3.8 for the exact statement). To see that, since im 𝑓𝑖 is a 𝐸-compressible perfect section for𝐸, there exists some Borel injections𝑇𝑖 such that 𝑇(𝑥) 𝐸 𝑥 for every𝑥 and𝑇𝑖(𝑋) =im 𝑓𝑖.

Let𝐸 be an aperiodic CBER on a Polish space𝑋. Then for every Borel action𝐺 ↷𝐵 𝑋/𝐸 there is a comeager 𝐸∨𝐺 invariant Borel subset𝑌 ⊆ 𝑋 such that 𝐺 ↷𝐵𝑌/𝐸 has a class bijective lift. Assume there is a free pmp action of a countable group 𝐺 on a standard probability space (𝑋, 𝜇) with the following properties:. i) 𝐺 is co-Hopfian (that is, injective morphisms of 𝐺 in itself are surjective) and 𝐺 has no non-trivial finite normal subgroups (e.g. SL3(Z)); ii).

Outer actions

For CBERs𝐸 ⊆ 𝐹 it is possible that 𝐸 is not normal in 𝐹, but there is still a Borel action 𝐺 ↷𝐵 𝑋/𝐸 such that 𝐹 = 𝐸∨𝐺, as shown in the example at the beginning of section 3.3. On the other hand, if there is a smooth compound 𝐿, then 𝐹 it must be compressible since it contains the aperiodic smooth 𝐿. Then there is a link given by the action of {0} ×Z, but there is no smooth link, because 𝐹 is not compressible.

By normality, any two 𝐸-classes contained in the same class 𝐹 have the same cardinality, so by dividing the space into invariant Borel groups, we can assume that there are some N} such that every class 𝐸 has cardinality𝑛. It is clear that if 𝐺 is a free set, then every external operation of 𝐺 has a raiser.

Outer actions of finite groups

Let𝐻 be a countable group, let (𝐺𝑛)𝑛be a countable family of countable groups, let𝐻 →𝐺𝑛 be morphisms, and let𝐺 be the merged free product of 𝐺𝑛over𝐻. If every outer action of𝐻 has a class-bijective lift, and every𝐻 →𝐺𝑛has the class-bijective lift property, then every outer action of𝐺 lifts. Thus, by the universal property of merged products, there is a lift 𝐺 →. Let 𝐺 be a countable group and let 𝑁 ⊳ 𝐺 be a finite normal subgroup such that every outer action of 𝐻 =𝐺/𝑁 has a class bijective lift.

This allows us to define an action of 𝐺 by setting 𝑔 𝑥 as the unique element in both [𝑥]𝐿𝐺 and . Let 𝐻 be a finite group, let (𝐺𝑛)𝑛<∞ be finite groups, let 𝐻 → 𝐺𝑛 be morphisms, and let 𝐺 be the aggregated free product of the 𝐺𝑛 over 𝐻.

Outer actions of amenable groups

Given CBERs 𝐸 ⊆ 𝐹, we say that 𝐹/𝐸 is hyperfinite if there is an increasing sequence(𝐹𝑛)𝑛of finite index expansions of𝐸 such that𝐹 =Ð. An A–quasi-tiling of 𝐴 is a tuple C = (𝐶𝐵)𝐵∈A of subsets of 𝐴 such that 𝐵𝑐 ⊆ 𝐴 for every 𝑐 ∈ 𝐶𝐵, and the family is disint} Let 𝐺 be an amenable group whose conjugation equivalence relation on its space of subgroups is smooth.

Let C be a transverse to the conjugation equivalence relation on the space of subgroups, and for each subgroup 𝐻 ≤ 𝐺, fix some 𝑔𝐻 ∈ 𝐺 such that 𝑔𝐻𝐻 𝑔−1. So if for every𝐻 ∈ C we let be the union of the ergodic components with a stabilizer𝐻, then there is a (𝐸 ↾ 𝑋𝐻, 𝐸∨𝐻 ↾ 𝑋𝐻)–.

Figure 3.1: The shaded regions are 𝑋 𝐵 for 𝐵 ∈ A 𝑛 , and the regions above each 𝑋 𝐵 are its translates 𝜑 𝑛

Summary of lifting results for outer actions

Then 𝐸0 is Borel reducible to the conjugation equivalence relation on 𝑋, which is therefore not smooth.

Additional topics

Given a pair 𝐸 ⊆ 𝐹 of CBERs, we say that From [RM21, Theorem 5], there exists a countable set of constraints to be generated by a Borel action. Two projections 𝑝and𝑞 are Murray-von Neumann equivalents, written

Given Polish𝑅 modules𝑀 and𝑁, we say that𝑀 embedsinto𝑁, denoted 𝑀 ⊑𝑅 𝑁, if there is a continuous linear injection of 𝑀in𝑁. We show that there is a countable basis of minimal uncountable Abelian Polish groups (one for each prime and one for characteristic 0).

Equidecomposition in cardinal algebras

Introduction

A classic theorem of Thorisson [Tho96] in probability theory states that if 𝑥and 𝛾then𝛾 𝑥 and𝑥′are equal in distribution. This characterization in terms of switching-merging has been applied to various areas of probability theory including graphs with random roots [Khe18], Brownian motion [PT15], and point processes [HS13]. Setting 𝜇𝛾 to be the measure on 𝑋 defined by 𝜇𝛾(𝐴):= 𝜇({𝛾} ×𝐴), we see that 𝜇and ∈Γon𝑋 such that𝜇=Í.

We thank Alexander Kechris for introducing the author to Thorisson's theorem and suggesting the use of cardinal algebras. We would also like to thank Ruiyuan (Ronnie) Chen for the statement in terms of equidecomposition as well as the reference to the Becker-Kechris lemma.

Preliminaries

More recently they have been used in [KM16] in the study of countable Borel equivalence relations. 17.2] The class of cardinals is a CA under addition (although we strictly require that a CA be a collection). If 𝑋 is a measurable space, then the set B (𝑋) of measurable sets is a GCA under disjoint union.

In CA induced by a 𝜎-complete, 𝜎-distributive network, ≤ is the partial order induced by the network, i.e. 𝑎 ≤ 𝑏iff𝑎=𝑎∧𝑏. For the cardinal class 𝜅 ≤ 𝜆iff there exists an injection 𝜅 ↩→𝜆 and the fact that this is a partial order is the Cantor-Schroeder-Bernstein theorem.

Proof of main theorem

Applications

This also implies a special case of [Mil12, Theorem 24], which says that if dim𝑅(𝑀) is uncountable, then there exists a linearly independent perfect set (see Corollary 5.5.2). 𝐹 ⟨𝐻𝑥⟩𝑥∈𝐹, where the union is adopted over everything finite𝐹 ⊆ 𝑋, so since 𝑋 is countable, there is a number of𝐹 for which 𝐻𝐹 :=⟨𝐻𝑥⟩𝑥∈𝐹 is not meager, and therefore open, since 𝐻𝐹 is analytical. If 𝑀0 and 𝑀1 are Polish 𝑅 modules with Baire measurable submodules 𝑁0 and 𝑁1, respectively, we write 𝑀0/𝑁0 ⊑𝑅 𝑀1/𝑁1 if there is a continuous linear map 𝑀0→ 𝑀1 descending to an injection𝑀0/𝑁 0 ↩→ 𝑀1/ 𝑁1 .

If 𝑆 ≤ 𝑅 is a closed subring, then there exists a corresponding norm on 𝑆 obtained by restricting the norm on 𝑅. By Theorem 5.1.1, there exists a minimal element ℓ1(R) that is bistable with the usual space ℓ1 of absolutely summable sequences.

A dichotomy for Polish modules

Introduction

A key feature of many of these theories (and all of the above examples) is the existence of dichotomy theorems, which state that either an object is simple or there is a canonical obstacle contained within it. This is usually stated in terms of preorders, saying that there is a natural basis for preordering objects that are not simple (recall that a basis for a preorder 𝑃 is a subset𝐵 ⊆ 𝑃such that for each 𝑝 ∈𝑃 there some𝑏 𝑏 ∉ 𝑏 ∈ ). A particularly good aspect of Polish modules is that the concept of "definable" reduction is much simpler than in the general case.

More precisely, we give a countable basis under⊑𝑅for Polish modules which are not countably generated. To contextualize this, we note that considering even very basic module homomorphisms (for example, inclusion of QintoRasQ vector spaces) naturally leads us to consider the broader class of quotients of Polish modules by sufficiently definable submodules.

Polish modules

Let 𝐺 be a Polish group, and let (𝐻𝑥)𝑥∈R be an analytic, independently generating family of subgroups of 𝐺. Let 𝑅 be a Polish ring with no non-trivial compact subgroups, and let 𝑀 be a locally compact Polish 𝑅 module. Let𝜋𝑛: 𝑅N → 𝑅𝑛denote the projection to the first𝑛 coordinates, and let𝑀𝑛=ker(𝜋𝑛◦ 𝑓), which is a closed submodule of𝑀.

Thus, 𝑀𝑛 is countable, so if we choose the root (𝑚𝑖)𝑖 <𝑛in𝑀 of standard base from 𝑅𝑛, then 𝑀 is generated by 𝑀𝑛∪ (𝑚𝑖)𝑖 <𝑛 and thus countably generated. We know nothing about the preorder⊑𝑅 bounded on locally compact modules, including the existence of a minimum or maximum element.

Proper normed rings

In general, we do not know whether every locally compact field ring admits a compatible regular norm. If 𝑅 is a finite regular normed ring, then ℓ1(𝑅) = 𝑅N, which in particular is homeomorphic to the Cantor space. To show this, we will use the characterization according to Dijkstra and van Mill [DM09, Theorem 1.1].

Then 𝑋 is homeomorphic to complete Erdő's space if there is a zero-dimensional metrizable topology𝜏on𝑋 coarser than the original topology such that every point in 𝑋 has a neighborhood basis (of the original topology) consisting of closed nowhere dense Polish subspaces of (𝑏, ) . It is enough to show that every closed sphere is a closed nowhere dense Polish subspace of (ℓ1(𝑅), 𝜏).

Special cases

Proof of the main theorems

To satisfy the second condition, when choosing 𝜀𝑘 you have to take into account that the set of (𝑟𝑖)𝑖 < 𝑘 is compact, so the set of Í. When choosing 𝑚𝑘, keep in mind that the first condition applies to a comeager set of 𝑚𝑘, since 𝑁+𝑅𝑚0+𝑅𝑚1 · · · +𝑅𝑚𝑘−1 is analytic and not open, because 𝑀/𝑁 would otherwise have a countable dimension. The second condition holds for an open set of𝑚𝑘, since the set of𝑟 with |𝑟𝑘!| ≤ 𝑘 is compact.

When choosing 𝑚𝑘, for the first condition, for fixed 𝑟 ∉𝐼 and 𝑚′ ∈ 𝑅𝑚0+ · · · +𝑅𝑚𝑘−1, we previously showed that {𝑟 𝑚 ∉𝑁+𝑚′} is small, i.e. by quantizing over countably many𝑟𝑚𝑚′ , the set 𝑚𝑘 satisfying the first condition is small. The second condition holds for the open set 𝑚𝑘, since the set 𝑟 with |𝑟𝑘!| ≤ 𝑘 is bounded.