Analogues of Amenability

Introduction

Chapter 4

Given a Polish space𝑋 acountable Borel equivalence relation is a Borel subset of 𝑋2which is an equivalence relation such that all equivalence classes are countable. Similarly, affiniteBorel equivalence relation is a subset of 𝑋2which is an equivalence relation whose equivalence classes are all finite.

Chapter 5

There is an irreducible basis for uncountable Polish modules over a PID: Let 𝑅 be a properly normed discrete PID and let 𝑀 be a Polish 𝑅-module. We recover [Mil12, Theorem 24] for proper normed division rings: [Miller] Let𝑅 be a proper normed division ring, and let 𝑀 be a Polish 𝑅-module.

Two

Introduction

The classical Choquet-Deny Theorem (which was first proved for Z𝑑 by Black-well [Bla55]) states that abelian groups are Choquet-Deny [CD60]; the same is true for practically nilpotent groups [DM61]. A countable discrete group𝐺 is Choquet-Negative if and only if it has no ICC quotients.

Proofs

Let 𝐺 be a countable group, and let (𝐹𝑛)𝑛 be an increasing set of finite subsets of 𝐺 with ∪𝑛𝐹𝑛 = 𝐺. Let𝐺be a countable group, and let(𝐹𝑛)𝑛be an increasing set of finite subsets of𝐺with∪𝑛𝐹𝑛=𝐺.

Three

Introduction

Note that characteristic measures are not necessarily invariant, and that invariant measures are not necessarily characteristic. Some shifts (Z,Σ) allow clearly characteristic measurements: these include unique ergodic shifts, shifts with a unique degree of maximum entropy, shifts with periodic ones. More generally, we do not know any countable group 𝐺 and a shift (𝐺,Σ) that does not allow for characteristic measures.

Theorem 3.1.2 is a consequence of the following, more general result that applies to finitely generated groups and relates the existence of characteristic measures to displacement growth. It is easy to construct a dynamical system (Z, 𝐶), which is not symbolic and has no characteristic measures: just leZact trivially on the Cantor group𝐶. An example of a topologically transitive Z-system without characteristic measures is Zaction with permutations in 𝐶Z, where 𝐶is the Cantor group.

Therefore, the following assertion highlights a tension that must be overcome in order to construct minimal systems without characteristic metrics.

Proofs

However, perhaps surprisingly, if Γ is a normal subgroup of 𝐺, then the equivalence relation 𝐺/Γ must have low Borel complexity: let 𝐺 be a Polish group and let Γ be a countable normal subgroup of 𝐺. When𝐺 is a compact group, Chapter 4.3 implies the following algebraic constraint onΓ: Let𝐺 be a compact group and letΓbe a countable normal subgroup of𝐺. We end with some open questions: let's be a Polish group and let's be a countable subgroup.

For abelian Polish groups, we obtain an irreducible basis (see Chapter 5.4): Let 𝐴 be an uncountable abelian Polish group. Most Polish modules that cannot be written as direct sums, even over a field: Let 𝑅 be a Polish ring, let 𝑀 be a Polish module of 𝑅, and let (𝑁𝑥)𝑥∈R be a family of submodules of 𝑀 such that the set {(𝑚, 𝑥) ∈ 𝑀 ×R : 𝑚 ∈ 𝑁𝑥} is analytic. Let 𝑅 be a normalized real splitting ring, let 𝑀 be a Polish vector space 𝑅 and let 𝑁 ⊆ 𝑀 be an analytic vector subspace.

Lad

Four

Introduction

The theory of definable equivalence relations focuses in particular on the study of Borel equivalence relations, where great progress has been made recently, in which each class is countable, the so-called harmful Borel equivalence relations. There is a natural preorder for Borel equivalence relations called Borel reduction, where 𝐸 reducing to 𝐹 is interpreted as 𝐸 being "easier" than 𝐹. The theory of harmful Borel equivalence relations has been used in many areas of mathematics.

In fact, they are equivalent to the universal countable Borel equivalence relation 𝐸∞, which is the most difficult countable Borel equivalence relation. Countable Borel equivalence relations can be characterized as equivalence relations arising from continuous actions of countable groups on Polish spaces, and thus interact very strongly with dynamics and group theory. Due to a fundamental result by Slaman-Steel and Weiss [SS88; Wei84], the equivalence relations arising from a continuous (or more generally Borel) action of Z are precisely the hyperfinite equivalence relations, which can be written as an increasing union of finite Borel equivalence relations.

By a theorem of Harrington-Kechris-Louveau [HKL90], the hyperfinite equivalence relations occupy only the first two levels of the hierarchy of countable Borel equivalence relations on uncountable Polish spaces under Borel reduction and are therefore considered to have low Borel complexity.

Preliminaries and examples

In general, if 𝐺 is a Polish group and Γ ≤ 𝐺 is a countable subgroup, then 𝐺/Γ can be quite complicated; we will give a non-hyperfinite example in Chapter 4.2. The proof continues by showing that the equivalence relation is generated by a Borel action of a countable abelian group, which is sufficient by the aforementioned theorem of Gao and Jackson. If 𝐺 is a compact group with a countable normal subgroupΓ⊳ 𝐺, then we also show that the algebraic structure of is strongly bounded: Let 𝐺 be a compact group and let be a countable normal subgroup of 𝐺.

Γ the trajectory equivalence relation of Γ y 𝑋, the Borel equivalence relation whose classes are the trajectories of the action. If this is clear from the context, we will abuse the notation and identify 𝐺/Γ with the coset equivalence relation induced by Γy𝐺 (technically 𝐺/Γ is caused by the right action𝐺 xΓ, but this is isomorphic to the left actionΓy𝐺 via inversion). If Γ is a countable group, then Inn(Γ) is a countable subgroup of Aut(Γ), a Polish group under the pointwise convergence topology, and we can consider the quotient Out(Γ) = Aut(Γ)/Inn (Γ) .

For example, when Γ = 𝑆fin (the set of finitely supported permutations on N), we have Out(𝑆fin) 𝑆∞/𝑆fin, which is hyperfinite and nonsmooth.

Proofs

Given Polish 𝑅 modules 𝑀 and 𝑁, we say that 𝑀 is embedded in 𝑁, denoted 𝑀 v𝑅 𝑁, if there is a continuous linear injection from 𝑀 into 𝑁. We show that there is a countable basis of minimal uncountable Abelian Polish groups (one for each prime number and one for characteristic 0). If 𝑀0 and 𝑀1 are Polish 𝑅-modules with Baire-measurable submodules 𝑁0 and 𝑁1, respectively, we write 𝑀0/𝑁0 v𝑅 𝑀1/𝑁1 if there is a continuous linear map.

If 𝑆 ≤ 𝑅 is a closed subring, then a good norm for 𝑆 is obtained by restricting the norm for 𝑅. If𝐼 ⊳ 𝑅 is a closed two-sided ideal, then there is a good norm for 𝑅/𝐼given by|𝑟+𝐼|=min𝑠∈𝑟+𝐼|𝑠|. We can assume that there exists a finite series (𝑟𝑘)𝑘 <𝑛 in 𝑅, such that 𝑈 is the set of series in ℓ1(𝑅), starting with (𝑟𝑘)𝑘 <𝑛.

However, it is unclear whether an intermediate vector space exists: Is there a PolishQ vector space 𝑉 such thatℓ1(Q) @Q𝑉 @Q R.

Five

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 the preordering of objects that are not simple (recall that a basis for a preorder 𝑃 is a subset 𝐵 ⊆ 𝑃 such that for every 𝑝 ∈ 𝑃 there are some 𝑏 ∈ 𝐵 with𝑏 ≤ 𝑝). A particularly good aspect of Polish modules is that the concept of "definable" reduction is much simpler than in the general case.

So there is no loss of generality in considering continuous homomorphisms rather than more general a priori Borel homomorphisms. More precisely, we give a countable basis under v𝑅 for Polish modules which are not countably generated. This also implies a special case of [Mil12, Theorem 24], which states that if dim𝑅(𝑀) is uncountable, then there exists a linearly independent perfect group (see Chapter 5.5).

To contextualize this, we note that considering homomorphisms of very basic modules (for example, involving QintoRasQ vector spaces) naturally leads us to consider the broader class of coefficients of Polish modules from .

Polish modules

They play a crucial role in [BLP20] in the form of "groups with a Polish cover", and they also form some of the most classical examples of countable Borel equivalence relations (e.g. the commensurability relation on the positive reals naturally). is endowed with an abelian group structure). In particular, this implies an unpublished result by Ben Miller showing that a Polish vector space with uncountable dimensions does not have an analytic basis. For certain rings, no locally compact module is embedded in𝑅N, and thus a minimal forv𝑅 cannot be locally compact: Let𝑅 be a Polish ring with no non-trivial compact subgroups, and let𝑀 be a locally compact Polish 𝑅-module.

Let𝜋𝑛: 𝑅N → 𝑅𝑛 denote the projection onto the first 𝑛 coordinates and let 𝑀𝑛=ker(𝜋𝑛◦ 𝑓), which is a closed submodule of 𝑀. Then the subgroup generated by 𝑚 is not compact, so there exists a minimal 𝑘 ∈ N with k𝑘 𝑚k ≥ 2, and thus 𝑘 𝑚 ∈𝐶, which is not possible. Thus, 𝑀𝑛 is countable, so if we choose a root (𝑚𝑖)𝑖 <𝑛 in 𝑀 of standard basis from 𝑅𝑛, then 𝑀 is generated by 𝑀𝑛∪ (𝑚𝑖)𝑖 <𝑛, and therefore countable.

We know nothing about the prior order v𝑅 restricted to locally compact modules, including the existence of a minimal or maximal element.

Proper normed rings

Then 𝑋 is homeomorphic to the completion of an Erdős space if there exists a zero-dimensional metric topology 𝜏on𝑋 coarser than the original topology such that every point in 𝑋 has a neighborhood basis (for the original topology) consisting of nowhere closed field subspaces (𝑋 , 𝜏) . It suffices to show that every closed sphere is nowhere a closed field subspace (ℓ1(𝑅), 𝜏).

Special cases

Since 𝑅/𝔭 is an integral domain, the annihilator of any nonzero element of ℓ1(𝑅/𝔭) is 𝔭, ​​and the same applies to. Then for every nonzero𝑥 ∈ℓ1(𝑅/𝔭) its map inℓ1(𝑅/‘)0 must have the same annihilator, since the map is injective, and hence 𝔭. Applying Chapter 5.4 with 𝑅 =Z gives an irreducible basis for uncountable abelian groups: Let 𝐴 be an uncountable abelian Polish group.

From Chapter 5.1, there exists a minimal elementℓ1(R), which is dually integrated with the usualℓ1 space of absolutely compact sequences.

Proof of the main theorems

When choosing𝜖𝑘, to satisfy the second condition, note that the set considered (𝑟𝑖)𝑖 < 𝑘 is compact, hence the set Í. When choosing𝑚𝑘, note that the first condition holds for a comer set of𝑚𝑘, since𝑁+𝑅𝑚0+𝑅𝑚1· · · +𝑅𝑚𝑘−1 is analytic, and it is not open, since it would otherwise have𝑁wo/. The second condition holds for an open set of𝑚𝑘, since the set of𝑟 with |𝑟𝑘!| ≤ 𝑘 is compact.

By replacing 𝑀 with the submodule generated by𝑈𝑛(which is not analytically lean and therefore open), we can assume that for each open𝑉 ⊆ 𝑀, we have {𝑟 ∈ 𝑅 : 𝑟𝑉 ⊆ 𝑁} = 𝐼. Choosing 𝑚𝑘, for the first condition, for a fixed𝑟 ∉ 𝐼 and𝑚0 ∈ 𝑅𝑚0+ · · · + 𝑅𝑚𝑘−1, we have previously shown that {𝑟 𝑚 ∉ 𝑁 +𝑚0} is lean, so by quantifying over the countable many 𝑟and𝑚0, the set of𝑚𝑘 that satisfies the first condition is comeager. The second condition holds for an open set of𝑚𝑘, since the set of𝑟 with |𝑘𝑟!| ≤ 𝑘 is finite.

Choquet–Deny theorem and distal properties of completely disconnected locally compact groups of polynomial growth”.