Elementary amenable groups and the space of marked groupsElementary amenable groups and the space of...

Elementary amenable groups and the space ofmarked groups

Phillip Wesolek (joint with Jay Williams)

Binghamton University

McGill Seminar on Logic, Category Theory, and Computation

Phillip Wesolek (joint with Jay Williams) Elementary amenable groups March, 2017 1 / 28

DefinitionA topological space is Polish if it is separable and admits a completecompatible metric.

DefinitionA (Polish) parameter space for a family of isomorphism types ofmathematical objects X is a (Polish) topological space X with asurjective function φ : X → X .

RemarkIf X is well-chosen, set-theoretic properties of X reflect mathematicalproperties of X and vice versa.

Let Fω be the free group on countably many generators and identifyP(Fω) with the cantor space {0,1}Fω =: 2Fω .

Definition (Grigorchuk)The space of marked groups is

Gω := {Fω/N | N E Fω}

with the subspace topology inherited by identifying Fω/N with N.

• Gω is a compact Polish space.• Gω parametrizes the class of countable groups.

• Gω is a compact Polish space.

• Gω parametrizes the class of countable groups.

DefinitionA discrete group is amenable if it admits a finitely additive left invariantprobability measure.

The class of these groups is denoted by A.

Examples: finite groups, abelian groups, Z× A5, Z o Z, Grigorchukgroup,...Non-examples: Non-abelian free groups, non-elementary hyperbolicgroups, SLn(Z),...

DefinitionA discrete group is amenable if it admits a finitely additive left invariantprobability measure. The class of these groups is denoted by A.

Examples: finite groups, abelian groups, Z× A5, Z o Z, Grigorchukgroup,...

Non-examples: Non-abelian free groups, non-elementary hyperbolicgroups, SLn(Z),...

FactThe class of amenable groups

1 contains all finite groups and all abelian groups and2 is closed under taking group extensions, subgroups, quotients,

and directed unions.

1 contains all finite groups and all abelian groups and

2 is closed under taking group extensions, subgroups, quotients,and directed unions.

Definition (∼57)The class of elementary amenable groups, denoted by EG, is thesmallest class of groups that

1 contains all finite groups and all abelian groups and

2 is closed under taking group extensions, subgroups, quotients,and directed unions.

Fact (folklore)The set of amenable marked groups is a Gδ set - i.e.

a countableintersection of open sets.

QuestionIs EG a Borel set in Gω?

Group-theoretic translationIs there a “nice” characterization of elementary amenable groups?

Fact (folklore)The set of amenable marked groups is a Gδ set - i.e. a countableintersection of open sets.

DefinitionLet X be a Polish space.

A set A ⊆ X is analytic if there is a Polishspace Y and a continuous map ψ : Y → X such that ψ(Y ) = A. A setB ⊆ X is coanalytic if X \ B is analytic.

• Borel sets are coanalytic• There are sets that are coanalytic but non-Borel. (Souslin)• Analytic or coanalytic sets require “uncountable information” to

define, while Borel sets are definable with “countable information.”

DefinitionLet X be a Polish space. A set A ⊆ X is analytic if there is a Polishspace Y and a continuous map ψ : Y → X such that ψ(Y ) = A.

A setB ⊆ X is coanalytic if X \ B is analytic.

DefinitionLet X be a Polish space. A set A ⊆ X is analytic if there is a Polishspace Y and a continuous map ψ : Y → X such that ψ(Y ) = A. A setB ⊆ X is coanalytic if X \ B is analytic.

• Borel sets are coanalytic

• There are sets that are coanalytic but non-Borel. (Souslin)• Analytic or coanalytic sets require “uncountable information” to

• Borel sets are coanalytic• There are sets that are coanalytic but non-Borel. (Souslin)

• Analytic or coanalytic sets require “uncountable information” todefine, while Borel sets are definable with “countable information.”

define,

while Borel sets are definable with “countable information.”

Theorem (W.–Williams, 15)The set of elementary amenable marked groups is coanalytic andnon-Borel.

Group-theoretic translationThere is no “nice” definition of elementary amenability. Indeed, weshow it is characterized by a chain condition.

Corollary (Grigorchuk, 83)EG $ A.

Group-theoretic translationThere is no “nice” definition of elementary amenability.

Indeed, weshow it is characterized by a chain condition.

Outline of proof

Let N<N denote the collection of finite sequences of natural numbers.

DefinitionA tree T ⊂ N<N is a subset closed under taking initial segments. Atree T is well-founded if there are no infinite branches.

ObservationThe set of trees, denoted by Tr, is a compact space

DefinitionA tree T ⊂ N<N is a subset closed under taking initial segments.

Atree T is well-founded if there are no infinite branches.

Outline of the proof

1 Via a decomposition procedure, we build a Borel map Φ : Gω → Tr.

2 We prove that G ∈ EG if and only if Φ(G) is well-founded.3 Applying classical results in descriptive set theory and group

theory, we deduce that EG is coanalytic and non-Borel.

1 Via a decomposition procedure, we build a Borel map Φ : Gω → Tr.2 We prove that G ∈ EG if and only if Φ(G) is well-founded.

3 Applying classical results in descriptive set theory and grouptheory, we deduce that EG is coanalytic and non-Borel.

1 Via a decomposition procedure, we build a Borel map Φ : Gω → Tr.2 We prove that G ∈ EG if and only if Φ(G) is well-founded.3 Applying classical results in descriptive set theory and group

theory, we deduce that EG is coanalytic and non-Borel.

Decomposition trees

QuestionHow can we take apart an elementary amenable group “from above?”

Observation (Chou, Osin)A non-trivial finitely generated elementary amenable group has anon-trivial finite or abelian quotient.

QuestionHow can we take apart an elementary amenable group “from above?”

Observation (Chou, Osin)A non-trivial finitely generated elementary amenable group has anon-trivial finite or abelian quotient.

A decomposition procedure

1 Exhaust the group by finitely generated subgroups. Considerthese subgroups.

2 Passing to the commutator subgroup of each eliminates theabelian quotient.

3 Intersecting some of the finite index subgroups of each takes careof some of the finite quotients. (Careful!)

4 Repeat.

1 Exhaust the group by finitely generated subgroups.

Considerthese subgroups.

4 Repeat.

3 Intersecting some of the finite index subgroups of each takes careof some of the finite quotients.

(Careful!)4 Repeat.

4 Repeat.

Suppose that G is a group with an enumeration.

1 Define Rn(G) := 〈g0, . . . ,gn〉.2 Put Nm(G) := {N E G | |G : N| ≤ m} and define

Sm(G) := [G,G] ∩⋂Nm(G).

NoteA marked group comes with a preferred enumeration.

Suppose that G is a group with an enumeration.1 Define Rn(G) := 〈g0, . . . ,gn〉.

2 Put Nm(G) := {N E G | |G : N| ≤ m} and define

Sm(G) := [G,G] ∩⋂Nm(G).

Suppose that G is a group with an enumeration.1 Define Rn(G) := 〈g0, . . . ,gn〉.2 Put Nm(G) := {N E G | |G : N| ≤ m} and

define

Sm(G) := [G,G] ∩⋂Nm(G).

Suppose that G is a group with an enumeration.1 Define Rn(G) := 〈g0, . . . ,gn〉.2 Put Nm(G) := {N E G | |G : N| ≤ m} and define

Sm(G) := [G,G] ∩⋂Nm(G).

Suppose that G is a group with an enumeration.1 Define Rn(G) := 〈g0, . . . ,gn〉.2 Put Nm(G) := {N E G | |G : N| ≤ m} and define

Sm(G) := [G,G] ∩⋂Nm(G).

For k ≥ 1 and G ∈ Gω, we define a tree T k (G) and associated markedgroups Gs ∈ Gω for each s ∈ T k (G) as follows:

• Put ∅ ∈ T k (G) and let G∅ := G.• Suppose we have s ∈ T k (G) and Gs. If Gs 6= {e}, put san ∈ T k (G)

and Gsan := S|s|+k (Rn (Gs)).

DefinitionT k (G) is the decomposition tree of G with offset k .

For k ≥ 1 and G ∈ Gω, we define a tree T k (G) and associated markedgroups Gs ∈ Gω for each s ∈ T k (G) as follows:• Put ∅ ∈ T k (G) and let G∅ := G.

• Suppose we have s ∈ T k (G) and Gs. If Gs 6= {e}, put san ∈ T k (G)and Gsan := S|s|+k (Rn (Gs)).

For k ≥ 1 and G ∈ Gω, we define a tree T k (G) and associated markedgroups Gs ∈ Gω for each s ∈ T k (G) as follows:• Put ∅ ∈ T k (G) and let G∅ := G.• Suppose we have s ∈ T k (G) and Gs. If Gs 6= {e},

put san ∈ T k (G)and Gsan := S|s|+k (Rn (Gs)).

For k ≥ 1 and G ∈ Gω, we define a tree T k (G) and associated markedgroups Gs ∈ Gω for each s ∈ T k (G) as follows:• Put ∅ ∈ T k (G) and let G∅ := G.• Suppose we have s ∈ T k (G) and Gs. If Gs 6= {e}, put san ∈ T k (G)

The decomposition tree T k(G)

G =: G∅

Sk(R0(G)) =: G0

R0(G0)

Sk+1(R0(G0)) =: G(0,0)

. . . R0(G1)

Sk+1(R0(G1)) =: G(1,0)

Sk(R1(G)) =: G1

The decomposition tree T k(G)

G =: G∅

Sk(R0(G)) =: G0

R0(G0)

Sk+1(R0(G0)) =: G(0,0)

. . . R0(G1)

Sk+1(R0(G1)) =: G(1,0)

Sk(R1(G)) =: G1

Theorem (W.–Williams, 15)For G a marked group, the following are equivalent:

1 G ∈ EG .2 T k (G) is well-founded for all k ≥ 1.3 T k (G) is well-founded for some k ≥ 1.

Idea of proof.(2)⇔ (3): This is an observation about the rank of well-foundeddecomposition trees.(1)⇒ (2): The collection of groups with well-founded decompositiontrees has the same closure properties as EG.(2)⇒ (1): Induction on the rank of a decomposition tree.

1 G ∈ EG .

2 T k (G) is well-founded for all k ≥ 1.3 T k (G) is well-founded for some k ≥ 1.

1 G ∈ EG .2 T k (G) is well-founded for all k ≥ 1.

3 T k (G) is well-founded for some k ≥ 1.

Idea of proof.(2)⇔ (3): This is an observation about the rank of well-foundeddecomposition trees.

(1)⇒ (2): The collection of groups with well-founded decompositiontrees has the same closure properties as EG.(2)⇒ (1): Induction on the rank of a decomposition tree.

Idea of proof.(2)⇔ (3): This is an observation about the rank of well-foundeddecomposition trees.(1)⇒ (2): The collection of groups with well-founded decompositiontrees has the same closure properties as EG.

(2)⇒ (1): Induction on the rank of a decomposition tree.

Recall thatNn(G) := {N E G | |G : N| ≤ n}

CorollaryA group G is elementary amenable if and only if there is no infinitedescending sequence of finitely generated subgroups

K1 ≥ K2 ≥ . . .

such that Kn 6= {e} and Kn+1 ≤ [Kn,Kn] ∩⋂Nn(Kn) for all n ≥ 1.

K1 ≥ K2 ≥ . . .

such that

Kn 6= {e} and Kn+1 ≤ [Kn,Kn] ∩⋂Nn(Kn) for all n ≥ 1.

K1 ≥ K2 ≥ . . .

such that Kn 6= {e} and

Kn+1 ≤ [Kn,Kn] ∩⋂Nn(Kn) for all n ≥ 1.

K1 ≥ K2 ≥ . . .

such that Kn 6= {e} and Kn+1 ≤ [Kn,Kn] ∩⋂Nn(Kn) for all n ≥ 1.

EG is coanalytic and non-Borel

• For fixed k , the map Φk : Gω → Tr by G 7→ T k (G) is Borel.

• Via our characterization, EG = Φ−1k (WF ).

PropositionEG is coanalytic in Gω.

Proof.The set WF ⊂ Tr is a coanalytic set.

• For fixed k , the map Φk : Gω → Tr by G 7→ T k (G) is Borel.• Via our characterization, EG = Φ−1

k (WF ).

Intuitive definitionA coanalytic rank is a ordinal-valued function which measures thecomplexity of A ⊆ X relative to the topological space X .

Intuitive boundedness theoremIf A ⊆ X is Borel, then any coanalytic rank admits a countable bound.

Intuitive definitionA coanalytic rank is a ordinal-valued function which measures thecomplexity of A ⊆ X relative to the topological space X .

Intuitive boundedness theoremIf A ⊆ X is Borel, then any coanalytic rank admits a countable bound.

• There is a coanalytic rank ρ : WF → ω1.

• The map ρ ◦ Φk : EG→ ω1 is then a coanalytic rank.

ConclusionTo show EG is non-Borel, it suffices to show the least upper bound ofρ ◦ Φk is ω1.

RemarkShowing ρ ◦Φk is unbounded does not require finding groups in A \EG.

• There is a coanalytic rank ρ : WF → ω1.• The map ρ ◦ Φk : EG→ ω1 is then a coanalytic rank.

We show ρ ◦ Φk is unbounded by induction.

Limit stage: Take direct sum.Successor: Use the Hall, Neumann–Neumann, Osin embeddingtheorem: Every elementary amenable group embeds in a twogenerated elementary amenable group.

Theorem (W.–Williams)ρ ◦ Φk is unbounded below ω1 on EGfg .

CorollaryEG is non-Borel in Gω.

We show ρ ◦ Φk is unbounded by induction.Limit stage: Take direct sum.

Successor: Use the Hall, Neumann–Neumann, Osin embeddingtheorem: Every elementary amenable group embeds in a twogenerated elementary amenable group.

We show ρ ◦ Φk is unbounded by induction.Limit stage: Take direct sum.Successor: Use the Hall, Neumann–Neumann, Osin embeddingtheorem:

Every elementary amenable group embeds in a twogenerated elementary amenable group.

We show ρ ◦ Φk is unbounded by induction.Limit stage: Take direct sum.Successor: Use the Hall, Neumann–Neumann, Osin embeddingtheorem: Every elementary amenable group embeds in a twogenerated elementary amenable group.

Questions

QuestionWhat is the least upper bound on the rank of finitely presentedelementary amenable groups?

QuestionIs the class of subexponentially amenable marked groups non-Borel?

Question (Cornulier)What is the Cantor-Bendixson rank of Gω or G2?

Thank you

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