On the Admissibility of a Polish Group Topologyand Other Things

Gianluca Paolini(joint work with Saharon Shelah)

Einstein Institute of MathematicsHebrew University of Jerusalem

Young Set Theory Workshop XI

Lausanne, 25-29 June 2018

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The Beginning of the Story

Question (Evans)

Can an uncountable free group be the automorphism group of acountable structure?

Answer (Shelah [Sh:744])1No uncountable free group can be the group of automorphisms ofa countable structure.

1S. Shelah. A Countable Structure Does Not Have a Free UncountableAutomorphism Group. Bull. London Math. Soc. 35 (2003), 1-7.

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Polish Groups

Question (Becker and Kechris)

Can an uncountable free group admit a Polish group topology?

Answer (Shelah [Sh:771])2No uncountable free group can admit a Polish group topology.

2Saharon Shelah. Polish Algebras, Shy From Freedom. Israel J. Math. 181(2011), 477-507.

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Some History/Literature

The question above was answered by Dudley3 “before the it wasasked”. In fact Dudley proved a more general result, but withtechniques very different from Shelah’s.

Inspired by the above question of Becker and Kechris, Solecki4

proved that no uncountable Polish group can be free abelian. AlsoSolecki’s proof used methods very different from Shelah’s.

3Richard M. Dudley. Continuity of Homomorphisms. Duke Math. J. 28(1961), 587-594.

4S lawomir Solecki. Polish Group Topologies. In: Sets and Proofs, LondonMath. Soc. Lecture Note Ser. 258. Cambridge University Press, 1999.

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The Completeness Lemma for Polish Groups

The crucial technical tool used by Shelah in his proof is what hecalls a Completeness (or Compactness) Lemma for Polish Groups.

This is a technical result stating that if G is a Polish group, thenfor every sequence d = (dn : n < ω) ∈ Gω converging to theidentity element eG = e, many countable sets of equations withparameters from d are solvable in G .

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Using The Completeness Lemma

The aim of our work was to extend the scope of applications of thetechniques from [Sh:771] to other classes of groups fromcombinatorial and geometric group theory, most notably:

I right-angled Artin and Coxeter groups;

I graph products of cyclic groups;

I graph products of groups.

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Our Papers (G.P. and Saharon Shelah)

I No Uncountable Polish Group Can be a Right-Angled ArtinGroup. Axioms 6 (2017), no. 2: 13.

I Polish Topologies for Graph Products of Cyclic Groups. IsraelJ. Math., to appear.

I Group Metrics for Graph Products of Cyclic Groups. TopologyAppl. 232 (2017), 281-287.

I Polish Topologies for Graph Products of Groups. Submitted.

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Right-Angled Artin Groups

DefinitionGiven a graph Γ = (E ,V ), the associated right-angled Artin group(a.k.a RAAG) A(Γ) is the group with presentation:

Ω(Γ) = 〈V | ab = ba : aEb〉.

If in the presentation Ω(Γ) we ask in addition that all thegenerators have order 2, then we speak of the right-angled Coxetergroup (a.k.a RACG) C (Γ).

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Examples

Let Γ1 be a discrete graph (no edges), then A(Γ1) is a free group.

Let Γ2 be a complete graph (a.k.a. clique), then A(Γ2) is a freeabelian group, and C (Γ2) is the abelian group

⊕α<|Γ| Z2.

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No Uncountable Polish group can be a RAAG

Theorem (P. & Shelah)

Let G = (G , d) be an uncountable Polish group and A a groupadmitting a system of generators whose associated length functionsatisfies the following conditions:

(i) if 0 < k < ω, then lg(x) 6 lg(xk);

(ii) if lg(y) < k < ω and xk = y , then x = e.

Then G is not isomorphic to A, in fact there exists a subgroup G ∗

of G of size b (the bounding number) such that G ∗ is notembeddable in A.

Corollary (P. & Shelah)

No uncountable Polish group can be a right-angled Artin group.

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What about right-angled Coxeter groups?

The structure M with ω many disjoint unary predicates of size 2 issuch that Aut(M) = (Z2)ω =

⊕α<2ω Z2, i.e. Aut(M) is the

right-angled Coxeter group on the complete graph K2ℵ0 .

QuestionWhich right-angled Coxeter groups admit a Polish group topology(resp. a non-Archimedean Polish group topology)?

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Graph Products of Cyclic Groups

DefinitionLet Γ = (V ,E ) be a graph and let:

p : V → pn : p prime and 1 6 n ∪ ∞

a vertex graph coloring (i.e. p is a function). We define a groupG (Γ, p) with the following presentation:

〈V | ap(a) = 1, bc = cb : p(a) 6=∞ and bEc〉.

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Examples

Let (Γ, p) be as above and suppose that ran(p) = ∞, thenG (Γ, p) is a right-angled Artin group.

Let (Γ, p) be as above and suppose that ran(p) = 2, thenG (Γ, p) is a right-angled Coxeter group.

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A Characterization

Theorem (P. & Shelah)

Let G = G (Γ, p). Then G admits a Polish group topology if only if(Γ, p) satisfies the following four conditions:

(a) there exists a countable A ⊆ Γ such that for every a ∈ Γ anda 6= b ∈ Γ− A, a is adjacent to b;

(b) there are only finitely many colors c such that the set ofvertices of color c is uncountable;

(c) there are only countably many vertices of color ∞;

(d) if there are uncountably many vertices of color c , then the setof vertices of color c has the size of the continuum.

Furthermore, if (Γ, p) satisfies conditions (a)-(d) above, then G canbe realized as the group of automorphisms of a countable structure.

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In Plain Words

Theorem (P. & Shelah)

The only graph products of cyclic groups G (Γ, p) admitting aPolish group topology are the direct sums G1 ⊕ G2 with G1 acountable graph product of cyclic groups and G2 a direct sum offinitely many continuum sized vector spaces over a finite field.

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Embeddability of Graph Products into Polish groups

FactThe free group on continuum many generators is embeddable intoSym(ω) (the symmetric group on a countably infinite set).

QuestionWhich graph products of cyclic groups G (Γ, p) are embeddableinto a Polish group?

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Another Characterization

Theorem (P. & Shelah)

Let G = G (Γ, p), then the following are equivalent:

(a) there is a metric on Γ which induces a separable topology inwhich EΓ is closed;

(b) G is embeddable into a Polish group;

(c) G is embeddable into a non-Archimedean Polish group.

The condition(s) above fail e.g. for the ℵ1-half graph Γ = Γ(ℵ1),i.e. the graph on vertex set aα : α < ℵ1 ∪ bβ : β < ℵ1 withedge relation defined as aαEΓbβ if and only if α < β.

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Even More...

Theorem (P. & Shelah)

Let Γ = (ωω,E ) be a graph and

p : V → pn : p prime, n > 1 ∪ ∞

a vertex graph coloring. Suppose further that E is closed in theBaire space ωω, and that p(η) depends5 only on η(0). ThenG = G (Γ, p) admits a left-invariant separable group ultrametricextending the standard metric on the Baire space.

This generalizes results of Gao et al. on left-invariant groupmetrics on free groups on continuum many generators.

5I.e., for η, η′ ∈ 2ω, we have: η(0) = η′(0) implies p(η) = p(η′). This isessentially a technical convenience.

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The Last Level of Generality

DefinitionLet Γ = (V ,E ) be a graph and Ga : a ∈ Γ a set of non-trivialgroups each presented with its multiplication table presentation andsuch that for a 6= b ∈ Γ we have eGa = e = eGb

and Ga ∩Gb = e.We define the graph product of the groups Ga : a ∈ Γ over Γ,denoted G (Γ,Ga), via the following presentation:

generators:⋃a∈Vg : g ∈ Ga,

relations:⋃a∈Vthe relations for Ga

∪⋃

a,b∈E

gg ′ = g ′g : g ∈ Ga and g ′ ∈ Gb.

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Examples

Let Γ be a graph and let, for a ∈ Γ, Ga be a primitive6 cyclicgroup. Then G (Γ,Ga) is a graph product of cyclic groups G (Γ, p).

6I.e. a cyclic group of order of the form pn or infinity.20 / 40

Some Notation

Notation

(1) We denote by G ∗∞ = Q the rational numbers, by G ∗p = Z∞p thedivisible abelian p-group of rank 1 (Prufer p-group), and byG ∗(p,k) = Zpk the finite cyclic group of order pk .

(2) We let S∗ = (p, k) : p prime and k > 1 ∪ ∞ andS∗∗ = S∗ ∪ p : p prime;

(3) For s ∈ S∗∗ and λ a cardinal, we let G ∗s,λ be the direct sum ofλ copies of G ∗s .

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The First Venue

Theorem (P. & Shelah)

Let G = G (Γ,Ga) and suppose that G admits a Polish grouptopology. Then for some countable A ⊆ Γ and 1 6 n < ω we have:

(a) for every a ∈ Γ and a 6= b ∈ Γ− A, a is adjacent to b;

(b) if a ∈ Γ− A, then Ga =⊕G ∗s,λa,s : s ∈ S∗;

(c) if λa,(p,k) > 0, then pk | n;

(d) if in addition A = ∅, then for every s ∈ S∗ we have that∑λa,s : a ∈ Γ is either 6 ℵ0 or 2ℵ0 .

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The Second Venue

Theorem (P. & Shelah)

Let G = G (Γ,Ga). Then there is a finite subset B1 of Γ such thatif we let B = Γ−B1 then the following conditions are equivalent:

(a) G (Γ B) admits a Polish group topology;

(b) for every s ∈ S∗ the cardinal:

λBs =∑λa,s : a ∈ B ∈ ℵ0, 2

ℵ0.

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The Third Venue

Corollary (P. & Shelah)

Let G = G (Γ,Ga) with all the Ga countable. Then G admits aPolish group topology if only if G admits a non-ArchimedeanPolish group topology if and only if there exist a countable A ⊆ Γand 1 6 n < ω such that:

(a) for every a ∈ Γ and a 6= b ∈ Γ− A, a is adjacent to b;

(b) if a ∈ Γ− A, then Ga =⊕G ∗s,λa,s : s ∈ S∗;

(c) if λa,(p,k) > 0, then pk | n;

(d) for every s ∈ S∗,∑λa,s : a ∈ Γ− A is either 6 ℵ0 or 2ℵ0 .

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The Third Venue (Cont.)

Corollary (P. & Shelah)

Let G be an abelian group which is a direct sum of countablegroups, then G admits a Polish group topology if only if G admitsa non-Archimedean Polish group topology if and only if thereexists a countable H 6 G and 1 6 n < ω such that:

G = H ⊕⊕α<λ∞

Q⊕⊕pk |n

⊕α<λ(p,k)

Zpk ,

with λ∞ and λ(p,k) 6 ℵ0 or 2ℵ0 .

Corollary (P. & Shelah, and independently Slutsky)

If G is an uncountable group admitting a Polish group topology,then G can not be expressed as a non-trivial free product.

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A Conjecture

Conjecture (Polish Direct Summand Conjecture)

Let G be a group admitting a Polish group topology.

(1) If G has a direct summand isomorphic to G ∗s,λ, for some

ℵ0 < λ 6 2ℵ0 and s ∈ S∗, then it has one of cardinality 2ℵ0 .

(2) If G = G1 ⊕ G2 and G2 =⊕G ∗s,λs : s ∈ S∗, then for some

G ′1,G′2 we have:

(i) G1 = G ′1 ⊕ G ′2;(ii) G ′1 admits a Polish group topology;(iii) G ′2 =

⊕G∗s,λ′

s: s ∈ S∗.

(3) If G = G1 ⊕ G2, then for some G ′1,G′2 we have:

(i) G1 = G ′1 ⊕ G ′2;(ii) G ′1 admits a Polish group topology;(iii) G ′2 =

⊕G∗s,λs

: s ∈ S∗.

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Reconstruction Theory

Reconstruction theory deals with the problem of reconstruction ofmodel-theoretic properties of a (countable) structure M from anaturally associated algebraic or topological object X (M), e.g. itsautomorphism group or its endomorphism monoid.

Prototipical results are:

X (M) ∼1 X (M ′) if and only if M ∼2 M ′,

for given equivalence relations ∼1 and ∼2 of interest.

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Techniques and Degrees of Reconstruction

Two independent techniques lead the scene in this field:

(A) the (strong) small index property;

(B) Rubin’s ∀∃-interpretation.

Furthermore, there are two classical degrees of reconstruction:

1. reconstruction up to bi-interpretability;

2. reconstruction up to bi-definability.

These correspond to:

1. abstract group ∼= =⇒ topological group ∼=;

2. abstract group ∼= =⇒ permutation group ∼=.

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The Strong Small Index Property

Let M be a countable structure.

I Small index property (SIP): every subgroup of Aut(M) ofindex less than the continuum contains the pointwisestabilizer of a finite set A ⊆ M.

I Strong small index property (SSIP): every subgroup ofAut(M) of index less than the continuum lies between thepointwise and the setwise stabilizer of a finite set A ⊆ M.

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State of the Art on First Degree of Reconstruction

Theorem (Rubin)

Let M and N be countable ℵ0-categorical structures and supposethat M has a ∀∃-interpretation. Then Aut(M) ∼= Aut(N) if andonly if M and N are bi-interpretable.

Theorem (Lascar (based on works of Ahlbrandt and Ziegler))

Let M and N be countable ℵ0-categorical structures and supposethat M has the small index property. Then Aut(M) ∼= Aut(N) ifand only if M and N are bi-interpretable.

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State of the Art on Second Degree of Reconstruction

Theorem (Rubin)

Let M and N be countable ℵ0-categorical structures with noalgebraicity and suppose that M has a ∀∃-interpretation. ThenAut(M) ∼= Aut(N) if and only if M and N are bi-definable.

Theorem

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Algebraicity

DefinitionLet M be a structure and G = Aut(M).

(1) We say that a is algebraic over A ⊆ M in M if the orbit of aunder G(A) is finite.

(2) The algebraic closure of A ⊆ M in M, denoted as aclM(A), isthe set of elements of M which are algebraic over A.

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The Missing Piece

Theorem (P. & Shelah)

Let K∗ be the class of countable structures M satisfying:

(1) M has the strong small index property;

(2) for every finite A ⊆ M, aclM(A) is finite;

(3) for every a ∈ M, aclM(a) = a.Then for M,N ∈ K∗, Aut(M) and Aut(N) are isomorphic asabstract groups if and only if (Aut(M),M) and (Aut(N),N) areisomorphic as permutation groups.

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The Missing Piece (Cont.)

Corollary (P. & Shelah)

Let M and N be countable ℵ0-categorical structures with thestrong small index property and no algebraicity. Then Aut(M) andAut(N) are isomorphic as abstract groups if and only if M and Nare bi-definable. Furthermore, if f : M → N witnesses thebi-definability of M and N, then f induces the isomorphism ofabstract groups πf : Aut(M) ∼= Aut(N) given by α 7→ f αf −1.

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Outer Automorphisms Groups

Given a group G we let:

Out(G ) = Aut(G )/Inn(G ).

We say that G is complete if it has trivial center andAut(G ) = Inn(G ) (which implies that G ∼= Aut(G )).

If M satisfies the conclusion of the theorem above, then anyf ∈ Aut(Aut(M)) is induced by a permutation of M.

With this it is easy to see:

(1) Letting Rn be the n-coloured random graph (n > 2) we havethat Out(Aut(Rn)) ∼= Sym(n).

(2) Letting Mn be the Kn-free random graph (n > 3) we have thatAut(Mn) is complete.

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Outer Automorphisms Groups (Cont.)

Theorem (P. & Shelah)

Let K be a finite group. Then there exists a countableℵ0-categorical homogeneous structure M with the strong smallindex property and no algebraicity such that K ∼= Out(Aut(M)).

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Automorphism Groups and Model-Theoretic Stability

One of the main tools in modern model theory are certain dividinglines which classify theories in function of how much combinatorialinformation can be coded in the class of its models.

This area of research was invented by Shelah and has had hugenumber of applications in the most disparate fields of mathematics.

The most well-known dividing lines are: ℵ0-stability, superstability,and stability. Each of these notions can be defined in terms ofbounds on the number of types over sets of parameters for modelsof a given theory, but there are many equivalent definitions.

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On a Problem of Rosendal

In a recent work Rosendal isolates a property of topological groupswhich he calls locally boundedness and proves that if M is thecountable, saturated model of an ℵ0-stable theory then Aut(M) islocally bounded.

Rosendal asks if the property of locally boundedness is satisfied bythe group of automorphisms of any countable model of anℵ0-stable theory. This was settled in the negative by Zielinski7

7Zielinski proved more, i.e. that the model can be taken to be atomic.38 / 40

An Impossibility Result

Theorem (P. & Shelah)

For every countable structure M there exists an ℵ0-stablecountable structure N such that Aut(M) and Aut(N) aretopologically isomorphic with respect to the naturally associatedPolish group topologies.

This shows that it is impossible to detect any form of stability of acountable structure M from the topological properties of the Polishgroup Aut(M).

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Our Other Papers (G.P. and Saharon Shelah)

I Reconstructing Structures with the Strong Small IndexProperty up to Bi-Definability. Fund. Math., to appear.

I The Automorphism Group of Hall’s Universal Group. Proc.Amer. Math. Soc. 146 (2018), 1439-1445.

I The Strong Small Index Property for Free HomogeneousStructures. Submitted.

I Automorphism Groups of Countable Stable Structures.Submitted.

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