On the Admissibility of a Polish Group Topology

Gianluca Paolini(joint work with Saharon Shelah)

Einstein Institute of MathematicsHebrew University of Jerusalem

A Descriptive Set Theory Day in Torino

Torino, 12 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

DefinitionA Polish group is a topological group whose topology is separableand completely metrizable.

FactGroups of automophisms of countable structures are Polish groups(such groups are called non-archimedean Polish groups).

Question (Becher and Kechris)

Can an uncountable free group admit a Polish group topology?

Answer (Shelah [Sh:771])2No uncountable free group admits 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 Becher 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 .

We will state a version of this lemma later in the talk.

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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 intothe automorphism group of the random graph (and any other freehomogeneous structure in a finite relational language, and also inHall’s universal locally finite group, etc.).

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.

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 / 27

Some Notation

Notation

(1) We denote by Q = G ∗∞ the rational numbers, by Z∞p = G ∗p thedivisible abelian p-group of rank 1, and by Zpk = G ∗(p,k) 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

TheoremLet 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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Bibliography

Gianluca Paolini and Saharon Shelah.No Uncountable Polish Group Can be a Right-Angled ArtinGroup.Axioms 6 (2017), no. 2: 13.

Gianluca Paolini and Saharon Shelah.Polish Topologies for Graph Products of Cyclic Groups.Israel J. Math., to appear.

Gianluca Paolini and Saharon Shelah.Group Metrics for Graph Products of Cyclic Groups.Topology Appl. 232 (2017), 281-287.

Gianluca Paolini and Saharon Shelah.Polish Topologies for Graph Products of Groups.Submitted.

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