Survey on the square and hexagonal model for random groupsWhat are random groups? Isoperimetric inequality Propery (T) and Haagerup Property Walls and hypergraphs Other models Survey

What are random groups?Isoperimetric inequality

Propery (T) and Haagerup PropertyWalls and hypergraphs

Other models

Survey on the square and hexagonal model forrandom groups

Tomasz Odrzygóźdź

Institute of Mathematics PAN

14th November 2016

Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups



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MotivationsDefinitionSquare model - introduction

We want to consider questions:

What does a typical group look like?

What properties are typical for groups?

Random groups:

have some „exotic” properties,

provide examples hard to construct without random groups




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We want to consider questions:

What does a typical group look like?

What properties are typical for groups?

Random groups:

have some „exotic” properties,

provide examples hard to construct without random groups




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Definition (General definition of random group)

G = 〈S |R〉S is a finite set of generators

R is a random set of relations

We need some distribution to draw |R| at random.

Question:What properties hold with high probability when |R| → ∞?




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Definition (General definition of random group)

G = 〈S |R〉S is a finite set of generators

R is a random set of relations

We need some distribution to draw |R| at random.

Question:What properties hold with high probability when |R| → ∞?




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Definition (The square model)

G (n, d) = 〈S |R〉S is a finite set of n generators

R is a set of (2n − 1)4d relators chosen uniformly at randomamong about (2n − 1)4 words of length 4.

d ∈ (0, 1) is called the density

A property P occurs with overwhelming probability (w.o.p.) if

P(P holds for G (n, d))→ 1,

as n→∞.




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The square model - basic results

d (density)0 1

412

hyperbolic and torsion-free trivial

free

For d < 12 hyperbolic, torsion-free, of dimension 2 [O. ’13]

For d > 12 w.o.p. trivial [O. ’13]

For d < 14 w.o.p. free [O. ’13]

We will show the idea of proving the hyperbolicity.




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The square model - basic results

d (density)0 1

412

hyperbolic and torsion-free trivial

free

For d < 12 hyperbolic, torsion-free, of dimension 2 [O. ’13]

For d > 12 w.o.p. trivial [O. ’13]

For d < 14 w.o.p. free [O. ’13]

We will show the idea of proving the hyperbolicity.




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van Kampen diagram

Definition

Let G = 〈S |R〉. For every relation r ∈ R we have a polygon with asmany edges as letters in r . On every edge there is letter s.t. the theboundary word is r . A van Kampen diagram is a planar diagramobtained by gluing this polygons along corresponding edges.

Example

G =⟨a, b|aba−1b−1⟩




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Theorem (O. ’13, based on Ollivier ’07)

For any ε > 0 in the square model at density d < 12 w.o.p. every

reduced van Kampen diagram D w.r.t. the group presentationsatisfies

|∂D| > 4(1− 2d − ε)|D| (1)

|∂D| - number of edges in the boundary of D|D| - number of 2-cells of D

Corollary

Random group in the square model at density d < 12 is w.o.p.

hyperbolic.




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Theorem (O. ’13, based on Ollivier ’07)

For any ε > 0 in the square model at density d < 12 w.o.p. every

reduced van Kampen diagram D w.r.t. the group presentationsatisfies

|∂D| > 4(1− 2d − ε)|D| (1)

|∂D| - number of edges in the boundary of D|D| - number of 2-cells of D

Corollary

Random group in the square model at density d < 12 is w.o.p.

hyperbolic.




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DefinitionsFurther results

Definition

A group G has Property (T) if for every real Hilbert space H everyaction of G on H via affine isometries has a fixed point.

Definition

A group G has a Haagerup Property if there exists a proper actionof this group via isometries on a real Hilbert space.

Haagerup property is a strong negation of Property (T)

If a group has Haagerup property then it satisfiesBaum-Connes conjecture and Novikov conjecture.

Property (T) was used to find an explicit family of expanders




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Definition


Definition








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Definition


Definition








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Definition


Definition








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Definition


Definition








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The square model - further results

d (density)0 3

1038

14

512

12

hyperbolic, torsion-free

lack of Property (T) (T) trivial

free Haagerup

For d < 38 w.o.p. no Property (T) [O. ’16]

For d < 310 w.o.p. Haagerup property [O. ’16]

For d > 512 w.o.p. has (T) [Przytycki, Orlef, O. ’16]




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Space with wallsCayley complexHypergraphs

Space with walls

Definition

Space with walls is a set Y with a nonempty family H ofnonempty subsets satisfying: for every h ∈ H, h′ (completion in Y )belongs to H.

Pairs {h, h′} for h ∈ H are called walls.

Example

Ai = {(x , y) : x < i}

Bi = {(x , y) : y < i}

For i ∈ Z pairs {Ai ,A′i} and {Bi ,B

′i }

form walls on R2.




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Space with walls

Definition



Example

Ai = {(x , y) : x < i}

Bi = {(x , y) : y < i}


′i }

form walls on R2.




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Space with walls

Definition



Example

Ai = {(x , y) : x < i}

Bi = {(x , y) : y < i}


′i }

form walls on R2.




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Metric on a space with walls

Definition

We say that a wall {h, h′} separates points x , y ∈ Y ifx ∈ h, y ∈ h′ or x ∈ h′, y ∈ h.

The distance between x and y in the wall metric dwall is thenumber of walls separating x and y .

Example

x

yWall distance between x and y is 5.




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Definition



Example

x





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Definition



Example

x





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Definition

A group G acts on a space with walls if the action preserves walls,i.e. for every g ∈ G and wall {h, h′} the pair {gh, gh′} is a wall.

Definition

Action of G on a space with walls is proper if it is metricallyproper, i.e. for every p ∈ Y and a sequence gn of elements G s.t.dG (e, gn)→∞ holds dwall(p, gn(p))→∞.

Theorem (Chatterji, Niblo ’04)

If a discrete group G acts properly on a space with walls then itacts properly on CAT(0) cube complex, so has Haagerup Property.




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Definition


Definition







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Definition


Definition







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Theorem (Niblo, Roller ’98)

If a group acts non-trivially on a CAT(0) cube complex then itdoes not have Property (T).

If a group has a subgroup with at least two relative ends then itacts nontrivially on CAT(0) cube complex.

Finding subgroup with 2 relative ends - using walls.




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Definition

The Cayley complex X̃ of a group G = 〈S |R〉 is the universal coverof the presentation complex of G .

In the square model Cayley complex consists of square 2-cells andhas dimension 2.




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Definition






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Definition






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Definition (Ollivier, Wise, ’11)

We define graph Γ:

V (Γ) - set of midpoints of edges of X̃ .

Vertices x , y are jointed if are antipodal points of a 2-cell ofX̃ .

A connected component of Γ is called hypergraph.




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We define graph Γ:







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We define graph Γ:







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Properties of hypergraphs

Action of group on X̃ preserves the system of hypergraphs

Theorem (O. ’13)

In the square model at density d < 13 w.o.p. hypergraphs are

embedded trees in X̃ .

Theorem (O. ’16)

In the square model at density d < 38 w.o.p. hypergraphs can be

corrected to be embedded trees in X̃ .

Embedded trees → split X into two connected components

Two connected components → structure of a space with wallson X




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Theorem (O. ’13)



Theorem (O. ’16)








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Theorem (O. ’13)



Theorem (O. ’16)








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Theorem (O. ’13)



Theorem (O. ’16)








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Theorem (O. ’13)



Theorem (O. ’16)








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Collared diagrams

Definition

A collared diagram is a disc diagram such that the correspondinghypergraph segment passes through all exterior 2-cells but nointernal 2-cells.




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Collared diagrams

Definition

A collared diagram is a disc diagram such that the correspondinghypergraph segment passes through all exterior 2-cells but nointernal 2-cells.




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Theorem (Ollivier, Wise ’11, generalization O. ’13)

A hypergraph is not an embedded tree iff there exists a reduceddiagram collared by this hypergraph.

Remark

For density d < 13 diagrams a), b) and c) violate Isoperimetric

Inequality ⇒ hypergarphs are embedded trees.




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Remark


Inequality

⇒ hypergarphs are embedded trees.




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Remark


Inequality ⇒ hypergarphs are embedded trees.




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If the density d < 38 only a) can occur. We can correct it to omit

self-intersection.




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If the density d < 38 only a) can occur. We can correct it to omit

self-intersection.




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To sum up

Hypergraphs are embedded trees

→They split X̃ into two connected components →Group acts on a space with walls →We check properties of this action (is it proper action?) →Subgroup having ¬ 2 relative ends is a stabilizer of somehypergraph →We conclude lack of (T) or Haagerup property.




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To sum up

Hypergraphs are embedded trees →They split X̃ into two connected components

→Group acts on a space with walls →We check properties of this action (is it proper action?) →Subgroup having ¬ 2 relative ends is a stabilizer of somehypergraph →We conclude lack of (T) or Haagerup property.




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To sum up

Hypergraphs are embedded trees →They split X̃ into two connected components →Group acts on a space with walls

→We check properties of this action (is it proper action?) →Subgroup having ¬ 2 relative ends is a stabilizer of somehypergraph →We conclude lack of (T) or Haagerup property.




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To sum up

Hypergraphs are embedded trees →They split X̃ into two connected components →Group acts on a space with walls →We check properties of this action (is it proper action?)

→Subgroup having ¬ 2 relative ends is a stabilizer of somehypergraph →We conclude lack of (T) or Haagerup property.




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To sum up

Hypergraphs are embedded trees →They split X̃ into two connected components →Group acts on a space with walls →We check properties of this action (is it proper action?) →Subgroup having ¬ 2 relative ends is a stabilizer of somehypergraph

→We conclude lack of (T) or Haagerup property.




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To sum up

Hypergraphs are embedded trees →They split X̃ into two connected components →Group acts on a space with walls →We check properties of this action (is it proper action?) →Subgroup having ¬ 2 relative ends is a stabilizer of somehypergraph →We conclude lack of (T) or Haagerup property.




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Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)

Definition (The hexagonal model)


R is a set of (2n − 1)6d relators chosen uniformly at randomamong about (2n − 1)6 words of length 6.




as n→∞.




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The hexagonal model - results

d (density)0 1

316

12

no (T) (T) trivial

free

For d < 16 free, and for d > 1

2 trivial [O. ’16]

For d < 13 w.o.p. no (T) [O. ’16]

For d > 13 w.o.p. Property (T) [easy observation]

Sharp threshold for Property (T) is 13 .




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Definition (The Gromov density model)

G (n, l , d) = 〈S |R〉S is a finite set of n generators

R is a set of (2n − 1)dl relators chosen uniformly at randomamong about (2n − 1)l words of length l.


n is fixed but l goes to infinity.



as l →∞.




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Results in the Gromov model

d (density)0 1

65

2413

12

no (T) (T) trivial

Haagerup

for d < 16 w.o.p. it has Haagerup property [Ollivier, Wise ’08]

for d < 524 w.o.p. it does not have (T) [Przytycki, Mackay ’14]

For d > 13 w.o.p. it has (T) [Żuk ’03, Kotowski and Kotowski

’13]




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Definition (The k-angular model model)


R is a set of (2n − 1)dk relators chosen uniformly at randomamong about (2n − 1)k words of length k.




as n→∞.




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Let k = mk ′.

If at density d Property (T) holds w.o.p. in the k ’-angularmodel it holds w.o.p. in the k-angular model

If at density d Property (T) does not hold w.o.p. in thek-angular model it w.o.p. does not hold in the k ′-angularmode

Definition

We say that dT ∈ (0, 1) is a sharp threshold in a random groupmodel for a property P if

for densities d < dT w.o.p. property P does not hold

for densities d > dT w.o.p. property P holds.




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Let k = mk ′.



Definition







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Let k = mk ′.



Definition







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Let k = mk ′.



Definition







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Sharp threshold for Property (T)

k

d(T)

12

381

3

524

3 4 5 6 7 8 9 109. . .

„∞”Gromov model




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The triangular model (k=3)

d (density)0 13

12

free (T) trivial

For d > 13 w.o.p. Property (T) [Żuk ’03, Kotowski and

Kotowski ’13]

More detailed picture - Antoniuk, Łuczak, Świątkowski,Friedgut (series of papers).


Documents

Survey on the square and hexagonal model for random groupsWhat are random groups? Isoperimetric inequality Propery (T) and Haagerup Property Walls and hypergraphs Other models Survey