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
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
MotivationsDefinitionSquare model - introduction
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| → ∞?
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
MotivationsDefinitionSquare model - introduction
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| → ∞?
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
MotivationsDefinitionSquare model - introduction
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→∞.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
MotivationsDefinitionSquare model - introduction
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
MotivationsDefinitionSquare model - introduction
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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⟩
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
DefinitionsFurther results
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]
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
If the density d < 38 only a) can occur. We can correct it to omit
self-intersection.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
If the density d < 38 only a) can occur. We can correct it to omit
self-intersection.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Space with wallsCayley complexHypergraphs
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
Definition (The hexagonal model)
G (n, d) = 〈S |R〉S is a finite set of n generators
R is a set of (2n − 1)6d relators chosen uniformly at randomamong about (2n − 1)6 words of length 6.
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→∞.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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 .
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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.
d ∈ (0, 1) is called the density
n is fixed but l goes to infinity.
A property P occurs with overwhelming probability (w.o.p.) if
P(P holds for G (n, d))→ 1,
as l →∞.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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]
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
Definition (The k-angular model model)
G (n, d) = 〈S |R〉S is a finite set of n generators
R is a set of (2n − 1)dk relators chosen uniformly at randomamong about (2n − 1)k words of length k.
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→∞.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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.
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
Sharp threshold for Property (T)
k
d(T)
12
381
3
524
3 4 5 6 7 8 9 109. . .
„∞”Gromov model
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups
What are random groups?Isoperimetric inequality
Propery (T) and Haagerup PropertyWalls and hypergraphs
Other models
Hexagonal model - definitionHexagonal model - resultsGromov modelk-angular model - definitionSharp threshold for Property (T)
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).
Tomasz Odrzygóźdź Survey on the square and hexagonal model for random groups