Embeddability and universal equivalence of partially commutative - - PowerPoint PPT Presentation

Embeddability and universal equivalence of partially commutative groups Montserrat Casals-Ruiz

Marie Curie Postdoctoral Fellow University of Oxford

GAGTA 2013 May 29, 2013

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 1 / 16

Partially commutative groups

Definition

Let Γ = (V(Γ), E(Γ)) be a (undirected) simplicial graph. The partially commutative group (pc group) G = G(Γ) defined by the commutation graph Γ is the group given by the following presentation, G = V(Γ) | [v1, v2] = 1, whenever (v1, v2) ∈ E(Γ).

Remark

Indeed, pc groups = right-angled Artin groups = graph groups =...

Remark

Dually, pc groups can be defined via its non-commutation graph Γ which is the complement of the commutation graph.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 2 / 16

Partially commutative groups

Definition

Let Γ = (V(Γ), E(Γ)) be a (undirected) simplicial graph. The partially commutative group (pc group) G = G(Γ) defined by the commutation graph Γ is the group given by the following presentation, G = V(Γ) | [v1, v2] = 1, whenever (v1, v2) ∈ E(Γ).

Remark

Indeed, pc groups = right-angled Artin groups = graph groups =...

Remark

Dually, pc groups can be defined via its non-commutation graph Γ which is the complement of the commutation graph.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 2 / 16

Partially commutative groups

Definition

Let Γ = (V(Γ), E(Γ)) be a (undirected) simplicial graph. The partially commutative group (pc group) G = G(Γ) defined by the commutation graph Γ is the group given by the following presentation, G = V(Γ) | [v1, v2] = 1, whenever (v1, v2) ∈ E(Γ).

Remark

Indeed, pc groups = right-angled Artin groups = graph groups =...

Remark

Dually, pc groups can be defined via its non-commutation graph Γ which is the complement of the commutation graph.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 2 / 16

Partially commutative groups

Definition

Let Γ = (V(Γ), E(Γ)) be a (undirected) simplicial graph. The partially commutative group (pc group) G = G(Γ) defined by the commutation graph Γ is the group given by the following presentation, G = V(Γ) | [v1, v2] = 1, whenever (v1, v2) ∈ E(Γ).

Remark

Indeed, pc groups = right-angled Artin groups = graph groups =...

Remark

Dually, pc groups can be defined via its non-commutation graph Γ which is the complement of the commutation graph.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 2 / 16

Examples

! ! ! a! b! a! b! Z2=!<!a,b!|![a,b]!>! a! b! a! b! !>! a! b! c! d! a! b! c! d! ,d]!>! a! b! d! c! e! a! b! c! d! e! e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! a! d! c! b! a! c! d! b! ,d]!>!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16

Examples

! ! ! a! b! a! b! Z2=!<!a,b!|![a,b]!>! a! b! a! b! !>! a! b! c! d! a! b! c! d! ,d]!>! a! b! d! c! e! a! b! c! d! e! e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! a! d! c! b! a! c! d! b! ,d]!>! ! ! ! ! ! a! b! a! b! F2=!<!a,b|!ø!>! a! b! c! d! a! b! c! d! ,d]!>! a! b! d! c! e! a! b! c! d! e! e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! a! d! c! b! a! c! d! b! ,d]!>!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16

Examples

a! b! c! d! a! b! c! d! F2!x!F2=!<!a,b,c,d!|[a,b],[a,c],[b,d],![c,d]!>! a! b! d! c! e! a! b! c! d! e! e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! a! d! c! b! a! c! d! b! ,d]!>!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16

Examples

a! b! c! d! a! b! c! d! F2!x!F2=!<!a,b,c,d!|[a,b],[a,c],[b,d],![c,d]!>! a! b! d! c! e! a! b! c! d! e! e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! a! d! c! b! a! c! d! b! ,d]!>! c! d! a! b! d! c! e! a! b! c! d! e! e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! a! d! c! b! a! c! d! b! Z2!*!Z2=!<!a,b,c,d!|![a,b][c,d]!>!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16

Examples

a! b! d! c! e! a! b! c! d! e! <!a,b,c,d,e!|![a,b],![b,c],![c,d],![d,e],![e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! c! b!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16

Examples

a! b! d! c! e! a! b! c! d! e! <!a,b,c,d,e!|![a,b],![b,c],![c,d],![d,e],![e,a]>! a! b! c! d! e! a! b! c! d! e! ]>! c! b! a! b! c! d! e! a! b! c! d! e! <!a,b,c,d,e!|![a,b],![b,c],![c,d],![d,e]>!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16

Algebraic vs. graph properties

Slogan

Many algebraic properties of G = G(Γ) are determined by properties of its defining graph Γ. G is freely decomposable if and only if Γ is not connected. G is directly decomposable if and only if Γ is not connected. The centraliser of a generator v is generated by the star of v in Γ. (Droms 1987) G(Γ) ≃ G(∆) if and only if Γ ≃ ∆.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 4 / 16

Algebraic vs. graph properties

Slogan

Many algebraic properties of G = G(Γ) are determined by properties of its defining graph Γ. G is freely decomposable if and only if Γ is not connected. G is directly decomposable if and only if Γ is not connected. The centraliser of a generator v is generated by the star of v in Γ. (Droms 1987) G(Γ) ≃ G(∆) if and only if Γ ≃ ∆.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 4 / 16

Algebraic vs. graph properties

Slogan

Many algebraic properties of G = G(Γ) are determined by properties of its defining graph Γ. G is freely decomposable if and only if Γ is not connected. G is directly decomposable if and only if Γ is not connected. The centraliser of a generator v is generated by the star of v in Γ. (Droms 1987) G(Γ) ≃ G(∆) if and only if Γ ≃ ∆.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 4 / 16

Algebraic vs. graph properties

Slogan

Many algebraic properties of G = G(Γ) are determined by properties of its defining graph Γ. G is freely decomposable if and only if Γ is not connected. G is directly decomposable if and only if Γ is not connected. The centraliser of a generator v is generated by the star of v in Γ. (Droms 1987) G(Γ) ≃ G(∆) if and only if Γ ≃ ∆.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 4 / 16

Algebraic vs. graph properties

Slogan

Many algebraic properties of G = G(Γ) are determined by properties of its defining graph Γ. G is freely decomposable if and only if Γ is not connected. G is directly decomposable if and only if Γ is not connected. The centraliser of a generator v is generated by the star of v in Γ. (Droms 1987) G(Γ) ≃ G(∆) if and only if Γ ≃ ∆.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 4 / 16

Embeddability between pc groups

Question

Can we characterise when G(∆) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 5 / 16

Warm-up examples

! ! ! ! a! b! c! d! a! b! c! d! e! a! b! c! d! e! a! b! c! d! bd! ad! a! b! c! d! e! ac! ec! a! b! c! d! f! e!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

Warm-up examples

! ! ! ! a! b! c! d! a! b! c! d! e! a! b! c! d! e! a! b! c! d! bd! ad! a! b! c! d! e! ac! ec! a! b! c! d! f! e! ! ! ! ! ! ! ! a! b! c! d! e! a! b! c! d! bd! ad! a! b! c! d! e! ac! ec! a! b! c! d! f! e!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

Warm-up examples

Paths(Kim-Koberda 2011)

If F is a forest, then G(F) < G(P3)

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

Warm-up examples

! ! ! ! ! ! ! a! b! c! d! e! ac! ec! a! b! c! d! f! e!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

Warm-up examples

! ! ! "! #! $! %! '! &! "$! &$! "! #! $! %! &! "!

("$)!

&!

("$)!

%!

("$)!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

Warm-up examples

Cycles (Kim-Koberda 2011)

If Cn is the cycle with n vertices, n ≥ 5, then G(Cn) < G(C5).

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

Extension Graph Conjecture

Definition

Let Γ be a simplicial graph, V(Γ) = {a1, . . . , ak}. We define the extension graph Γe to be the graph whose

1

set of vertices V(Γe) are labelled by elements aw

i , w ∈ G(Γ) and

2

the set of edges E(Γe) are pairs of different vertices (awi

i , a wj j ) such that

[awi

i , a wj j ] = 1 in G(Γ).

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 7 / 16

Extension Graph Conjecture

Definition

Let Γ be a simplicial graph, V(Γ) = {a1, . . . , ak}. We define the extension graph Γe to be the graph whose

1

set of vertices V(Γe) are labelled by elements aw

i , w ∈ G(Γ) and

2

the set of edges E(Γe) are pairs of different vertices (awi

i , a wj j ) such that

[awi

i , a wj j ] = 1 in G(Γ).

Extension Graph Conjecture (Kim-Koberda)

G(∆) < G(Γ) if and only if ∆ < Γe.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 7 / 16

Extension Graph Conjecture

Extension Graph Conjecture (Kim-Koberda)

G(∆) < G(Γ) if and only if ∆ < Γe.

Theorem (Kim-Koberda 2011)

If ∆ < Γe then G(∆) < G(Γ).

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 7 / 16

Extension Graph Conjecture

Extension Graph Conjecture (Kim-Koberda)

G(∆) < G(Γ) if and only if ∆ < Γe.

Theorem (Kim-Koberda 2011)

If ∆ < Γe then G(∆) < G(Γ).

Theorem (Kim-Koberda 2011)

If Γ is triangle-free, then the Extension Graph conjecture holds.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 7 / 16

Extension Graph Conjecture

Extension Graph Conjecture (Kim-Koberda)

G(∆) < G(Γ) if and only if ∆ < Γe.

Theorem (Kim-Koberda 2011)

If ∆ < Γe then G(∆) < G(Γ).

Theorem (Kim-Koberda 2011)

If Γ is triangle-free, then the Extension Graph conjecture holds.

Theorem (C.-Duncan- Kazachkov 2013)

If Γ is triangle-built (i.e. C4, P3-free) then the Extension Graph conjecture holds.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 7 / 16

From a proof to a counterexample

Fact (Kim-Koberda)

One can assume that the embedding from G(∆) to G(Γ) sends: vi ∈ V(∆) → wi1 . . . wiri, wij ∈ V(Γe) and wij pair-wise commute.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 8 / 16

From a proof to a counterexample

Fact (Kim-Koberda)

One can assume that the embedding from G(∆) to G(Γ) sends: vi ∈ V(∆) → wi1 . . . wiri, wij ∈ V(Γe) and wij pair-wise commute.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 8 / 16

Counterexample to the Extension Graph conjecture

! ! ! ! ! ! ! ! ! "! #! $! "%! "&! "'! #%! #&! #'! $%! $&! $'!

()!*! ()"*!

+! "+,!#+,!$+!#!-)".*! "! #! $! "%! "&! "'! #%! #&! #'! $%! $&! $'!

() *! () *!

+! "! #! $! "%! "&! "'! #%! #&! #'! $%! $&! $'!

() *! () *!

1%! 0,/!#%!#&!#',!$ +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

! ! ! ! ! ! ! ! ! "! #! $! "%! "&! "'! #%! #&! #'! $%! $&! $'!

()!*! ()"*!

+! "! #! $! "%! "&! "'! #%! #&! #'! $%! $&! $'!

() *! () *!

1%! 0,/!#%!#&!#',!$ +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

! ! ! ! ! ! ! ! ! ! ! ! "! #! $! "%! "&! "'! #%! #&! #'! $%! $&! $'!

()!*! ()"*!

/!/",#0,!/#,$0!0!1%! ! /!/"%!"&!"'!,!#%!#&!#'0,/!#%!#&!#',!$%!$&!$'00!2! /!/"%,!#&0,/!#'!,$'002%! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

! "! #! $! "%! "&! "'! #%! #&! #'! $%! $&! $'!

()!*! ()"*!

+!

() *!

"%! "&! "'! #%! #&! #'! $%! $&! $'!

() *!

"! #! $! +!

() *!

"!

!

#%! #&! $!

() *!

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() *! () *!

"! #%! #&! $! "! #! $! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

()"*!

"%! "&! "'! #%! #&! #'! $%! $&! $'!

()!*!

"! #! $! +!

() *!

"!

!

#%! #&! $!

() *!

"! #! $! +!

() *! () *!

"! #%! #&! $! "! #! $! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

()"*!

"!

!

#%! #&! $!

()!*!

"! #! $! +!

() *! () *!

"! #%! #&! $! "! #! $! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

$

()"*! ()!*!

"! #%! #&! $! "! #! $!

()"*!

"! #%! #&! $"! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

! "! #! $!

%&!'! %&"'!

(! "! #)! #*! $! (! +!

%& '! %& '!

"! #! $! (! "! #* $"! (! #)! +!

%& '! %& '!

"! #! $! (! "! #)! $! (! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

%&!'! %&"'!

"! #! $! (! "! #*! $"! (! #)! +!

%& '! %& '!

"! #! $! (! "! #)! #* $! (! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

%&!'! %&"'!

"! #! $! (! "! #)! #*! $! (! +!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

! ! "! #! $! %! &! "! $! %! & & "! #! $! %! &! "! $! %'! %(!

)*!+! )*"+!

"! #! %! $! &! "! #'! #(! %! $! &! ,!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Counterexample to the Extension Graph conjecture

! ! "! #! $! %! &! "! $! %! & & "! #! $! %! &! "! $! %'! %(!

)*!+! )*"+!

"! #! %! $! &! "! #'! #(! %! $! &! ,!

Theorem(C.-Duncan-Kazachkov)

Let ∆ and Γ be the above non-commutation graphs. Then G(∆) < G(Γ) and ∆ < Γe.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 9 / 16

Extension Graph conjecture

!

Free! Triangular,free!!!! “few!commutation”! Free!Abelian! C4!P3,!free! “a!lot!of!commutation”!

Extension!Graph! conjecture!holds! Extension!Graph! conjecture!holds! Extension!Graph! conjecture!fails!

Counterexample!is!a! coherent!pc!group!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 10 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)? Droms-Servatius-Servatius ’89: Gave first examples of π1(S) < G(Γ). Crisp-Wiest ’04: The fundamental group of (almost) all surfaces embed in some G(Γ). Röver ’07: π1(S) < G(C5), for all S orientable.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)? Droms-Servatius-Servatius ’89: Gave first examples of π1(S) < G(Γ). Crisp-Wiest ’04: The fundamental group of (almost) all surfaces embed in some G(Γ). Röver ’07: π1(S) < G(C5), for all S orientable.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)? Droms-Servatius-Servatius ’89: Gave first examples of π1(S) < G(Γ). Crisp-Wiest ’04: The fundamental group of (almost) all surfaces embed in some G(Γ). Röver ’07: π1(S) < G(C5), for all S orientable.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Gordon-Long-Reid Question ’04

If π1(S) < G(Γ) then Cn < Γ ?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Gordon-Long-Reid Question ’04

If π1(S) < G(Γ) then Cn < Γ ?

Theorem (Crisp-Sageev-Sapir ’08, Kim ’08)

The answer to GLR’s question is NO: there are π1(S) < G(Γ) so that Cn < Γ

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Updated Question

If π1(S) < G(Γ) then G(Cn) < G(Γ) ?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Updated Question

If π1(S) < G(Γ) then G(Cn) < G(Γ) ?

Weakly Chordal Conjecture (Kim-Koberda)

If Γ is a weakly chordal graph (Cn and Cn-free, n ≥ 5), then G(Cn) < G(Γ).

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Updated Question

If π1(S) < G(Γ) then G(Cn) < G(Γ) ?

Weakly Chordal Conjecture (Kim-Koberda)

If Γ is a weakly chordal graph (Cn and Cn-free, n ≥ 5), then G(Cn) < G(Γ). If the Weakly Chordal conjecture is true, we have a negative answer to the Updated Question.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Updated Question

If π1(S) < G(Γ) then G(Cn) < G(Γ) ?

Weakly Chordal Conjecture (Kim-Koberda)

If Γ is a weakly chordal graph (Cn and Cn-free, n ≥ 5), then G(Cn) < G(Γ). The Extension Graph Conjecture implies the Weakly Chordal conjecture. In particular, the Weakly Chordal conjecture is true for triangle-free/ triangle-built graphs.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Question

Let S be a closed surface. Can it be characterised when π1(S) < G(Γ)?

Updated Question

If π1(S) < G(Γ) then G(Cn) < G(Γ) ?

Weakly Chordal Conjecture (Kim-Koberda)

If Γ is a weakly chordal graph (Cn and Cn-free, n ≥ 5), then G(Cn) < G(Γ).

Theorem (C.-Duncan-Kazachkov)

The Weakly Chordal conjecture does not hold.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Theorem (C.-Duncan-Kazachkov)

The Weakly Chordal conjecture does not hold.

a! b! d! c! e! a! b! d! c! e1! e2! a! b! d! c! e! a! b1! d! c1! e1! e2! c2! b

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Theorem (C.-Duncan-Kazachkov)

The Weakly Chordal conjecture does not hold.

a! b! d! c! e! a! b1! d! c1! e1! e2! c2! b2!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Theorem (C.-Duncan-Kazachkov)

The Weakly Chordal conjecture does not hold.

Remark

The question: If π1(S) < G(Γ) then G(Cn) < G(Γ) ? is still open.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Theorem (C.-Duncan-Kazachkov)

The Weakly Chordal conjecture does not hold.

Remark

The question: If π1(S) < G(Γ) then G(Cn) < G(Γ) ? is still open. The non-commutation graph P6 is an interesting/annoying case.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Surface Embeddability

Theorem (C.-Duncan-Kazachkov)

The Weakly Chordal conjecture does not hold.

Remark

The question: If π1(S) < G(Γ) then G(Cn) < G(Γ) ? is still open. The non-commutation graph P6 is an interesting/annoying case. It stresses the interest in solving the embeddability problem between pc groups.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 11 / 16

Trying to learn from mistakes...

Question

Given a map vi → wi1 . . . wiri, wij ∈ Γe from G(∆) to G(Γ), is it sufficient to check injectivity in a finite family of commutators to assure injectivity? I proved it! or I thought so, because..

Obvious counterexample

Take any map from Z2 to Z

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 12 / 16

Trying to learn from mistakes...

Question

Given a map vi → wi1 . . . wiri, wij ∈ Γe from G(∆) to G(Γ), is it sufficient to check injectivity in a finite family of commutators to assure injectivity? I proved it! or I thought so, because..

Obvious counterexample

Take any map from Z2 to Z

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 12 / 16

Trying to learn from mistakes...

Question

Given a map vi → wi1 . . . wiri, wij ∈ Γe from G(∆) to G(Γ), is it sufficient to check injectivity in a finite family of commutators to assure injectivity? I proved it! or I thought so, because..

Obvious counterexample

Take any map from Z2 to Z

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 12 / 16

Theorem (C.)

Given a map ϕ : vi → wi1 . . . wiri, wij ∈ Γe from G(∆) to G(Γ) such that wij = wkl , if (i, j) = (kl), there exists n = n(Γ) ∈ N such that if ϕ is injective in the set of commutators of weight less than or equal to n, then ϕ is injective. Let’s not give up...

Question

Maybe if there exists an embedding from G(∆) to G(Γ), there exists one satisfying conditions of the theorem? No, the theorem is not strong enough for a characterisation...

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 13 / 16

Theorem (C.)

Given a map ϕ : vi → wi1 . . . wiri, wij ∈ Γe from G(∆) to G(Γ) such that wij = wkl , if (i, j) = (kl), there exists n = n(Γ) ∈ N such that if ϕ is injective in the set of commutators of weight less than or equal to n, then ϕ is injective. Let’s not give up...

Question

Maybe if there exists an embedding from G(∆) to G(Γ), there exists one satisfying conditions of the theorem? No, the theorem is not strong enough for a characterisation...

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 13 / 16

Theorem (C.)

Given a map ϕ : vi → wi1 . . . wiri, wij ∈ Γe from G(∆) to G(Γ) such that wij = wkl , if (i, j) = (kl), there exists n = n(Γ) ∈ N such that if ϕ is injective in the set of commutators of weight less than or equal to n, then ϕ is injective. Let’s not give up...

Question

Maybe if there exists an embedding from G(∆) to G(Γ), there exists one satisfying conditions of the theorem? No, the theorem is not strong enough for a characterisation...

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 13 / 16

Logic point of view

Faking generators

We define G(ΓN) to be the graph product G(Γ; ZN) with underlying graph Γ and free abelian groups of rank N as vertex groups. Note that G(ΓN) is a pc group.

Theorem (C.)

G(∆) ≡∀ G(Γ) ⇐ ⇒ G(∆) < G(ΓN) and G(Γ) < G(∆N), for some N ∈ N

Corollary

G(∆) is a limit group over G(Γ) if and only if G(∆) < G(ΓN), for some N ∈ N.

Theorem (C.)

There is an algorithm so that given two simplicial graphs ∆ and Γ determines whether or not G(∆) ≡∀ G(Γ).

Weak Extension Graph Question

G(∆) < G(Γ) and G(Γ) < G(∆) ⇐ ⇒

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 14 / 16

Logic point of view

Faking generators

We define G(ΓN) to be the graph product G(Γ; ZN) with underlying graph Γ and free abelian groups of rank N as vertex groups. Note that G(ΓN) is a pc group.

Theorem (C.)

G(∆) ≡∀ G(Γ) ⇐ ⇒ G(∆) < G(ΓN) and G(Γ) < G(∆N), for some N ∈ N

Corollary

G(∆) is a limit group over G(Γ) if and only if G(∆) < G(ΓN), for some N ∈ N.

Theorem (C.)

There is an algorithm so that given two simplicial graphs ∆ and Γ determines whether or not G(∆) ≡∀ G(Γ).

Weak Extension Graph Question

G(∆) < G(Γ) and G(Γ) < G(∆) ⇐ ⇒

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 14 / 16

Logic point of view

Faking generators

We define G(ΓN) to be the graph product G(Γ; ZN) with underlying graph Γ and free abelian groups of rank N as vertex groups. Note that G(ΓN) is a pc group.

Theorem (C.)

G(∆) ≡∀ G(Γ) ⇐ ⇒ G(∆) < G(ΓN) and G(Γ) < G(∆N), for some N ∈ N

Corollary

G(∆) is a limit group over G(Γ) if and only if G(∆) < G(ΓN), for some N ∈ N.

Theorem (C.)

There is an algorithm so that given two simplicial graphs ∆ and Γ determines whether or not G(∆) ≡∀ G(Γ).

Weak Extension Graph Question

G(∆) < G(Γ) and G(Γ) < G(∆) ⇐ ⇒

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 14 / 16

Logic point of view

Theorem (C.)

G(∆) ≡∀ G(Γ) ⇐ ⇒ G(∆) < G(ΓN) and G(Γ) < G(∆N), for some N ∈ N

Corollary

G(∆) is a limit group over G(Γ) if and only if G(∆) < G(ΓN), for some N ∈ N.

Theorem (C.)

There is an algorithm so that given two simplicial graphs ∆ and Γ determines whether or not G(∆) ≡∀ G(Γ).

Weak Extension Graph Question

G(∆) < G(Γ) and G(Γ) < G(∆) ⇐ ⇒ ∆ < Γe and Γ < ∆e.

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 14 / 16

Maybe an interesting connection: Quasi-isometries

1

Given an n-free group Fn, n ≥ 2, we have G(Γ) ∼qi Fn, if and only if G(Γ) ≃ Fm, m ≥ 2;

2

G(Γ) ∼qi Zn if and only if G(Γ) ≃ Zn;

3

Theorem (Behrstock-Neumann 2008): G(Γ) ∼qi G(Pn) if and only if G(Γ) ≃ G(Pm), n, m > 2, where Pi is a tree of diameter i;

4

Theorem (Bestvina-Kleiner-Sageev 2008): G(A1) ∼qi G(A2), A1, A2 atomic graphs if and only if A1 ≃ A2; Atomic graphs are connected graphs with no valence one vertices, no cycles of length less than five, and no separating closed vertex stars

5

Theorem (Behrstock-Januszkiewicz-Neumann 2010): G(T1) ∼qi G(T2), T1, T2 n-trees if and only if T1 and T2 are bisimilar.

Question

Let Γ and ∆ be connected graphs. Is it true G(Γ) ∼qi G(∆) if and only if G(Γ) < G(∆) and G(∆) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 15 / 16

Maybe an interesting connection: Quasi-isometries

1

Given an n-free group Fn, n ≥ 2, we have G(Γ) ∼qi Fn, if and only if G(Γ) ≃ Fm, m ≥ 2;

2

G(Γ) ∼qi Zn if and only if G(Γ) ≃ Zn;

3

Theorem (Behrstock-Neumann 2008): G(Γ) ∼qi G(Pn) if and only if G(Γ) ≃ G(Pm), n, m > 2, where Pi is a tree of diameter i;

4

Theorem (Bestvina-Kleiner-Sageev 2008): G(A1) ∼qi G(A2), A1, A2 atomic graphs if and only if A1 ≃ A2; Atomic graphs are connected graphs with no valence one vertices, no cycles of length less than five, and no separating closed vertex stars

5

Theorem (Behrstock-Januszkiewicz-Neumann 2010): G(T1) ∼qi G(T2), T1, T2 n-trees if and only if T1 and T2 are bisimilar.

Question

Let Γ and ∆ be connected graphs. Is it true G(Γ) ∼qi G(∆) if and only if G(Γ) < G(∆) and G(∆) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 15 / 16

Maybe an interesting connection: Quasi-isometries

1

Given an n-free group Fn, n ≥ 2, we have G(Γ) ∼qi Fn, if and only if G(Γ) ≃ Fm, m ≥ 2;

2

G(Γ) ∼qi Zn if and only if G(Γ) ≃ Zn;

3

Theorem (Behrstock-Neumann 2008): G(Γ) ∼qi G(Pn) if and only if G(Γ) ≃ G(Pm), n, m > 2, where Pi is a tree of diameter i;

4

Theorem (Bestvina-Kleiner-Sageev 2008): G(A1) ∼qi G(A2), A1, A2 atomic graphs if and only if A1 ≃ A2; Atomic graphs are connected graphs with no valence one vertices, no cycles of length less than five, and no separating closed vertex stars

5

Theorem (Behrstock-Januszkiewicz-Neumann 2010): G(T1) ∼qi G(T2), T1, T2 n-trees if and only if T1 and T2 are bisimilar.

Question

Let Γ and ∆ be connected graphs. Is it true G(Γ) ∼qi G(∆) if and only if G(Γ) < G(∆) and G(∆) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 15 / 16

Maybe an interesting connection: Quasi-isometries

1

Given an n-free group Fn, n ≥ 2, we have G(Γ) ∼qi Fn, if and only if G(Γ) ≃ Fm, m ≥ 2;

2

G(Γ) ∼qi Zn if and only if G(Γ) ≃ Zn;

3

Theorem (Behrstock-Neumann 2008): G(Γ) ∼qi G(Pn) if and only if G(Γ) ≃ G(Pm), n, m > 2, where Pi is a tree of diameter i;

4

Theorem (Bestvina-Kleiner-Sageev 2008): G(A1) ∼qi G(A2), A1, A2 atomic graphs if and only if A1 ≃ A2; Atomic graphs are connected graphs with no valence one vertices, no cycles of length less than five, and no separating closed vertex stars

5

Theorem (Behrstock-Januszkiewicz-Neumann 2010): G(T1) ∼qi G(T2), T1, T2 n-trees if and only if T1 and T2 are bisimilar.

Question

Let Γ and ∆ be connected graphs. Is it true G(Γ) ∼qi G(∆) if and only if G(Γ) < G(∆) and G(∆) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 15 / 16

Maybe an interesting connection: Quasi-isometries

1

Given an n-free group Fn, n ≥ 2, we have G(Γ) ∼qi Fn, if and only if G(Γ) ≃ Fm, m ≥ 2;

2

G(Γ) ∼qi Zn if and only if G(Γ) ≃ Zn;

3

Theorem (Behrstock-Neumann 2008): G(Γ) ∼qi G(Pn) if and only if G(Γ) ≃ G(Pm), n, m > 2, where Pi is a tree of diameter i;

4

Theorem (Bestvina-Kleiner-Sageev 2008): G(A1) ∼qi G(A2), A1, A2 atomic graphs if and only if A1 ≃ A2; Atomic graphs are connected graphs with no valence one vertices, no cycles of length less than five, and no separating closed vertex stars

5

Theorem (Behrstock-Januszkiewicz-Neumann 2010): G(T1) ∼qi G(T2), T1, T2 n-trees if and only if T1 and T2 are bisimilar.

Question

Let Γ and ∆ be connected graphs. Is it true G(Γ) ∼qi G(∆) if and only if G(Γ) < G(∆) and G(∆) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 15 / 16

Maybe an interesting connection: Quasi-isometries

1

Given an n-free group Fn, n ≥ 2, we have G(Γ) ∼qi Fn, if and only if G(Γ) ≃ Fm, m ≥ 2;

2

G(Γ) ∼qi Zn if and only if G(Γ) ≃ Zn;

3

Theorem (Behrstock-Neumann 2008): G(Γ) ∼qi G(Pn) if and only if G(Γ) ≃ G(Pm), n, m > 2, where Pi is a tree of diameter i;

4

Theorem (Bestvina-Kleiner-Sageev 2008): G(A1) ∼qi G(A2), A1, A2 atomic graphs if and only if A1 ≃ A2; Atomic graphs are connected graphs with no valence one vertices, no cycles of length less than five, and no separating closed vertex stars

5

Theorem (Behrstock-Januszkiewicz-Neumann 2010): G(T1) ∼qi G(T2), T1, T2 n-trees if and only if T1 and T2 are bisimilar.

Question

Let Γ and ∆ be connected graphs. Is it true G(Γ) ∼qi G(∆) if and only if G(Γ) < G(∆) and G(∆) < G(Γ)?

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 15 / 16

THANK YOU!

Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 16 / 16