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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,

  1. 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

  2. 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) | [ v 1 , v 2 ] = 1 , whenever ( v 1 , v 2 ) ∈ 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

  3. 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) | [ v 1 , v 2 ] = 1 , whenever ( v 1 , v 2 ) ∈ 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

  4. 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) | [ v 1 , v 2 ] = 1 , whenever ( v 1 , v 2 ) ∈ 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

  5. 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) | [ v 1 , v 2 ] = 1 , whenever ( v 1 , v 2 ) ∈ 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

  6. Examples Z 2 =!<!a,b!|![a,b]!>! a! b! a! b! ! ! ! !>! a! b! a! b! a! b! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16 a! d! ,d]!>! b! c! c! d! a! d! a! b! ,d]!>! c! d! c! b! b! d! a! c! a! b! e,a]>! e! e! d! c! d! b! a! b! c! d! e! ]>! a! e! c!

  7. Examples Z 2 =!<!a,b!|![a,b]!>! a! b! a! b! ! ! ! ! ! ! F 2 =!<!a,b|! ø !>! !>! a! a! b! b! a! a! b! b! ! ! a! b! a! b! a! d! a! d! ,d]!>! Montserrat Casals-Ruiz (Oxford) Embeddability ,d]!>! GAGTA 2013 May 29, 2013 3 / 16 b! c! b! c! c! c! d! d! a! d! a! d! a! b! a! b! ,d]!>! ,d]!>! c! d! c! d! c! b! c! b! d! b! d! b! a! c! a! b! a! c! a! b! e,a]>! e,a]>! c! e! e! d! e! e! d! c! d! b! d! b! a! b! c! d! e! b! ]>! a! c! d! e! ]>! a! e! a! e! c! c!

  8. Examples a! b! a! d! F 2 ! x! F 2 =!<!a,b,c,d!|[a,b],[a,c],[b,d],![c,d]!>! b! c! c! d! a! d! a! b! ,d]!>! c! d! c! b! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16 d! b! a! c! a! b! e,a]>! c! e! e! d! d! b! a! b! c! d! e! ]>! a! e! c!

  9. Examples a! b! a! d! F 2 ! x! F 2 =!<!a,b,c,d!|[a,b],[a,c],[b,d],![c,d]!>! b! c! c! d! c! d! a! d! a! b! a! d! a! b! ,d]!>! Z 2! *!Z 2 =!<!a,b,c,d!|![a,b][c,d]!>! c! d! c! d! c! b! c! b! d! b! b! d! a! c! a! b! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16 e,a]>! a! b! a! c! e,a]>! c! e! e! d! e! e! d! c! d! b! d! b! a! b! c! d! e! ]>! a! b! c! d! e! ]>! a! e! a! e! c! c!

  10. Examples c! b! d! b! a! b! a! c! <!a,b,c,d,e!|![a,b],![b,c],![c,d],![d,e],![e,a]>! e! e! d! c! d! b! a! b! c! d! e! ]>! a! e! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16 c!

  11. Examples c! b! d! b! a! c! a! b! <!a,b,c,d,e!|![a,b],![b,c],![c,d],![d,e],![e,a]>! e! e! d! c! d! b! d! b! a! b! c! d! e! <!a,b,c,d,e!|![a,b],![b,c],![c,d],![d,e]>! a! b! c! d! e! ]>! a! e! a! e! c! c! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 3 / 16

  12. 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

  13. 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

  14. 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

  15. 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

  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

  17. Embeddability between pc groups Question Can we characterise when G (∆) < G (Γ) ? Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 5 / 16

  18. Warm-up examples a! b! c! d! a! b! c! d! e! ! ! ! ! a! b! c! d! a! b! c! d! e! b d! a d! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16 c! b! b! a c! a! c! a! d! e c! f! e! e! d!

  19. Warm-up examples a! b! c! d! a! b! c! d! e! ! ! ! ! ! ! a! b! c! d! e! a! b! c! d! ! a! b! c! d! a! b! c! d! e! b d! ! a d! ! b d! a d! ! ! c! b! b! a c! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16 c! a! b! c! b! a! d! a c! a! c! e c! a! d! f! e! d! e! e c! f! e! e! d!

  20. Warm-up examples Paths(Kim-Koberda 2011) If F is a forest, then G ( F ) < G ( P 3 ) Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

  21. Warm-up examples ! ! c! b! b! ! a c! a! c! ! a! d! ! e c! f! e! d! ! e! ! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

  22. Warm-up examples ! ! (" $ )! "! ! $! (" $ )! #! &! #! "! %! " $! "! $! (" $ )! %! & $! '! &! &! %! Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

  23. Warm-up examples Cycles (Kim-Koberda 2011) If C n is the cycle with n vertices, n ≥ 5, then G ( C n ) < G ( C 5 ) . Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 6 / 16

  24. Extension Graph Conjecture Definition Let Γ be a simplicial graph, V (Γ) = { a 1 , . . . , a k } . We define the extension graph Γ e to be the graph whose set of vertices V (Γ e ) are labelled by elements a w i , w ∈ G (Γ) and 1 w j the set of edges E (Γ e ) are pairs of different vertices ( a w i i , a j ) such that 2 w j [ a w i i , a j ] = 1 in G (Γ) . Montserrat Casals-Ruiz (Oxford) Embeddability GAGTA 2013 May 29, 2013 7 / 16

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