An obstruction to the existence of embeddings between right-angled - PowerPoint PPT Presentation

An obstruction to the existence of embeddings between right-angled Artin groups Takuya Katayama Hiroshima University Nihon University, December 21, 2016 Takuya Katayama Obstruction to embedding RAAGs

Right-angled Artin groups Γ: a finite (simplicial) graph V (Γ) = { v 1 , v 2 , · · · , v n } : the vertex set of Γ E (Γ): the edge set of Γ Definition The right-angled Artin group (RAAG) G (Γ) on Γ is the group given by the following presentation: G (Γ) = ⟨ v 1 , v 2 , . . . , v n | [ v i , v j ] = 1 if { v i , v j } ̸∈ E (Γ) ⟩ . G (Γ 1 ) ∼ = G (Γ 2 ) if and only if Γ 1 ∼ = Γ 2 . Takuya Katayama Obstruction to embedding RAAGs

Example ) ∼ = Z 3 G ( ) ∼ G ( = Z × F 2 = Z 2 ∗ Z ) ∼ G ( ) ∼ G ( = F 3 Note: G (Γ) = ⟨ v 1 , v 2 , . . . , v n | [ v i , v j ] = 1 if { v i , v j } ̸∈ E (Γ) ⟩ Takuya Katayama Obstruction to embedding RAAGs

Motivation and main results Problem (Crisp-Sageev-Sapir, 2008) For given two finite graphs Λ and Γ, decide whether G (Λ) can be embedded into G (Γ). The following is standard. Proposition Λ , Γ: finite graphs If Λ ≤ Γ, then G (Λ) ֒ → G (Γ). Takuya Katayama Obstruction to embedding RAAGs

Proposition Λ , Γ: finite graphs If Λ ≤ Γ, then G (Λ) ֒ → G (Γ). A subgraph Λ of a graph Γ is said to be full if E (Λ) contains every e ∈ E (Γ) whose end points both lie in V (Λ). We denote by Λ ≤ Γ if Λ is a full subgraph of Γ. Takuya Katayama Obstruction to embedding RAAGs

In general, the converse implication “ G (Λ) ֒ → G (Γ)” ⇒ “Λ ≤ Γ” is false. Example ) ∼ → F 2 ∼ G ( = F 3 ֒ = G ( ). Proposition (cf. Charney-Vogtmann, 2009) K c n : the edgeless graph on n vertices Γ: a finite graph Then ( Z n ∼ =) G ( K c → G (Γ) if and only if K c n ) ֒ n ≤ Γ. m , the above theorem just says “ Z n ֒ → Z m if In the case where Γ = K c and only if n ≤ m ”. Takuya Katayama Obstruction to embedding RAAGs

Question Which finite graph Λ satisfies the following property: for any finite graph Γ, “ G (Λ) ֒ → G (Γ)” ⇒ “Λ ≤ Γ”? The following gives a complete answer to the above question. Theorem A (K.) Let Λ be a finite graph. (1) If Λ is a linear forest, then Λ has the above property, i.e., ∀ Γ, if G (Λ) ֒ → G (Γ), then Λ ≤ Γ. (2) If Λ is not a linear forest, then Λ does not have the above property, i.e., ∃ Γ such that G (Λ) ֒ → G (Γ), though Λ ̸≤ Γ. A finite graph Λ is said to be a linear forest if each connected component of Λ is a path graph. P n P n : the path graph consisting of n vertices Takuya Katayama Obstruction to embedding RAAGs

Theorem A(1) Suppose that Λ is a linear forest. Then ∀ Γ, G (Λ) ֒ → G (Γ) implies Λ ≤ Γ. Application of Thm A(1) to concrete embedding problems • ¬ ( Z 2 ∗ Z ֒ → F 2 × F 2 × · · · × F 2 ). = Z 2 ∗ Z , G ( P 2 ⊔ · · · ⊔ P 2 ) ∼ Note: G ( P 3 ) ∼ = F 2 × · · · × F 2 . Proof) Suppose to the contrary that Z 2 ∗ Z ֒ → F 2 × F 2 × · · · × F 2 . Then since P 3 is a linear forest, Theorem A(1) implies P 3 ≤ P 2 ⊔ P 2 ⊔ · · · ⊔ P 2 , a contradiction. Q.E.D. Takuya Katayama Obstruction to embedding RAAGs

Appl of Thm A(1) (cont’d). • ¬ ( G (Λ 1 ) ֒ → G (Λ 2 )). Proof) Suppose to the contrary that G (Λ 1 ) ֒ → G (Λ 2 ). Then since P 1 ⊔ P 4 ≤ Λ 1 , we have G ( P 1 ⊔ P 4 ) ֒ → G (Λ 1 ). Hence, G ( P 1 ⊔ P 4 ) ֒ → G (Λ 2 ). This together with Theorem A(1) implies P 1 ⊔ P 4 ≤ Λ 2 , which is impossible. Q.E.D. Theorem A(1) is sometimes valid to find that the RAAG, on a graph which is not a linear forest, cannot embed into another RAAG. Takuya Katayama Obstruction to embedding RAAGs

Moreover, we obtain the following as a consequence of Theorem A(1). Theorem Λ: a linear forest → M (Σ g , n ), then Λ ≤ C c (Σ g , n ). If G (Λ) ֒ This is a partial converse of the following embedding theorem. Theorem (Koberda, 2012) Λ: a finite graph If Λ ≤ C c (Σ g , n ), then G (Λ) ֒ → M (Σ g , n ) Takuya Katayama Obstruction to embedding RAAGs

Proof of Theorem A(1) Theorem A(1) Λ: a linear forest Γ: a finite graph If G (Λ) ֒ → G (Γ), then Λ ≤ Γ. Sketch of proof. Step 1. Prove Λ ≤ Γ e , where Γ e is a graph such that • V (Γ e ) = { g − 1 ug ∈ G (Γ) | u ∈ V (Γ) , g ∈ G (Γ) } . • u g and v h span an edge ⇔ u g and v h are not commutative. Theorem (Casals-Ruiz, 2015) For a forest Λ and a finite graph Γ, if G (Λ) ֒ → G (Γ), then Λ ≤ Γ e . Step 2. Prove that Λ ≤ Γ e implies Λ ≤ Γ. Takuya Katayama Obstruction to embedding RAAGs

Step 2. Prove that Λ ≤ Γ e implies Λ ≤ Γ. Use the “finiteness” of Γ e . Theorem (Kim-Koberda, 2013) If Λ ≤ Γ e , then there exists a sequence of consecutive “co-doubles” Γ = Γ 0 ≤ Γ 1 ≤ Γ 2 ≤ · · · ≤ Γ n ≤ Γ e such that Γ i = D (Γ i − 1 ) and Λ ≤ Γ n . Here, for a finite graph ∆, D (∆) := ( D (∆ c )) c . The operation c : “taking the complement graph” The operation D : “taking a double graph” Takuya Katayama Obstruction to embedding RAAGs

Step 2. Prove that Λ ≤ Γ e implies Λ ≤ Γ (cont’d). Use the “finiteness” of Γ e . Theorem (Kim-Koberda, 2013) If Λ ≤ Γ e , then there exists a sequence of consecutive “co-doubles” Γ = Γ 0 ≤ Γ 1 ≤ Γ 2 ≤ · · · ≤ Γ n ≤ Γ e such that Γ i = D (Γ i − 1 ) and Λ ≤ Γ n . Proposition (K.) Λ: a linear forest ∆: a finite graph If Λ ≤ D (∆), then Λ ≤ ∆. This completes the proof. Takuya Katayama Obstruction to embedding RAAGs

For a graph Γ, the complement graph Γ c is the graph consisting of • V (Γ c ) = V (Γ) and • E (Γ c ) = {{ u , v } | u , v ∈ V (Γ) , { u , v } / ∈ E (Γ) } . = c P 5 P 5 P 5 St ( v , Γ): the full subgraph induced by v and the vertices adjacent to v . D v (Γ): the double of Γ along the full subgraph St ( v , Γ), namely, D v (Γ) is obtained by taking two copies of Γ and gluing them along copies of St ( v , Γ). = Takuya Katayama Obstruction to embedding RAAGs

Proposition Λ: the complement graph of a linear forest, Γ: a finite graph If Λ ≤ D v (Γ), then Λ ≤ Γ. Example Takuya Katayama Obstruction to embedding RAAGs

RAAGs in mapping class groups—future work— Σ g , n : the orientable compact surface of genus g with n punctures We assume χ (Σ g , n ) < 0. The mapping class group of Σ g , n is defined as follows. M (Σ g , n ) := π 0 ( Homeo + (Σ g , n )) The complement graph of the curve graph C c (Σ g , n ) is a graph such that • V ( C c (Σ g , n )) = { isotopy classes of esls on Σ g , n } • esls α, β span an edge iff α, β CANNOT be realized disjointly. Takuya Katayama Obstruction to embedding RAAGs

Theorem (Koberda, 2012) Λ: a finite graph If Λ ≤ C c (Σ g , n ), G (Λ) ֒ → M (Σ g , n ). Theorem (K.) Λ: a linear forest → M (Σ g , n ), then Λ ≤ C c (Σ g , n ). If G (Λ) ֒ Takuya Katayama Obstruction to embedding RAAGs

Theorem (Koberda + K.) Λ: a linear forest Then G (Λ) ֒ → M (Σ g , n ) if and only if Λ ≤ C c (Σ g , n ). We can regard the above theorem as a generalization of the following classical result. Theorem (Birman-Lubotzky-McCarthy, 1983) Z n ֒ → M (Σ g , n ) if and only if n does not exceed the number of simple closed curves needed in the pants-decomposition of Σ g , n (= 3 g + n − 3). Takuya Katayama Obstruction to embedding RAAGs

Theorem (Koberda + K.) Λ: a linear forest → M (Σ g , n ) if and only if Λ ≤ C c (Σ g , n ). Then G (Λ) ֒ Theorem (BLM in our terminology) Λ: the disjoint union of finitely many copies of P 1 → M (Σ g , n ) if and only if Λ ≤ C c (Σ g , n ). Then G (Λ) ֒ Takuya Katayama Obstruction to embedding RAAGs

Bering IV, Conant and Gaster proved that P 2 ⊔ P 2 ⊔ · · · ⊔ P 2 ≤ C c (Σ g , n ) if and only if the number of the copies of P 2 is at most g + ⌊ g + n 2 ⌋ − 1 in this September... Proposition F 2 × F 2 × · · · × F 2 ֒ → M (Σ g , n ) if and only if the number of the direct factors F 2 is at most g + ⌊ g + n 2 ⌋ − 1. Question (Kim-Koberda, 2014) Given a right-angled Artin group, what is the simplest surface for which there is an embedding of the right-angled Artin group into the mapping class group? e.g. for F 2 × F 2 × F 2 , the simplest surface(s) are Σ 2 , 2 , Σ 3 , 0 ... Takuya Katayama Obstruction to embedding RAAGs

References • E. Bering IV, G. Conant, J. Gaster, ‘On the complexity of finite subgraphs of the curve graph’, preprint (2016), available at arXiv:1609.02548. • J. Birman, A. Lubotzky and J. McCarthy, ‘Abelian and solvable subgroups of the mapping class groups’, Duke Math. J. 50 (1983) 1107–1120. • M. Casals-Ruiz, ‘Embeddability and universal equivalence of partially commutative groups’, Int. Math. Res. Not. (2015) 13575–13622. • R. Charney and K. Vogtmann, ‘Finiteness properties of automorphism groups of right-angled Artin groups’, Bull. Lond. Math. Soc. 41 (2009) 94–102. • J. Crisp, M. Sageev and M. Sapir, ‘Surface subgroups of right-angled Artin groups’, Internat. J. Algebra Comput. 18 (2008) 443–491. Takuya Katayama Obstruction to embedding RAAGs