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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) = {v1, v2, · · · , vn}: 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) = ⟨ v1, v2, . . . , vn | [vi, vj] = 1 if {vi, vj} ̸∈ E(Γ) ⟩. G(Γ1) ∼ = G(Γ2) if and only if Γ1 ∼ = Γ2.

Takuya Katayama Obstruction to embedding RAAGs

Example

G( ) ∼ = Z3 G( ) ∼ = Z × F2 G( ) ∼ = Z2 ∗ Z G( ) ∼ = F3 Note: G(Γ) = ⟨ v1, v2, . . . , vn | [vi, vj] = 1 if {vi, vj} ̸∈ 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

G( ) ∼ = F3 ֒ → F2 ∼ = G( ).

Proposition (cf. Charney-Vogtmann, 2009)

K c

n : the edgeless graph on n vertices

Γ: a finite graph Then (Zn ∼ =)G(K c

n ) ֒

→ G(Γ) if and only if K c

n ≤ Γ.

In the case where Γ = K c

m, the above theorem just says “Zn ֒

→ Zm if 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. Pn: the path graph consisting of n vertices

Pn

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

• ¬(Z2 ∗ Z ֒

→ F2 × F2 × · · · × F2). Note: G(P3) ∼ = Z2 ∗ Z, G(P2 ⊔ · · · ⊔ P2) ∼ = F2 × · · · × F2. Proof) Suppose to the contrary that Z2 ∗ Z ֒ → F2 × F2 × · · · × F2. Then since P3 is a linear forest, Theorem A(1) implies P3 ≤ P2 ⊔ P2 ⊔ · · · ⊔ P2, a contradiction. Q.E.D.

Takuya Katayama Obstruction to embedding RAAGs

Appl of Thm A(1) (cont’d).

• ¬(G(Λ1) ֒

→ G(Λ2)). Proof) Suppose to the contrary that G(Λ1) ֒ → G(Λ2). Then since P1 ⊔ P4 ≤ Λ1, we have G(P1 ⊔ P4) ֒ → G(Λ1). Hence, G(P1 ⊔ P4) ֒ → G(Λ2). This together with Theorem A(1) implies P1 ⊔ P4 ≤ Λ2, 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 If G(Λ) ֒ → M(Σg,n), then Λ ≤ Cc(Σg,n). This is a partial converse of the following embedding theorem.

Theorem (Koberda, 2012)

Λ: a finite graph If Λ ≤ Cc(Σg,n), then G(Λ) ֒ → M(Σg,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 ≤ Γe, where Γe is a graph such that

• V (Γe) = {g −1ug ∈ G(Γ) | u ∈ V (Γ), g ∈ G(Γ)}.
• ug and v h span an edge ⇔ ug 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 ≤ Γe. Step 2. Prove that Λ ≤ Γe implies Λ ≤ Γ.

Takuya Katayama Obstruction to embedding RAAGs

Step 2. Prove that Λ ≤ Γe implies Λ ≤ Γ. Use the “finiteness” of Γe.

Theorem (Kim-Koberda, 2013)

If Λ ≤ Γe, then there exists a sequence of consecutive “co-doubles” Γ = Γ0 ≤ Γ1 ≤ Γ2 ≤ · · · ≤ Γn ≤ Γe such that Γi = D(Γi−1) and Λ ≤ Γn. 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 ≤ Γe implies Λ ≤ Γ (cont’d). Use the “finiteness” of Γe.

Theorem (Kim-Koberda, 2013)

If Λ ≤ Γe, then there exists a sequence of consecutive “co-doubles” Γ = Γ0 ≤ Γ1 ≤ Γ2 ≤ · · · ≤ Γn ≤ Γe such that Γi = D(Γi−1) and Λ ≤ Γn.

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 Γc is the graph consisting of

• V (Γc) = V (Γ) and
• E(Γc) = {{u, v} | u, v ∈ V (Γ), {u, v} /

∈ E(Γ)}.

=

P5 P5 P5

c

St(v, Γ): the full subgraph induced by v and the vertices adjacent to v. Dv(Γ): the double of Γ along the full subgraph St(v, Γ), namely, Dv(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 ≤ Dv(Γ), then Λ ≤ Γ.

Example

Takuya Katayama Obstruction to embedding RAAGs

RAAGs in mapping class groups—future work—

Σg,n: the orientable compact surface of genus g with n punctures We assume χ(Σg,n) < 0. The mapping class group of Σg,n is defined as follows. M(Σg,n) := π0(Homeo+(Σg,n)) The complement graph of the curve graph Cc(Σg,n) is a graph such that

• V (Cc(Σg,n)) = {isotopy classes of esls on Σg,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 ≤ Cc(Σg,n), G(Λ) ֒ → M(Σg,n).

Theorem (K.)

Λ: a linear forest If G(Λ) ֒ → M(Σg,n), then Λ ≤ Cc(Σg,n).

Takuya Katayama Obstruction to embedding RAAGs

Theorem (Koberda + K.)

Λ: a linear forest Then G(Λ) ֒ → M(Σg,n) if and only if Λ ≤ Cc(Σg,n). We can regard the above theorem as a generalization of the following classical result.

Theorem (Birman-Lubotzky-McCarthy, 1983)

Zn ֒ → M(Σg,n) if and only if n does not exceed the number of simple closed curves needed in the pants-decomposition of Σg,n (= 3g + n − 3).

Takuya Katayama Obstruction to embedding RAAGs

Theorem (Koberda + K.)

Λ: a linear forest Then G(Λ) ֒ → M(Σg,n) if and only if Λ ≤ Cc(Σg,n).

Theorem (BLM in our terminology)

Λ: the disjoint union of finitely many copies of P1 Then G(Λ) ֒ → M(Σg,n) if and only if Λ ≤ Cc(Σg,n).

Takuya Katayama Obstruction to embedding RAAGs

Bering IV, Conant and Gaster proved that P2 ⊔ P2 ⊔ · · · ⊔ P2 ≤ Cc(Σg,n) if and only if the number of the copies

• f P2 is at most g + ⌊ g+n

2 ⌋ − 1 in this September...

Proposition

F2 × F2 × · · · × F2 ֒ → M(Σg,n) if and only if the number of the direct factors F2 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 F2 × F2 × F2, the simplest surface(s) are Σ2,2, Σ3,0...

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

• S. Kim and T. Koberda, ‘Embedability between right-angled Artin

groups’, Geom. Topol. 17 (2013) 493–530.

• T. Koberda, ‘Right-angled Artin groups and a generalized

isomorphism problem for finitely generated subgroups of mapping class groups’, Geom. Funct. Anal. 22 (2012) 1541–1590. This talk is based on:

• T. Katayama, ‘Right-angled Artin groups and full subgraphs of

graphs’, preprint, available at arXiv:1612.01732.

Takuya Katayama Obstruction to embedding RAAGs

Thank you very much for your attention!

Takuya Katayama Obstruction to embedding RAAGs