An obstruction to the existence of embeddings between right-angled Artin groups

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

, v

, · · · , v

} : 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

, v

, . . . , v

| [v

, v

] = 1 if { v

, v

} ̸∈ E (Γ) ⟩ .

G(Γ

) ∼ = G(Γ

) if and only if Γ

∼ = Γ

.

Example

G( ) ∼ = Z

G( ) ∼ = Z × F

G( ) ∼ = Z

∗ Z G( ) ∼ = F

Note: G (Γ) = ⟨ v

, v

, . . . , v

| [v

, v

] = 1 if { v

, v

} ̸∈ 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).

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( ) ∼ = F

, → F

∼ = G ( ).

Proposition (cf. Charney-Vogtmann, 2009)

K

: the edgeless graph on n vertices Γ: a finite graph

Then ( Z

∼ =)G (K

) , → G (Γ) if and only if K

≤ Γ.

In the case where Γ = K

, the above theorem just says “ Z

, → Z

if

and only if n ≤ m”.

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

: the path graph consisting of n vertices ^P

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

∗ Z , → F

× F

× · · · × F

).

Note: G (P

) ∼ = Z

∗ Z , G (P

⊔ · · · ⊔ P

) ∼ = F

× · · · × F

.

Proof) Suppose to the contrary that Z

∗ Z , → F

× F

× · · · × F

. Then since P

is a linear forest, Theorem A(1) implies

P

≤ P

⊔ P

⊔ · · · ⊔ P

, a contradiction. Q.E.D.

Appl of Thm A(1) (cont’d).

• ¬ (G (Λ

) , → G (Λ

)).

Proof) Suppose to the contrary that G (Λ

) , → G (Λ

).

Then since P

⊔ P

≤ Λ

, we have G(P

⊔ P

) , → G (Λ

).

Hence, G (P

⊔ P

) , → G (Λ

).

This together with Theorem A(1) implies P

⊔ P

≤ Λ

, 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 (Σ

), then Λ ≤ C

(Σ

).

This is a partial converse of the following embedding theorem.

Theorem (Koberda, 2012)

Λ: a finite graph

If Λ ≤ C

(Σ

), then G (Λ) , → M (Σ

)

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

, where Γ

is a graph such that

• V (Γ

) = { g

ug ∈ G (Γ) | u ∈ V (Γ), g ∈ G (Γ) } .

• u

and v

span an edge ⇔ u

and v

are not commutative.

Theorem (Casals-Ruiz, 2015)

For a forest Λ and a finite graph Γ, if G (Λ) , → G (Γ), then Λ ≤ Γ

. Step 2. Prove that Λ ≤ Γ

implies Λ ≤ Γ.

Takuya Katayama Obstruction to embedding RAAGs

Step 2. Prove that Λ ≤ Γ

implies Λ ≤ Γ.

Use the “finiteness” of Γ

.

Theorem (Kim-Koberda, 2013)

If Λ ≤ Γ

, then there exists a sequence of consecutive “co-doubles”

Γ = Γ

≤ Γ

≤ Γ

≤ · · · ≤ Γ

≤ Γ

such that Γ

= D(Γ

) and Λ ≤ Γ

.

Here, for a finite graph ∆,

D(∆) := (D(∆

))

.

The operation c: “taking the complement graph”

Step 2. Prove that Λ ≤ Γ

implies Λ ≤ Γ (cont’d).

Use the “finiteness” of Γ

.

Theorem (Kim-Koberda, 2013)

If Λ ≤ Γ

, then there exists a sequence of consecutive “co-doubles”

Γ = Γ

≤ Γ

≤ Γ

≤ · · · ≤ Γ

≤ Γ

such that Γ

= D(Γ

) and Λ ≤ Γ

.

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

is the graph consisting of

• V (Γ

) = V (Γ) and

• E (Γ

) = {{ u, v } | u, v ∈ V (Γ), { u, v } ∈ / E (Γ) } .

= P

P

P

c

St(v , Γ): the full subgraph induced by v and the vertices adjacent to v .

D

(Γ): the double of Γ along the full subgraph St(v , Γ),

namely, D

(Γ) is obtained by taking two copies of Γ and gluing them along copies of St(v , Γ).

=

Proposition

Λ: the complement graph of a linear forest, Γ: a finite graph If Λ ≤ D

(Γ), then Λ ≤ Γ.

Example

Takuya Katayama Obstruction to embedding RAAGs

RAAGs in mapping class groups—future work—

Σ

: the orientable compact surface of genus g with n punctures We assume χ(Σ

) < 0.

The mapping class group of Σ

is defined as follows.

M (Σ

) := π

(Homeo

(Σ

))

The complement graph of the curve graph C

(Σ

) is a graph such that

• V ( C

(Σ

)) = { isotopy classes of esls on Σ

}

• esls α, β span an edge iff α, β CANNOT be realized disjointly.

Theorem (Koberda, 2012)

Λ: a finite graph

If Λ ≤ C

(Σ

), G (Λ) , → M (Σ

).

Theorem (K.)

Λ: a linear forest

If G(Λ) , → M (Σ

), then Λ ≤ C

(Σ

).

Takuya Katayama Obstruction to embedding RAAGs

Theorem (Koberda + K.)

Λ: a linear forest

Then G (Λ) , → M (Σ

) if and only if Λ ≤ C

(Σ

).

We can regard the above theorem as a generalization of the following classical result.

Theorem (Birman-Lubotzky-McCarthy, 1983)

Z

, → M (Σ

) if and only if n does not exceed the number of

simple closed curves needed in the pants-decomposition of Σ

(= 3g + n − 3).

Theorem (Koberda + K.)

Λ: a linear forest

Then G (Λ) , → M (Σ

) if and only if Λ ≤ C

(Σ

).

Theorem (BLM in our terminology)

Λ: the disjoint union of finitely many copies of P

Then G (Λ) , → M (Σ

) if and only if Λ ≤ C

(Σ

).

Takuya Katayama Obstruction to embedding RAAGs

Bering IV, Conant and Gaster proved that

P

⊔ P

⊔ · · · ⊔ P

≤ C

(Σ

) if and only if the number of the copies of P

is at most g + ⌊

⌋ − 1 in this September...

Proposition

F

× F

× · · · × F

, → M (Σ

) if and only if the number of the direct factors F

is at most g + ⌊

⌋ − 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

× F

× F

, the simplest surface(s) are Σ

, Σ

...

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.

Thank you very much for your attention!

Takuya Katayama Obstruction to embedding RAAGs