Embeddability between the right-angled Artin groups of surfaces - PowerPoint PPT Presentation

Embeddability between the right-angled Artin groups of surfaces Takuya Katayama Hiroshima University (JSPS Research Fellow PD) Mathematics of Knots II Nihon University, December 19, 2019 Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 1 / 23

Table of contents I. Introduction II. Main Theorem Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 2 / 23

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) A (Γ) on Γ is the group given by the following presentation: A (Γ) = ⟨ v 1 , v 2 , . . . , v n | [ v i , v j ] = 1 if { v i , v j } ∈ E (Γ) ⟩ . A (Γ 1 ) ∼ = A (Γ 2 ) if and only if Γ 1 ∼ = Γ 2 . ) ∼ A ( = F 3 ) ∼ = Z ∗ Z 2 A ( ) ∼ A ( = Z × F 2 ) ∼ = Z 3 A ( Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 3 / 23

Previous studies and motivation Problem (Crisp-Sageev-Sapir, 2006) For given two finite graphs Λ and Γ, decide whether A (Λ) can be embedded into A (Γ). Graph theory ⇝ Embeddability between RAAGs Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 4 / 23

Full graph embedding of the defining graph A graph embedding ι : Λ ֒ → Γ is said to be full if { u , v } ∈ E (Λ) ⇔ { u , v } ∈ E (Γ) for all u , v ∈ V (Λ). We denote by Λ ≤ Γ if Λ has a full graph embedding into Γ. Λ is called full subgraph of Γ. Theorem (van der Lek, 1983) Λ , Γ: finite graphs If Λ ≤ Γ, then A (Λ) ֒ → A (Γ). ι Natural map v �→ ι ( v ) extends to a homomorphism, and is injective (non-trivial). Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 5 / 23

Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 6 / 23

Extension graph The extension graph Γ e of a finite graph Γ is a graph that V (Γ e ) = { v g | v ∈ V (Γ) , g ∈ A (Γ) } and E (Γ e ) = {{ u h , v g } | [ u h , v g ] = 1 in A (Γ) } . v g := g − 1 vg ⇒ P e E.g. P 4 = 4 is a locally infinite tree. valence ∞ valence 1 Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 7 / 23

valence ∞ valence 1 Note: If [ u , v ] = uvu − 1 v − 1 = 1, then [ u w , v w ] = ( w − 1 uw )( w − 1 vw )( w − 1 u − 1 w )( w − 1 v − 1 w ) = w − 1 uvu − 1 v − 1 w = 1. Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 8 / 23

Theorem (Kim–Koberda, 2013) Λ , Γ: finite graphs If Λ ≤ Γ e , then A (Λ) ֒ → A (Γ). Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 9 / 23

The existence of embeddings not derived from the extension graph Theorem (Kim–Koberda, 2013) C 5 ̸≤ ( P c 8 ) e . Theorem (Casals-Ruiz, Duncan, Kazachkov, 2015) → A ( P c A ( C 5 ) ֒ 8 ). Kim–Koberda (2015) gave infinitely many examples. Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 10 / 23

Some results on non-existence of embeddings Kim–Koberda (2013): Every embedding A (Λ) ֒ → A (Γ) has a “normal form” described by Λ and Γ e . Theorem (Kim–Koberda, 2013) Λ , Γ: finite graphs w/o triangles → A (Γ), then Λ ≤ Γ e . If A (Λ) ֒ Theorem (K., 2018) Λ: the complement graph of a liner forest Γ: a finite graph → A (Γ), then Λ ≤ Γ e . If A (Λ) ֒ Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 11 / 23

Problem (recall) For given two finite graphs Λ and Γ, decide whether A (Λ) can be embedded into A (Γ). Graph theory ⇝ Embeddability between RAAGs How about topology?? Motivation Topology ⇝ ?? Embeddability between RAAGs Theorem (Kim–Koberda, 2013) Λ: a circle Γ: a finite graph → A (Γ), then Γ contains a circle. If A (Λ) ֒ Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 12 / 23

In order to consider manifolds together with graphs, we need the following concept. Definition Λ: a finite graph The flag complex of Λ is a simplicial complex that 1:1 { n -simplex in X Λ } ↔ { complete subgraph on ( n + 1) vertices in Λ } X (1) = Λ . Λ Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 13 / 23

E.g. ) ∼ = Z ∗ Z 2 A ( ) ∼ = Z 3 A ( Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 14 / 23

Main Theorem Main Thereom Λ , Γ: finite graphs w/ A (Λ) ֒ → A (Γ). If X Λ ∼ = S 2 , then X Γ contains a subcomplex ∼ = S 2 . Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 15 / 23

Main Thereom (recall) Λ , Γ: finite graphs w/ A (Λ) ֒ → A (Γ). If X Λ ∼ = S 2 , then X Γ contains a subcomplex ∼ = S 2 . Main Theorem and the fact that a 2-sphere S 2 is not contained in any other surfaces imply the following. Corollary Λ , Γ: finite graphs = S 2 and X Γ ∼ Suppose that X Λ ∼ =“a surface other than S 2 ”. Then A (Λ) ̸ ֒ → A (Γ). Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 16 / 23

Main Thereom (recall) Λ , Γ: finite graphs w/ A (Λ) ֒ → A (Γ). If X Λ ∼ = S 2 , then X Γ contains a subcomplex ∼ = S 2 . We divide the proof into the following two steps. Step A. Λ , Γ: finite graphs w/ A (Λ) ֒ → A (Γ). Assuming X Λ ∼ = S 2 , prove that X Γ e contains a subcomplex ∼ = S 2 . Here Γ e is the extension graph of Γ. Step B. Assuming X Γ e contains a subcomplex ∼ = S 2 , prove that X Γ contains a subcomplex ∼ = S 2 . Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 17 / 23

Step B. Assuming X Γ e contains a subcomplex ∼ = S 2 , prove that X Γ contains a subcomplex ∼ = S 2 . Lemma B Γ: a finite graph If X Γ e contains a subcomplex ∼ = S 2 , then so does X Γ . In order to understand the extension graph well, we use the “double” of graphs. Definition (recall) Let Γ be a graph and v a vertex of Γ. The star subgraph St ( v , Γ) of Γ is a full subgraph induced by { u ∈ V (Γ) | { u , v } ∈ E (Γ) } ∪ { v } . The double D ( v , Γ) of Γ is Γ ∪ St ( v , Γ) Γ. = Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 18 / 23

Lemma B (recall) Γ: a finite graph If X Γ e contains a subcomplex ∼ = S 2 , then so does X Γ . Lemma (Kim–Koberda) Λ , Γ: finite graphs If Λ ⊂ Γ e , there is a sequence of doubles Γ = D 1 ≤ D 2 ≤ . . . ≤ D n such that Λ ⊂ D n and that D i +1 is a double of D i . By the above Lemma, it is enough to prove the following. Lemma B’ Γ: a finite graph, v ∈ V (Γ) If X D ( v , Γ) contains a subcomplex ∼ = S 2 , then so does X Γ . S ⊂ X Γ e w/ S ∼ = S 2 ⇒ X Γ = X D 1 ≤ ∃ X D 2 ≤ . . . ≤ ∃ X D n ; S ⊂ X D n . Lemma (KK) Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 19 / 23

Lemma B’ (recall) Γ: a finite graph, v ∈ V (Γ) If X D ( v , Γ) contains a subcomplex ∼ = S 2 , then so does X Γ . Idea of the proof) Note: X D ( v , Γ) = X Γ ∪ X St ( v , Γ) X Γ . Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 20 / 23

How to find a disk V in the sphere with ∂ V ⊂ X St and V ⊂ X Γ ? Claim Suppose that S 2 ̸⊂ “copies of X Γ ”. Then S 2 ∩ X St contains a circle. Claim V : a disk in S 2 with ∂ V ⊂ X St and V ̸⊂ “copies of X Γ ”. For any edge-path P joining opposite points, ∃ C a circle; C ⊂ V ∩ X St and C ∩ P ̸ = ∅ . Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 21 / 23

Future work Main Thereom (recall) Λ , Γ: finite graphs w/ A (Λ) ֒ → A (Γ). If X Λ has a subcomplex ∼ = S 2 , then X Γ has a subcomplex ∼ = S 2 . Asphericity Question Λ , Γ: finite graphs w/ A (Λ) ֒ → A (Γ). dim ( X Γ ) = 2. Then does π 2 ( X Λ ) ̸ = 1 imply π 2 ( X Γ ) ̸ = 1? Note: For 2-complexes, π 2 = 1 ⇔ asphericity. Whitehead’s Asphericity Conjecture for simp.cpx., 1941 X : an aspherical 2-dim simplicial complex Every connected subcomplex of X is aspherical. Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 22 / 23

Thank you very much for your attention! Takuya Katayama (Hiroshima Univ.) RAAGs of surfaces 23 / 23