Co-contractions of graphs and right-angled Artin groups

Kim, Sang-Hyun

doi:10.2140/agt.2008.8.849

Cited by 25 publications

(36 citation statements)

References 20 publications

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“…Our main technical tool is the disk diagrams introduced in [CW04] (as "dissections of a disk") and further detailed in [Kim08]. We present the general theory here for the convenience of the reader.…”

Section: Disk Diagrams In Raagsmentioning

confidence: 99%

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The geometry of purely loxodromic subgroups of right-angled Artin groups

Koberda

Mangahas

Taylor

2017

Trans. Amer. Math. Soc.

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Abstract. We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder condition for a finitely generated subgroup of A(Γ) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Γ. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups.

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Section: Disk Diagrams In Raagsmentioning

confidence: 99%

“…It is shown in [Kim08] that any disk diagram for w is dual in S 2 to a van Kampen diagram with boundary word w, and vice versa. In particular, given a disk diagram, one obtains a van Kampen diagram by simply taking its dual complex; see Figure 2.…”

Section: Disk Diagrams In Raagsmentioning

confidence: 99%

The geometry of purely loxodromic subgroups of right-angled Artin groups

Koberda

Mangahas

Taylor

2017

Trans. Amer. Math. Soc.

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“…Theorem 1.3(1) also generalises the result of Kim-Koberda [10], which states that the linear forests P c 3 = P 1 P 2 (the symbol means the disjoint union), P c 4 = P 4 and C c 4 = P 2 P 2 have property ( * ). As a consequence of Theorem 1.3(1) and a result of Kim [9], we obtain the following result concerning embeddability between RAAGs on finite graphs whose underlying spaces are connected 1-manifolds. Theorem 1.4.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 66%

“…Several studies have demonstrated that the embeddability of RAAGs can be understood via certain graph theoretical concepts (e.g. [4], [5], [9], [10], [12] and [14]). In fact, theorems due to Kim and Koberda [10] state that the following for any finite graphs Λ and Γ.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Right-angled Artin groups and full subgraphs of graphs

Katayama

2017

J. Knot Theory Ramifications

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For a finite graph Γ, let G(Γ) be the right-angled Artin group defined by the complement graph of Γ. We show that, for any linear forest Λ and any finite graph Γ, G(Λ) can be embedded into G(Γ) if and only if Λ can be realised as a full subgraph of Γ. We also prove that if we drop the assumption that Λ is a linear forest, then the above assertion does not hold, namely, for any finite graph Λ, which is not a linear forest, there exists a finite graph Γ such that G(Λ) can be embedded into G(Γ), though Λ cannot be embedded into Γ as a full subgraph.2010 Mathematics Subject Classification. 20F36 (primary).

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“…In fact, Kim [Kim1] showed that A(P 1 (6)) contains A(C 5 ). On the other hand it is known [GS 2 ] that diagram groups cannot contain A(C n ) for odd n > 3.…”

Section: Introductionmentioning

confidence: 99%

Surface Subgroups of Right-Angled Artin Groups

Crisp

Sageev

Sapir

2008

Int. J. Algebra Comput.

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We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight "forbidden" graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or contains a hole of length ≥ 5, or has the property that A(K) does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a right angled Artin group to contain no hyperbolic surface subgroups.We prove that for one of these "forbidden" subgraphs P 2 (6), the right angled Artin group A(P 2 (6)) is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).

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Co-contractions of graphs and right-angled Artin groups

Cited by 25 publications

References 20 publications

The geometry of purely loxodromic subgroups of right-angled Artin groups

The geometry of purely loxodromic subgroups of right-angled Artin groups

Right-angled Artin groups and full subgraphs of graphs

Surface Subgroups of Right-Angled Artin Groups

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