Residual finiteness of certain 2-dimensional Artin groups

Jankiewicz, Kasia

doi:10.48550/arxiv.2006.04717

Cited by 2 publications

(6 citation statements)

References 24 publications

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“…We now state a version for HNN extension. Similarly as above, combining Thm 2.12 and Lem 2.6 from [Jan20], we obtain the following. Theorem 3.3 ([Jan20, Thm 2.12]).…”

Section: Residual Finitenessmentioning

confidence: 80%

“…We now assume that the inclusion of free groups C → A is induced by a map f : X C → X A of graphs, and the automorphism β : C → C is induced by some graph automorphism X C → X C . The following theorem was proven in [Jan20].…”

Section: Residual Finitenessmentioning

confidence: 95%

“…We do so separately in the case where M, N are both even (Theorem 3.6), and where exactly one of M, N is odd (Theorem 3.7). We start with recalling a criterion for residual finiteness of amalgamated products and HNN extensions of finite rank free groups, proven in [Jan20], which relies on Wise's result on residual finiteness of algebraically clean graphs of free groups [Wis02]. In the second subsection we compute the intersections of conjugates of the amalgamating subgroup in the factor groups.…”

Section: Residual Finitenessmentioning

confidence: 99%

“…The theorem above is a combination of Thm 2.9 and Lem 2.6 in [Jan20]. Condition (2) in the statement of [Jan20, Thm 2.9] is that π(C) is malnormal in Ā.…”

Section: Residual Finitenessmentioning

confidence: 99%

“…The cases (3, 3, 3), (2, 4, 4) and (2, 3, 6) follows from [Squ87]. The cases where M, N, P ≥ 3, except the case of (3, 3, 2m + 1), were covered by [Jan20].…”

Section: Residual Finitenessmentioning

confidence: 99%

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Splittings of triangle Artin groups

Jankiewicz

2021

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We show that a triangle Artin group Art MN P where M ≤ N ≤ P splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either M > 2, or N > 3. We also prove that all even three generator Artin groups are residually finite.An Artin group Art M N P is even if all M, N, P are even. The above theorem combined with our result in [Jan20] (and the fact that Art 22P = Z × Art P is linear) gives us the following.Corollary C. All even Artin groups on three generators are residually finite.All linear groups are residually finite [Mal40], so residual finiteness can be viewed as testing for linearity. Spherical Artin groups are known to be linear ([Kra02], [Big01] for braid groups, and [CW02], [Dig03] for other spherical Artin groups). The right-angled Artin groups are also well known to be linear, but not much more is known about linearity of Artin groups. In last years, a successful approach in proving that groups are linear is by showing that they are virtually special. Artin groups whose defining graphs are forests are the fundamental groups of graph manifolds with boundary [Bru92], [HM99], and so they are virtually special [Liu13], [PW14]. Many Artin groups in certain classes (including 2-dimensional, or three generator) are not cocompactly cubulated even virtually, unless they are sufficiently similar to RAAGs [HJP16], [Hae20b]. In particular, the only (virtually) cocompactly cubulated three generator Artin group are Art 22M = Z × Art M , Art M N ∞ where M, N are both even, Art M ∞∞ = Z * Art M , and Art ∞∞∞ = F 3 . Some triangle-free Artin groups act properly but not cocompactly on locally finite, finite dimensional CAT(0) cube complexes [Hae20a].In [Jan20] we showed that Art M N P are residually finite when M, N, P ≥ 3, except for the cases where (M, N, P ) = (3, 3, 2p + 1) with p ≥ 2. Few more families of Artin groups are known to be residually finite, e.g. even FC type Artin groups [BGMPP19], and certain triangle-free Artin groups [BGJP18].Organization. In Section 1 we provide some background. In Section 2 we prove Theorem A as Proposition 2.5 and Corollary 2.9. We also show that the only irreducible spherical Artin groups splitting as graph of finite rank free groups are dihedral. In Section 3 we recall a criterion for residual finiteness of amalgamated products and HNN extensions of free groups from [Jan20] and prove Theorem B.

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“…We now state a version for HNN extension. Similarly as above, combining Thm 2.12 and Lem 2.6 from [Jan20], we obtain the following. Theorem 3.3 ([Jan20, Thm 2.12]).…”

Section: Residual Finitenessmentioning

confidence: 80%

Section: Residual Finitenessmentioning

confidence: 95%

Section: Residual Finitenessmentioning

confidence: 99%

“…The theorem above is a combination of Thm 2.9 and Lem 2.6 in [Jan20]. Condition (2) in the statement of [Jan20, Thm 2.9] is that π(C) is malnormal in Ā.…”

Section: Residual Finitenessmentioning

confidence: 99%

“…The cases (3, 3, 3), (2, 4, 4) and (2, 3, 6) follows from [Squ87]. The cases where M, N, P ≥ 3, except the case of (3, 3, 2m + 1), were covered by [Jan20].…”

Section: Residual Finitenessmentioning

confidence: 99%

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Splittings of triangle Artin groups

Jankiewicz

2021

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Asymptotic dimension of Artin groups and a new upper bound for Coxeter groups

Tselekidis¹

2022

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If A Γ (W Γ ) is the Artin (Coxeter) group with defining graph Γ we denote by Sim(Γ) the number of vertices of the largest clique in Γ. We show that asdimA Γ ≤ Sim(Γ), if Sim(Γ) = 2. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that asdimW Γ ≤ Sim(Γ) for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptotic dimension of any Artin group of large type with Sim(Γ) = 3 is exactly two.

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Residual finiteness of certain 2-dimensional Artin groups

Cited by 2 publications

References 24 publications

Splittings of triangle Artin groups

Splittings of triangle Artin groups

Asymptotic dimension of Artin groups and a new upper bound for Coxeter groups

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