Intersection of parabolic subgroups in even Artin groups of FC-type

Antolín, Yago; Foniqi, Islam

doi:10.1017/s0013091522000414

Cited by 3 publications

(10 citation statements)

References 24 publications

(57 reference statements)

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“…𝑖 ∈ {2, … , 𝑝}, we have 𝑥 𝑎 1 1 ∈ 𝐷 𝑌 ∩ g 0 𝐷 𝑌 ′ g −1 0 , hence, by the case 𝑝 = 1 previously proved, 𝑥 1 ∈ 𝑍. □ Lemma 6.2.…”

Section: A C K N O W L E D G M E N T Smentioning

confidence: 73%

“…The following theorem gathers together some results proved in Dyer [8]. Part (1) Theorem 3. 1 Dyer [8].…”

Section: Dyer Systemsmentioning

confidence: 74%

“…The following consequence of Theorem 2.10 is widely used in the theory of Coxeter groups (see Krammer [10], for example). On the other hand, an equivalent statement for Artin groups is one of the central questions in the field (see Cumplido–Gebhardt–González‐Meneses–Wiest [5], Morris‐Wright [13], Cumplido–Martin–Vaskou [6], Möller–Paris–Varghese [12], and Antolín–Foniqi [1]). Corollary Let

(D,X)

be a Dyer system of finite type and let

A

be a subset of

D

.…”

Section: Definitions and Statementsmentioning

confidence: 99%

“…a reduced syllabic expression for g, let (𝑦 𝑏1 1 , … , 𝑦 𝑏 𝑞 𝑞 ) be a reduced syllabic expression for g ′ ,…”

mentioning

confidence: 99%

“…1 , 𝑡 1 ] 𝑞 ⋅ 𝑢 to [𝑡 1 , 𝑠 1 ] 𝑞 ⋅ 𝑢 through a single elementary M-operation of type II. So, we can go from 𝑤 to 𝑤 ′ through a finite sequence of elementary M-operations of type II.It remains to show that, if a syllabic word 𝑤 = (𝑥 We argue by induction on the length 𝓁 of the syllabic word.…”

mentioning

confidence: 99%

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Word problem and parabolic subgroups in Dyer groups

Paris,

Soergel

2023

Bulletin of London Math Soc

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One can observe that Coxeter groups and right‐angled Artin groups share the same solution to the word problem. On the other hand, in his study of reflection subgroups of Coxeter groups, Dyer introduces a family of groups, which we call Dyer groups, which contains both, Coxeter groups and right‐angled Artin groups. We show that all Dyer groups have this solution to the word problem, we show that a group which admits such a solution belongs to a little more general family of groups that we call quasi‐Dyer groups, and we show that this inclusion is strict. Then we show several results on parabolic subgroups in quasi‐Dyer groups and in Dyer groups. Notably, we prove that any intersection of parabolic subgroups in a Dyer group of finite type is a parabolic subgroup.

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“…𝑖 ∈ {2, … , 𝑝}, we have 𝑥 𝑎 1 1 ∈ 𝐷 𝑌 ∩ g 0 𝐷 𝑌 ′ g −1 0 , hence, by the case 𝑝 = 1 previously proved, 𝑥 1 ∈ 𝑍. □ Lemma 6.2.…”

Section: A C K N O W L E D G M E N T Smentioning

confidence: 73%

“…The following theorem gathers together some results proved in Dyer [8]. Part (1) Theorem 3. 1 Dyer [8].…”

Section: Dyer Systemsmentioning

confidence: 74%

(D,X)

be a Dyer system of finite type and let

A

be a subset of

D

.…”

Section: Definitions and Statementsmentioning

confidence: 99%

“…a reduced syllabic expression for g, let (𝑦 𝑏1 1 , … , 𝑦 𝑏 𝑞 𝑞 ) be a reduced syllabic expression for g ′ ,…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Word problem and parabolic subgroups in Dyer groups

Paris,

Soergel

2023

Bulletin of London Math Soc

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Subgroups of even Artin groups of FC type

Antolín,

Foniqi

2024

Journal of Group Theory

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We prove a Tits alternative theorem for subgroups of finitely presented even Artin groups of FC type, stating that there exists a finite index subgroup such that every subgroup of it is either finitely generated abelian, or maps onto a non-abelian free group. Parabolic subgroups play a pivotal role in our proofs, and we show that parabolic subgroups of even Artin groups of FC type are closed under taking roots.

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