Subgroup growth of right‐angled Artin and Coxeter groups

Baik, Hyungryul; Petri, Bram; Raimbault, Jean

doi:10.1112/jlms.12277

Cited by 3 publications

(3 citation statements)

References 23 publications

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“…In our previous paper [1], we considered the factorial growth rate of s n pΓq for right-angled Artin and Coxeter groups. That is, we studied limits of the form lim nÑ8 logps n pΓqq n logpnq .…”

Section: Introductionmentioning

confidence: 99%

“…where A r " A Γ Cox pAr l q , B r " B Γ Cox pAr l q and C r " C Γ Cox pAr l q . The values for some low complexity cases are 1 Recall that for functions f, g : N Ñ R the notation f pnq " gpnq as n Ñ 8 indicates that f pnq{gpnq Ñ 1 as n Ñ 8.…”

Section: Introductionmentioning

confidence: 99%

“…This means that p2r, 1q, p2r `1, 1q and p2r `2, 0q are the first vertices of the Newton polytope and hence that γ 1 " 1{2. So y " 1 `c1 t 1{2 `O ´tγ 1 ¯for some γ 1 ą 1 2 . To obtain the coefficient c 1 , we now solve equate the terms that lie on the line of slope ´γ1 supporting the Newton polygon.…”

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confidence: 99%

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Subgroup Growth of Virtually Cyclic Right-Angled Coxeter Groups and Their Free Products

2019

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Subgroup Growth of Virtually Cyclic Right-Angled Coxeter Groups and Their Free Products

2019

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Congruence subgroups and crystallographic quotients of small Coxeter groups

Kumar,

Naik,

Singh

2023

Forum Mathematicum

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Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group S n {S_{n}} has three natural extensions, namely the braid group B n {B_{n}} , the twin group T n {T_{n}} and the triplet group L n {L_{n}} . The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander–Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both T n {T_{n}} and L n {L_{n}} when n ≥ 4 {n\geq 4} . We also determine subquotients of principal congruence subgroups of T n {T_{n}} , and identify the pure twin group P ⁢ T n {PT_{n}} and the pure triplet group P ⁢ L n {PL_{n}} with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that T n / P ⁢ T n ′ {T_{n}/PT_{n}^{\prime}} , T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} and L n / P ⁢ L n ′ {L_{n}/PL_{n}^{\prime}} are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} .

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Statistics of finite degree covers of torus knot complements

Baker,

Petri

2023

Annales Henri Lebesgue

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In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine their Benjamini-Schramm limit and the linear growth rate of the Betti numbers of these covers. All these results generalise to a larger class of lattices in PLS(2, R) × R. As a by-product of our proofs, we obtain analogous limit theorems for high index random subgroups of non-uniform Fuchsian lattices with torsion.Résumé. -Dans la première partie de cet article, nous déterminons la croissance de sousgroupes asymptotique des groupes fondamentaux des compléments des noe uds toriques. Dans la deuxième partie, nous utilisons cela pour étudier des revêtements aléatoires de degré fini des compléments de noe uds toriques. Nous déterminons leurs limites au sens de Benjamini-Schramm et le taux de croissance linéaire de leur nombres de Betti. Tous ces résultats se généralisent à une classe plus large de réseaux de PLS(2, R) × R. Comme conséquence de nos preuves, nous obtenons également des théorèmes limites analogues pour des sous-groupes aléatoires de grand indice des réseaux Fuchsiens non-uniformes.

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Subgroup growth of right‐angled Artin and Coxeter groups

Cited by 3 publications

References 23 publications

Subgroup Growth of Virtually Cyclic Right-Angled Coxeter Groups and Their Free Products

Subgroup Growth of Virtually Cyclic Right-Angled Coxeter Groups and Their Free Products

Congruence subgroups and crystallographic quotients of small Coxeter groups

Statistics of finite degree covers of torus knot complements

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