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Congruence subgroups and crystallographic quotients of small Coxeter groups

  • Pravin Kumar EMAIL logo , Tushar Kanta Naik and Mahender Singh
Published/Copyright: October 27, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 36 Issue 1

Abstract

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 has three natural extensions, namely the braid group B n , the twin group T n and the triplet group 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 and L n when n ≥ 4 . We also determine subquotients of principal congruence subgroups of T n , and identify the pure twin group P ⁢ T n and the pure triplet group P ⁢ L 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 / T n ′′ and L n / P ⁢ L n ′ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ .

Keywords: Bieberbach group; braid group; congruence subgroup property; crystallographic group; Tits representation; triplet group; twin group
MSC 2020: 20F55; 20H15; 20F36

Communicated by Manfred Droste

Funding source: Department of Science and Technology, Ministry of Science and Technology, India

Award Identifier / Grant number: DST/SJF/MSA-02/2018-19

Award Identifier / Grant number: SB/SJF/2019-20/04

Funding statement: Pravin Kumar is supported by the PMRF fellowship at IISER Mohali. He also gives thanks to NISER Bhubaneswar for the warm hospitality during his visit, where a part of this project was carried out. Mahender Singh is supported by the SwarnaJayanti Fellowship grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20/04.

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Received: 2023-03-26
Published Online: 2023-10-27
Published in Print: 2024-01-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On Mukhin’s necessary and sufficient condition for the validity of the local limit theorem
  3. Non-commutative Khinchine-type inequality for dependent random variables and overview of its applications in data science
  4. Connections and genuinely ramified maps of curves
  5. Caustics of pseudo-spherical surfaces in the Euclidean 3-space
  6. On liftings of modular forms and Weil representations
  7. Approximation via statistical measurable convergence with respect to power series for double sequences
  8. Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems
  9. Gem-induced trisections of compact PL 4-manifolds
  10. Categories of modules, comodules and contramodules over representations
  11. On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions
  12. On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms
  13. Congruence subgroups and crystallographic quotients of small Coxeter groups
  14. Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2
  15. Hardy and BMO spaces on Weyl chambers
  16. De Branges–Rovnyak spaces and local Dirichlet spaces of higher order
https://doi.org/10.1515/forum-2023-0103
Keywords for this article
Bieberbach group; braid group; congruence subgroup property; crystallographic group; Tits representation; triplet group; twin group

Articles in the same Issue

  1. Frontmatter
  2. On Mukhin’s necessary and sufficient condition for the validity of the local limit theorem
  3. Non-commutative Khinchine-type inequality for dependent random variables and overview of its applications in data science
  4. Connections and genuinely ramified maps of curves
  5. Caustics of pseudo-spherical surfaces in the Euclidean 3-space
  6. On liftings of modular forms and Weil representations
  7. Approximation via statistical measurable convergence with respect to power series for double sequences
  8. Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems
  9. Gem-induced trisections of compact PL 4-manifolds
  10. Categories of modules, comodules and contramodules over representations
  11. On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions
  12. On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms
  13. Congruence subgroups and crystallographic quotients of small Coxeter groups
  14. Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2
  15. Hardy and BMO spaces on Weyl chambers
  16. De Branges–Rovnyak spaces and local Dirichlet spaces of higher order
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