No CrossRef data available.
Article contents
TORIC REFLECTION GROUPS
Published online by Cambridge University Press: 18 October 2023
Abstract
Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some 
$k\geq 2$, and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J-groups of Achar and Aubert [‘On rank 2 complex reflection groups’, Comm. Algebra 36(6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of ‘toric reflection groups’ that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter’s truncations of the 
$3$-strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest.
Keywords
MSC classification
Primary:
20F55: Reflection and Coxeter groups
Secondary:
20F36: Braid groups; Artin groups
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by Michael Giudici
References
Abramenko, P. and Brown, K. S., Buildings. Theory and Applications, Graduate Texts in Mathematics, 248 (Springer, New York, 2008).CrossRefGoogle Scholar
Achar, P. and Aubert, A.-M., ‘On rank 2 complex reflection groups’, Comm. Algebra 36(6) (2008), 2092–2132.CrossRefGoogle Scholar
Baader, S., Blair, R. and Kjuchukova, A., ‘Coxeter groups and meridional rank of links’, Math. Ann. 379(3–4) (2021), 1533–1551.CrossRefGoogle Scholar
Baader, S., Blair, R., Kjuchukova, A. and Misev, P., ‘The bridge number of arborescent links with many twigs’, Algebr. Geom. Topol. 23(1) (2023), 75–85.CrossRefGoogle Scholar
Bannai, E., ‘Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension 2’, J. Math. Soc. Japan 28(3) (1976), 447–454.CrossRefGoogle Scholar
Bessis, D., ‘The dual braid monoid’, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 647–683.CrossRefGoogle Scholar
Birman, J., Ko, K. H. and Lee, S. J., ‘A new approach to the word and conjugacy problems in the braid groups’, Adv. Math. 139 (1998), 322–353.CrossRefGoogle Scholar
Bourbaki, N., Groupes et algèbres de Lie, Éléments de Mathématique, Fasc. XXXIV (Hermann, Paris, 1968), Ch. IV–VI.Google Scholar
Brieskorn, E., ‘Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe’, Invent. Math. 12 (1971), 57–61.CrossRefGoogle Scholar
Brink, B. and Howlett, R., ‘A finiteness property and an automatic structure for Coxeter groups’, Math. Ann. 296 (1993), 179–190.CrossRefGoogle Scholar
Broué, M., Introduction to Complex Reflection Groups and their Braid Groups, Lecture Notes in Mathematics, 1988 (Springer-Verlag, Berlin, 2010).CrossRefGoogle Scholar
Broué, M., Malle, G. and Rouquier, R., ‘Complex reflection groups, braid groups, Hecke algebras’, J. reine angew. Math. 500 (1998), 127–190.Google Scholar
Burde, G., Zieschang, H. and Heusener, M., Knots, 3rd edn, De Gruyter Studies in Mathematics, 5 (De Gruyter, Berlin, 2014).Google Scholar
Coxeter, H. S. M., ‘The abstract groups
${{G}}^{\mathit{m},\mathit{n},\mathit{p}}$
’, Trans. Amer. Math. Soc. 45(1) (1939), 73–150.Google Scholar
${{G}}^{\mathit{m},\mathit{n},\mathit{p}}$
’, Trans. Amer. Math. Soc. 45(1) (1939), 73–150.Google ScholarCoxeter, H. S. M., ‘Factor groups of the Braid group’, in: Proceedings of the Fourth Canadian Mathematical Congress, Banff, 1957 (ed. M. S. Macphail) (University of Toronto Press, Toronto, 1959), 95–122.Google Scholar
Dehornoy, P., Digne, F., Krammer, D., Godelle, E. and Michel, J., Foundations of Garside Theory, Tracts in Mathematics, 22 (European Mathematical Society, Zürich, 2015).CrossRefGoogle Scholar
Dehornoy, P. and Paris, L., ‘Gaussian groups and Garside groups, two generalisations of Artin groups’, Proc. Lond. Math. Soc. (3) 79(3) (1999), 569–604.CrossRefGoogle Scholar
Dyer, M. J., ‘Reflection subgroups of Coxeter systems’, J. Algebra 135(1) (1990), 57–73.CrossRefGoogle Scholar
Dyer, M. J., ‘On the Bruhat graph of a Coxeter system’, Compos. Math. 78(2) (1991), 185–191.Google Scholar
Gobet, T., ‘On some torus knot groups and submonoids of the braid groups’, J. Algebra 607 (2022), 260–289.CrossRefGoogle Scholar
Kassel, C. and Turaev, V., Braid Groups, Graduate Texts in Mathematics, 247 (Springer, New York, 2008).CrossRefGoogle Scholar
Krammer, D., ‘The conjugacy problem for Coxeter groups’, Groups Geom. Dyn. 3(1) (2009), 71–171.CrossRefGoogle Scholar
Lehrer, G. I. and Taylor, D. E., Unitary Reflection Groups, Australian Mathematical Society Lecture Series, 20 (Cambridge University Press, Cambridge, 2009).Google Scholar
Magnus, W., Karras, A. and Solitar, D., Combinatorial Group Theory (Dover Publications, Inc., Mineola, New York, 1976).Google Scholar
Malle, G. and Michel, J., ‘Constructing representations of Hecke algebras for complex reflection groups’, LMS J. Comput. Math. 13 (2010), 426–450.CrossRefGoogle Scholar
Maxwell, G., ‘The normal subgroups of finite and affine Coxeter groups’, Proc. Lond. Math. Soc. (3) 76(2) (1998), 359–382.CrossRefGoogle Scholar
Paris, L., ‘Irreducible Coxeter groups’, Internat. J. Algebra Comput. 17(3) (2007), 427–447.CrossRefGoogle Scholar
Qi, D., ‘On irreducible, infinite, nonaffine Coxeter groups’, Fund. Math. 193(1) (2007), 79–93.CrossRefGoogle Scholar
Rolfsen, D., Knots and Links, Mathematics Lecture Series, 7 (Publish or Perish Inc., Berkeley, CA, 1976).Google Scholar
Rost, M. and Zieschang, H., ‘Meridional generators and plat presentations of torus links’, J. Lond. Math. Soc. (2) 35(3) (1987), 551–562.CrossRefGoogle Scholar
Schreier, O., ‘Über die Gruppen
${\mathit{A}}^{\mathit{a}}{\mathit{B}}^{\mathit{b}}=1$
’, Abh. Math. Sem. Hamburg 3 (1923), 167–169.CrossRefGoogle Scholar
${\mathit{A}}^{\mathit{a}}{\mathit{B}}^{\mathit{b}}=1$
’, Abh. Math. Sem. Hamburg 3 (1923), 167–169.CrossRefGoogle Scholar