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Parabolic subgroups of complex braid groups

Juan González-Meneses, Ivan Marin

TL;DR

This work generalizes the curve complex and lattice of parabolic subgroups from real Artin groups to generalized braid groups attached to irreducible complex reflection groups. It builds a purely topological (local-fundamental-group) definition of parabolic subgroups and then aligns it with robust Garside-theoretic machinery, including interval/Lcm-Garside structures and a novel swap transport mechanism. The main results prove the parabolic subgroups form a lattice, guarantee parabolic closures for all elements (except the exceptional $G_{31}$ case), and construct a curve complex whose vertices are irreducible parabolic subgroups with adjacency governed by commutation of canonical elements $z_{B}$. These developments yield a unifying framework for parabolic subgroups across all well-generated complex braid groups and establish hyperbolicity for key families, enriching the geometric and combinatorial understanding of complex braid groups.

Abstract

In this paper we introduce a class of `parabolic' subgroups for the generalized braid group associated to an arbitrary irreducible complex reflection group, which maps onto the collection of parabolic subgroups of the reflection group. Except for one case, which is proven separately elsewhere, we prove that this collection forms a lattice, so that intersections of parabolic subgroups are parabolic subgroups. In particular, every element admits a parabolic closure, which is the smallest parabolic subgroup containing it. We furthermore prove that it provides a simplicial complex which generalizes the curve complex of the usual braid group. In the case of real reflection groups, this complex generalizes the one previously introduced by Cumplido, Gebhardt, González-Meneses and Wiest for Artin groups of spherical type. We show that it shares similar properties, and similarly conjecture its hyperbolicity, with a few additional results in this direction.

Parabolic subgroups of complex braid groups

TL;DR

This work generalizes the curve complex and lattice of parabolic subgroups from real Artin groups to generalized braid groups attached to irreducible complex reflection groups. It builds a purely topological (local-fundamental-group) definition of parabolic subgroups and then aligns it with robust Garside-theoretic machinery, including interval/Lcm-Garside structures and a novel swap transport mechanism. The main results prove the parabolic subgroups form a lattice, guarantee parabolic closures for all elements (except the exceptional case), and construct a curve complex whose vertices are irreducible parabolic subgroups with adjacency governed by commutation of canonical elements . These developments yield a unifying framework for parabolic subgroups across all well-generated complex braid groups and establish hyperbolicity for key families, enriching the geometric and combinatorial understanding of complex braid groups.

Abstract

In this paper we introduce a class of `parabolic' subgroups for the generalized braid group associated to an arbitrary irreducible complex reflection group, which maps onto the collection of parabolic subgroups of the reflection group. Except for one case, which is proven separately elsewhere, we prove that this collection forms a lattice, so that intersections of parabolic subgroups are parabolic subgroups. In particular, every element admits a parabolic closure, which is the smallest parabolic subgroup containing it. We furthermore prove that it provides a simplicial complex which generalizes the curve complex of the usual braid group. In the case of real reflection groups, this complex generalizes the one previously introduced by Cumplido, Gebhardt, González-Meneses and Wiest for Artin groups of spherical type. We show that it shares similar properties, and similarly conjecture its hyperbolicity, with a few additional results in this direction.
Paper Structure (32 sections, 97 theorems, 109 equations, 1 figure)

This paper contains 32 sections, 97 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Parabolic subgroups
  3. Normal rays
  4. Local fundamental groups
  5. Parabolic subgroups of complex braid groups
  6. Subgroups of parabolic subgroups and products
  7. The curve graph and the curve complex
  8. Description of the parabolic subgroups
  9. Real reflection groups and Shephard groups
  10. The groups $G(de,e,n)$, $d > 1$
  11. Interval monoids
  12. Groups $G(e,e,n)$
  13. Well-generated 2-reflection groups
  14. Exceptional groups of rank 2
  15. Parabolic closures in Garside groups
  16. ...and 17 more sections

Key Result

Theorem 1.1

For every element $x \in B$, there exists a unique minimal parabolic subgroup $\mathop{\mathrm{PC}}\nolimits(x)$ of $B$ containing it, and we have $\mathop{\mathrm{PC}}\nolimits(x^m) = \mathop{\mathrm{PC}}\nolimits(x)$ for every $m \neq 0$.

Figures (1)

  • Figure 1: Defining relations of the dual braid monoid in type $G_{24}$.

Theorems & Definitions (189)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1
  • Theorem 1.4
  • Proposition 1
  • proof
  • Proposition 2
  • Definition 1
  • Lemma 1
  • ...and 179 more