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Canonical Reduction Systems in Artin-Tits groups of spherical type

María Cumplido, Juan González-Meneses, Davide Perego

TL;DR

This work defines a fully algebraic canonical reduction system CRS($\alpha$) for Artin-Tits groups of spherical type, extending Birman–Lubotzky–McCarthy’s braid-based CRS. It builds a robust framework using the complex of irreducible parabolic subgroups and Garside theory to prove CRS$$(\alpha)$$ has analogous invariance and reducibility properties, and it provides an algorithm to compute CRS via finite reduction simplices and periodic centralizers. A key theoretical contribution is the result that the centralizer sequence $Z(\gamma),Z(\gamma^2),Z(\gamma^3),\dots$ is periodic with computable period, enabling practical CRS computation. The paper also specializes the approach to braids, offering faster CRS detection for curves through component gluing and centralizer analysis, and it delivers a concrete algorithm with complexity analyses for computing CRS in braids, linking geometric and algebraic perspectives in spherical Artin–Tits groups.

Abstract

We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period.

Canonical Reduction Systems in Artin-Tits groups of spherical type

TL;DR

This work defines a fully algebraic canonical reduction system CRS() for Artin-Tits groups of spherical type, extending Birman–Lubotzky–McCarthy’s braid-based CRS. It builds a robust framework using the complex of irreducible parabolic subgroups and Garside theory to prove CRS has analogous invariance and reducibility properties, and it provides an algorithm to compute CRS via finite reduction simplices and periodic centralizers. A key theoretical contribution is the result that the centralizer sequence is periodic with computable period, enabling practical CRS computation. The paper also specializes the approach to braids, offering faster CRS detection for curves through component gluing and centralizer analysis, and it delivers a concrete algorithm with complexity analyses for computing CRS in braids, linking geometric and algebraic perspectives in spherical Artin–Tits groups.

Abstract

We introduce the canonical reduction system of an element in an Artin-Tits group of spherical type, which generalizes the similar notion for braids (and mapping classes) introduced by Birman, Lubotzky and McCarthy. We show its basic properties, which coincide with those satisfied in braid groups, and we provide an algorithm to compute it. We improve the algorithm in the case of braid groups, and discuss its complexity in this case. As a necessary result for obtaining the general algorithm, we prove that the centralizers of positive powers of an element form a periodic sequence and we show how to compute its period.

Paper Structure

This paper contains 15 sections, 27 theorems, 22 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and background
  3. Curve complex and canonical reduction system
  4. Complex of irreducible parabolic subgroups
  5. Garside Theory and $\mathcal{C}(G)$
  6. Canonical reduction systems in Artin-Tits groups
  7. Detecting vertices of the canonical reduction system
  8. Restricting the search to a finite set
  9. Computing standard reduction simplices
  10. Discarding vertices which are not essential
  11. Some conjectures
  12. The case of braid groups: detecting curves from the canonical reduction system
  13. Canonical reduction system and centralizers
  14. Gluing components
  15. An algorithm to compute the canonical reduction system of a braid

Key Result

Theorem 1

Let $G$ be an Artin-Tits group of spherical type. Two irreducible parabolic subgroups $P_1$ and $P_2$ of $G$ are adjacent if and only if one of the following properties is satisfied:

Figures (3)

  • Figure 1: The complete classification of irreducible Artin-Tits groups of spherical type.
  • Figure 2: The $5$-punctured disk with an example of a CRS (dashed) and of $D_C$ (the gray area and the gray curve $C$). In this case $D_C$ is homeomorphic to the $2$-punctured disk simply by collapsing to a puncture the region enclosed in the curve in the innermost dashed curve.
  • Figure 3: A possible component of $\alpha$ with respect to a curve $C$ (solid circle). In this case $\alpha_C$ is the full twist on $2$-strands.

Theorems & Definitions (62)

  • Definition 1: CGGW
  • Theorem 1: CGGW
  • Definition 2: Elrifai1994
  • Definition 3: Gebhardt2010a
  • Definition 4: Gebhardt2010a
  • Definition 5: Gebhardt2010a
  • Theorem 2: Benardete1995, Lee2008
  • Theorem 3: Lee2008
  • Theorem 4: Cumplido2019b
  • Corollary 1
  • ...and 52 more