The dual Artin isomorphism for Artin groups of XXL type

Abstract: We show that an Artin group $A_Γ$ of XXL type (with all defining integers satisfying $m_{ij}\geq 5$) is isomorphic to the corresponding dual Artin group for any choice of Coxeter element. Our proof involves the set of Hurwitz words $Q$ which arise from the Hurwitz action on tuples of elements of the free group. We show that the canonical epimorphism from $A_Γ$ to the dual Artin group is an isomorphism if and only if the projection from $A_Γ$ to the Coxeter group is injective on the image of $Q$. Then, using geometric properties of $Q$ and the solution to the word problem for Coxeter groups, we show this to be the case when all $m_{ij} \geq 5$.

• The paper establishes a canonical isomorphism between standard and dual Artin groups for XXL-type Coxeter graphs using Hurwitz set analysis.
• It employs geometric-combinatorial techniques to control alternating subwords and address the word problem.
• The result unifies earlier cases, offering algorithmic insights and extending Garside theory applications for Artin groups.

The Dual Artin Isomorphism for Artin Groups of XXL Type

Introduction and Motivation

Artin groups, a central object in geometric group theory, generalize braid groups and are defined in close analogy to Coxeter groups via labeled graphs. Despite several decades of intense study, the structure and properties of general Artin groups remain elusive, with many basic questions unresolved, including conjectures concerning torsion-freeness, center triviality, and the $K(\pi,1)$ property. Among the influential advances, dual presentations for Artin groups—arising from dual Garside structures and Hurwitz actions—have provided significant new perspectives on word and conjugacy problems, $K(\pi,1)$, and Garside theoretic properties.

This paper addresses the isomorphism problem between the standard Artin group $A_\Gamma$ and associated dual Artin groups for Coxeter diagrams $\Gamma$ of so-called XXL type, that is, with all $m_{ij}\geq5$. The dual Artin group, constructed with respect to a Coxeter element, plays a pivotal role in recent breakthroughs for diverse classes (e.g., finite, affine, rank 3) of Artin groups. The author establishes that for any such XXL-type Coxeter graph, the standard and dual Artin groups are canonically isomorphic, irrespective of the Coxeter element, thus extending and unifying much of the prior theory.

Preliminaries and Main Definitions

Coxeter groups $W_\Gamma$ and their associated Artin groups $A_\Gamma$ are defined from a labeled simplicial graph $\Gamma$ with vertices indexed by $\{1,\ldots,n\}$ and edge labels $m_{ij}\in\{3,4,\ldots\}\cup\{\infty\}$. The standard Artin group has relations

$K(\pi,1)$0

whereas the corresponding Coxeter group further imposes $K(\pi,1)$1. The dual Artin group, central to this work, is defined via generators corresponding to the orbit of a distinguished tuple of elements under the Hurwitz action of the braid group $K(\pi,1)$2—a generalization of the classical Hurwitz action on reduced words.

Canonical Map and Reduction to the Hurwitz Set

There is an explicit canonical epimorphism $K(\pi,1)$3 for each Coxeter element $K(\pi,1)$4. The isomorphism question is reduced to the injectivity of the standard projection $K(\pi,1)$5 on the Hurwitz set $K(\pi,1)$6—the set of all words in the free group $K(\pi,1)$7 that occur in any position in the Hurwitz orbit of the standard generators. The paper establishes that the canonical map is an isomorphism if and only if the map $K(\pi,1)$8 is injective on the image of $K(\pi,1)$9.

Analysis of the Hurwitz Set and the Word Problem

A crucial contribution is a geometric-combinatorial analysis of the Hurwitz set $A_\Gamma$0. By utilizing the realization of $A_\Gamma$1 as the fundamental group of the $A_\Gamma$2-punctured disk, $A_\Gamma$3 is shown to correspond to homotopy classes of simple non-crossing loops, and hence the associated words are characterized by square-freeness and explicit constraints on alternating subwords.

The analysis leverages the detailed understanding of alternating subwords: maximal alternating subwords in elements of $A_\Gamma$4 enjoy rigid structure with exactly one change of sign, tightly constrained location of this change, and controlled overlap. An essential technical result is that, in XXL type (i.e., $A_\Gamma$5 for all $A_\Gamma$6), these structural properties restrict how alternating words and possible Artin (braid) relations interact.

Using Tits's solution to the Coxeter group's word problem, the author proves that if two elements of $A_\Gamma$7 project to the same Coxeter group element, then they must be related by a sequence of "commuting" toggling operations associated with specific generators, each of which can be realized using a subgroup of the braid group $A_\Gamma$8 generated by suitable powers of the Birman–Ko–Lee generators.

Main Theorem and Proof Structure

The main theorem states:

Theorem: For any Coxeter group $A_\Gamma$9 with all $\Gamma$0 and any Coxeter element $\Gamma$1, the dual Artin group $\Gamma$2 is canonically isomorphic to the standard Artin group $\Gamma$3.

The proof proceeds by showing:

• For any element in the Hurwitz set $\Gamma$4, it can be moved (using the aforementioned braid subgroup) to a canonical form with controlled alternating subwords.
• Any two Hurwitz words mapping to the same Coxeter group element are related by a sequence of toggling operations, all realized within this braid subgroup, and the corresponding Artin group elements agree.
• Therefore, the Artin group projection is injective on $\Gamma$5, yielding the canonical isomorphism via the dual Artin presentation.

Comparison to Prior Work

Isomorphism results for standard and dual Artin groups were previously known for finite and affine Coxeter types [brady_watt_kp_2002a], [bessis_dual_2003], [mccammond_sulway_artin_2017], certain rank 3 Artin groups [delucchi_etal_dual_2024], and specific free products [resteghini_free_2024]. The present result unifies and extends these cases to arbitrary XXL type, for arbitrary Coxeter elements, via a cohesive geometric and combinatorial method centered on the Hurwitz set. Notably, the proof does not require case-by-case analysis and instead uses induction on the combinatorics of simple loops and the geometry of (square-free) word representatives in the fundamental group of the punctured disk.

Implications, Contrasts, and Future Directions

Rigidity of Presentations: For Artin groups of XXL type, all dual presentations—regardless of Coxeter element—coincide with the standard presentation, supporting a strong rigidity conjecture for dual Garside structures in this class.

Algorithmic Control: The methods give explicit control over the reduction of words, providing direct insight into solutions of the word and conjugacy problems for XXL Artin groups, complementing the CAT(0) and acylindrical hyperbolicity properties established in [haettel_xxl_2022].

Garside Theory: In combination with structural results (e.g., the existence of dual Garside structures when $\Gamma$6 [mccammond_introduction_2005]), this isomorphism extends the reach of Garside-theoretic techniques for establishing key properties (e.g., the $\Gamma$7 conjecture, center triviality, torsion-freeness) to a large family of Artin groups.

Limitations and Open Problems: The combinatorics become intractable in the presence of $\Gamma$8 or $\Gamma$9, as maximal alternating subwords can interact more pathologically. The current proof’s restriction to $m_{ij}\geq5$0 yields an explicit and uniform argument, but the extension to "XL" type ($m_{ij}\geq5$1) or large type ($m_{ij}\geq5$2) remains an important open problem, motivating further exploration of Hurwitz orbits, geometric models, and possible new functorial or poset-topological methods.

Broader Significance: The techniques and structural results prompt several future directions, including generalizations to complex and non-real reflection groups, potential connections to lattice properties, and new invariants for Artin groups modulo their Garside/dual presentations.

Conclusion

This work settles the isomorphism problem between standard and dual Artin groups for Coxeter graphs of XXL type. By deploying a geometric analysis of Hurwitz words and braid group actions, it provides a transparent, uniform framework for understanding the structure, presentation rigidity, and algorithmic properties of this essential class of Artin groups. The results substantiate the fundamental role of the dual presentation and Hurwitz group techniques within Artin group theory and open several avenues toward resolving the general isomorphism problem in broader Coxeter-theoretic and Garside-theoretic settings.

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References

(2606.13296) See also [haettel_xxl_2022], [bessis_dual_2003], [brady_watt_kp_2002a], [mccammond_sulway_artin_2017], [delucchi_etal_dual_2024], [resteghini_free_2024].