Mathematics > Group Theory
arXiv:2606.13296 (math)
[Submitted on 11 Jun 2026 (v1), last revised 28 Jul 2026 (this version, v2)]
Abstract:We show that an Artin group $A_\Gamma$ 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 $A_\Gamma$ to the dual Artin group is an isomorphism if and only if the projection from $A_\Gamma$ 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 $m_{ij} \geq 5$.
| Comments: | 29 pages, 15 figures. All comments welcome! v2: adds acknowledgement of an unpublished preprint with overlapping material |
| Subjects: | Group Theory (math.GR) |
| MSC classes: | 20F36 (Primary) 20F65, 20F55 (Secondary) |
| Cite as: | arXiv:2606.13296 [math.GR] |
| (or arXiv:2606.13296v2 [math.GR] for this version) | |
| https://doi.org/10.48550/arXiv.2606.13296 arXiv-issued DOI via DataCite |
Submission history
From: Sean O'Brien [view email]
[v1]
Thu, 11 Jun 2026 12:48:38 UTC (64 KB)
[v2]
Tue, 28 Jul 2026 17:31:48 UTC (64 KB)
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