Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.13296

The dual Artin isomorphism for Artin groups of XXL type

Sean O'Brien
View original source ↗Read source document ↗Export metadata ↓

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 $m_{ij} \geq 5$.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sean O'Brien. 2026-07-28. The dual Artin isomorphism for Artin groups of XXL type. https://arxiv.org/abs/2606.13296

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Word Measures on Wreath Products II

Every word $w$ in $F_r$, the free group of rank $r$, induces a probability measure (the $w$-measure) on every finite group $G$, by substitution of random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $E_w[χ]$ of the irreducible characters of $G$. For every finite group $G$, every stable character $χ$ of $G\wr S_n$ (trace of a finitely generated $FI_G$-module), and every word $w\in F_r$, we approximate $E_w[χ]$ up to an error term of $O(n^{-π(w)})$, where $π(w)$ is the primitivity rank of $w$. This generalizes previous works by Puder, Hanany, Magee and the author. As an application we show that random Schreier graphs of representation-stable actions of $G\wr S_n$ are close-to-optimal expanders. The paper reveals a surprising relation between stable representation theory of wreath products and not-necessarily connected Stallings core graphs.

math.GR↗
arXiv2026

Robust quasi-isometric embeddings inapproximable by Anosov representations

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For all but finitely many $m\in \mathbb{N}$, we exhibit the first examples of non-locally rigid, Zariski dense, robust quasi-isometric embeddings of hyperbolic groups in $\mathsf{SL}_m(\mathbb{K})$ which are not limits of Anosov representations. As a consequence, we show that higher rank analogues of Sullivan's structural stabilty theorem and of the density theorem for Kleinian groups fail for Anosov representations in $\mathsf{SL}_m(\mathbb{C}), m\geq 30$.

math.GR↗
arXiv2026

$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products

We provide a necessary and sufficient condition for the restricted wreath product $A\wr B$ to be $\mathcal{C}$-hereditarily conjugacy separable where $\mathcal{C}$ is an extension-closed pseudovariety of finite groups. Moreover, we prove that the Grigorchuk group is 2-hereditarily conjugacy separable. As an application, we demonstrate that the lamplighter groups and $\mathbb{Z} \wr \mathbb{Z}$ are hereditarily conjugacy separable (but not $p$-conjugacy separable for any prime $p$). This provides infinitely many new examples of solvable, non-polycyclic hereditarily conjugacy separable groups. Furthermore, we study wreath products of cyclic subgroup separable groups and the derived length of iterated wreath products of solvable groups with an abelian base group and, as an application, we give an explicit construction of non-polycyclic hereditarily conjugacy separable groups of arbitrary derived length as iterated wreath products of abelian groups.

math.GR↗