The dual approach to the $K(π, 1)$ conjecture

TL;DR

The paper surveys the $K(\pi,1)$ conjecture for Artin groups and develops a detailed dual Garside framework based on intervals $[1,w]_R$ of reflections and Coxeter elements. It explains how this dual perspective successfully underpins the affine $K(\pi,1)$ proof via a three-step construction involving the interval complex $K_W$, a finite subcomplex $X_W'$, and a deformation retraction, while outlining open questions for extending beyond spherical/affine cases. By contrasting the standard and dual approaches, the work lays foundational questions about lattice properties, EL-shellability, and the isomorphism between dual and standard Artin groups in general. The discussion highlights potential generalizations and identifies rank-3 Coxeter groups as a promising testbed for extending the dual method to broader classes of Artin groups.

Abstract

Dual presentations of Coxeter groups have recently led to breakthroughs in our understanding of affine Artin groups. In particular, they led to the proof of the $K(π, 1)$ conjecture and to the solution of the word problem. Will the "dual approach" extend to more general classes of Coxeter and Artin groups? In this paper, we describe the techniques used to prove the $K(π, 1)$ conjecture for affine Artin groups and we ask a series of questions that are mostly open beyond the spherical and affine cases.

Figures (8)

• Figure 1: The symmetric group $\mathfrak{S}_3$ on a three-element set is a spherical Coxeter group with presentation $\mathfrak{S}_3 = \langle a,b \mid a^2 = b^2 = (ab)^3 = 1 \rangle$. It can be represented as the group of linear isometries of $\mathbb{R}^3$ that permutes the three coordinates. This representation can be made essential by restricting to the plane $\{x_1+x_2+x_3=0\} \subseteq \mathbb{R}^3$. Left: the reflection lines (corresponding to the transpositions of $\mathfrak{S}_3$) in the plane $\{x_1+x_2+x_3=0\}$. Right: the Salvetti complex $X_{\mathfrak{S}_3}$, which has one $0$-cell, two $1$-cells (labeled $a$ and $b$), and one hexagonal $2$-cell. The fundamental group of $X_{\mathfrak{S}_3}$ is the braid group $G_{\mathfrak{S}_3} = \langle a, b \mid aba = bab \rangle$.
• Figure 2: Reflection arrangements of some Coxeter groups of rank $3$ (also known as triangle groups). The dashed line is the axis of the Coxeter element $w = abc$ (see \ref{['sec:coxeter-elements']}). Left: the spherical $(2, 3, 3)$ triangle group, a.k.a. the symmetric group $\mathfrak{S}_4$. Center: the affine $(3, 3, 3)$ triangle group, a.k.a. the affine symmetric group of type $\tilde{A}_3$. Right: the hyperbolic $(4, 3, 3)$ triangle group. In all three cases, the triple $(p, q, r)$ consists of the upper-triangular entries of the Coxeter matrix. The sphere $\mathbb{S}^2$ (left), Euclidean plane $\mathbb{R}^2$ (center), and the hyperbolic plane $\mathbb{H}^2$ (right) are tiled by triangles with angles $\frac{\pi}{p}, \frac{\pi}{q}, \frac{\pi}{r}$.
• Figure 3: A loop in the orbit configuration space $Y_W$ for $W = \mathfrak{S}_3$. In this case, $Y_W$ is the space of configurations of $3$ points in $\mathbb{R}^2$. Elements of the fundamental group $G_{\mathfrak{S}_3} = \pi_1(Y_W)$ are homotopy classes of loops, also known as braids.
• Figure 4: Left: the interval $[1, \delta]_S$ for the symmetric group $\mathfrak{S}_3$. The edges are labeled by the simple reflections $a$ and $b$. The longest element is given by $\delta = aba = bab$. Right: the labeled order complex of $[1, \delta]_S$. Its quotient $K$ is a classifying space for the braid group $G_{\mathfrak{S}_3}$ and consists of the following simplices: the $0$-simplex $[\,]$; the five $1$-simplices $[a]$, $[b]$, $[ab]$, $[ba]$, and $[\delta]$; the six $2$-simplices $[a|b]$, $[b|a]$, $[a|ba]$, $[ab|a]$, $[ba|b]$, and $[b|ab]$; the two $3$-simplices $[a|b|a]$ and $[b|a|b]$.
• Figure 5: Left: the "dual" interval $[1, w]_R$ for the symmetric group $\mathfrak{S}_3$ with $w = ab$ as the chosen Coxeter element. The edges are labeled by the three reflections $a,b,c$ (note that the third reflection $c$ is equal to the longest element $\delta = aba = bab$, but this is just a coincidence). Right: the labeled order complex of $[1, w]_R$. Its quotient $K$ is a classifying space for the (dual) braid group $G_{\mathfrak{S}_3}$ and consists of the following simplices: the $0$-simplex $[\,]$; the four $1$-simplices $[a]$, $[b]$, $[c]$, and $[w]$; the three $2$-simplices $[a|b]$, $[b|c]$, and $[c|a]$.

Theorems & Definitions (6)

• Conjecture 1.1: $K(\pi, 1)$ conjecture
• Lemma 2.1
• proof
• Lemma 3.4
• proof
• Theorem 6.1: Delucchi-Paolini-Salvetti delucchi2022dual