The $K(\pi,1)$-conjecture for Artin groups

Equivalent formulations 6

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The $K(\pi,1)$ conjecture for Artin groups

   Let $M$ be a Coxeter matrix, let $A$ be the corresponding Artin group, and let $N(M)$ be the topological space associated with a representation of the corresponding Coxeter group. A classifying space for $A$ is a space whose fundamental group is $A$ and whose universal cover is contractible. The $K(\pi,1)$ conjecture. The space $N(M)$ is a classifying space for the Artin group $A$ corresponding to $M$. The space $N(M)$ is homotopy equivalent to the Salvetti complex, so the conjecture is equivalently that the corresponding Salvetti complex is a classifying space for $A$. It is a central open question concerning the topology of Artin groups; the supplied text gives no resolution.

   source: Giovanni Paolini, “On the classifying space of Artin monoids”, arXiv:1511.02062 (2018).

2. The $K(\pi, 1)$ conjecture for Artin groups

   Let $W$ be a Coxeter group acting on its Tits cone $I\subseteq\mathbb{R}^n$, and let $\mathcal{A}$ be the set of fixed hyperplanes of reflections of $W$. Define

   $$Y=(I\times I)\setminus\bigcup_{H\in\mathcal{A}}(H\times H).$$

   The $K(\pi, 1)$ conjecture. The space $Y$ is a $K(\pi,1)$ space.

   This conjecture asserts that the topological space associated with the Coxeter group's action on the Tits cone is a classifying space for the corresponding Artin group. It is known for finite and affine Coxeter groups, while the general case remains open.

   source: Giovanni Paolini, “The dual approach to the K(π, 1) conjecture”, arXiv:2112.05255 (2025).

3. The $K(\pi,1)$-conjecture for Artin groups

   Let $(W,S)$ be a Coxeter system, let $A$ be the associated Artin group, and let $\mathcal H(W)$ be the complexified hyperplane arrangement associated to $W$. The group $W$ acts on $\mathcal H(W)$, and consider the orbit space $\mathcal H(W)/W$.

   The $K(\pi,1)$-conjecture. The orbit space $\mathcal H(W)/W$ is a $K(\pi,1)$ for the Artin group $A$ associated to $W$.

   The paper proves that this conjecture implies the center conjecture for Artin groups: every Artin group without a spherical factor that satisfies the $K(\pi,1)$-conjecture has trivial center. The conjecture is known for FC-type and $2$-dimensional Artin groups, but remains open in general.

   source: Kasia Jankiewicz and Kevin Schreve, “The K(π,1)-conjecture implies the center conjecture for Artin groups”, arXiv:2201.06591 (2022).

4. The $K(\pi,1)$ conjecture for Artin groups

   Let $(W,S)$ be a Coxeter system, let $G_W$ be its associated Artin group, and let $Y_W$ be the orbit configuration space associated with $W$. A classifying space for $G_W$ is a space whose fundamental group is $G_W$ and whose higher homotopy groups vanish. The $K(\pi,1)$ conjecture. The orbit configuration space $Y_W$ is a classifying space for the corresponding Artin group $G_W$. This conjecture is a central open problem for general Artin groups, although it has been proved for spherical Artin groups by Deligne and in the rank-three cases treated in this paper.

   source: Emanuele Delucchi, Giovanni Paolini and Mario Salvetti, “Dual structures on Coxeter and Artin groups of rank three”, arXiv:2206.14518 (2025).

5. The $K(\pi,1)$-Conjecture for Artin groups

   Let $W$ be a Coxeter group and $A$ its associated Artin group. Let $Y_W$ be the corresponding hyperplane complement, with the action of $W$ on $Y_W$ as above. A CW-complex is aspherical if its only nontrivial homotopy group is its fundamental group, and a $K(A,1)$ space is an aspherical CW-complex with fundamental group $A$.

   The $K(\pi,1)$-Conjecture. The quotient $Y_W/W$ is aspherical with fundamental group $A$; equivalently, it is a $K(A,1)$ space.

   The conjecture is one of the central open problems about Artin groups. It is known for spherical Artin groups, right-angled Artin groups, Artin groups of FC-type, and two-dimensional Artin groups. An equivalent formulation uses the Salvetti complex as a model for $K(A,1)$.

   source: Jone Lopez de Gamiz Zearra and Conchita Martínez Pérez, “Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case”, arXiv:2405.16641 (2025).

6. The $K(\pi,1)$ conjecture for Artin groups

   Let $\Gamma$ be a finite-type Coxeter graph with associated Artin group $A_\Gamma$, Coxeter group $W_\Gamma$, and complexified hyperplane complement $\mathcal H_\Gamma$. The quotient $\mathcal H_\Gamma/W_\Gamma$ has fundamental group $A_\Gamma$. The $K(\pi,1)$ conjecture. The space $\mathcal H_\Gamma/W_\Gamma$ is a classifying space for $A_\Gamma$, or a $K(A_\Gamma,1)$ space. This is one of the central conjectures in the theory of Artin groups; it is known for many classes of groups but remains open in general.

   source: Rachael Boyd, “An introduction to the geometric and combinatorial group theory of Artin groups”, arXiv:2601.08658 (2026).