---
id: AMR-010-0209
classification: OPEN-TRIAGE
wording_corrected: no
---

# AMR-010-0209 — Does every Artin group have a finite K(G,1)?

## Problem (corrected statement if needed)

The dataset transcription matches the source exactly; no correction was needed. The
original wording, from Mladen Bestvina's "Questions in Geometric Group Theory"
(updated July 2004), Question 2.9, reads:

> **Q 2.9.** Does every Artin group have a finite $K(G,1)$?
> *Yes for Artin groups of finite type (meaning that the associated Coxeter group is finite) by the work of [Del72].*

Source: [questions-updated.pdf](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf)
(verified directly; the question appears in Section 2, "CAT(0) groups", PDF page 8).

Here an Artin group is given by generators $s_1,\dots,s_n$ with relations
$\underbrace{s_i s_j s_i \cdots}_{m_{ij}\ \text{factors}} = \underbrace{s_j s_i s_j \cdots}_{m_{ij}\ \text{factors}}$
for $m_{ij} \in \{2,3,\dots,\infty\}$ encoded by a Coxeter matrix/diagram, and
"finite $K(G,1)$" means a classifying space that is a finite CW complex.

## Status / Literature

**Open in general.** This is a weak form of (and is implied by) the famous
$K(\pi,1)$ conjecture for Artin groups, attributed to Arnol'd, Brieskorn, Pham and
Thom, which remains unresolved for general Artin groups as of 2024–2025 survey
literature (see the Oberwolfach report [Boyd–Heng–Ozornova, OWR 21 (2024), 203–234](https://ems.press/journals/owr/articles/14298160):
"the $K(\pi,1)$-conjecture for Artin groups remains open except for certain special
families"). The relation to the question asked here is explained below.

Verified known cases (all checked against Crossref/arXiv):

- **Finite (spherical) type: YES.** P. Deligne, *Les immeubles des groupes de
  tresses généralisés*, Invent. Math. 17 (1972), 273–302,
  [doi:10.1007/BF01406236](https://doi.org/10.1007/bf01406236) — verified via
  Crossref. This is the "[Del72]" cited in Bestvina's own remark.
- **Right-angled Artin groups: YES** via the Salvetti complex (a finite CW complex,
  the "Salvetti blow-up" of the standard presentation complex); M. Salvetti,
  *The homotopy type of Artin groups*, Math. Res. Lett. 1 (1994), 565–577,
  [doi:10.4310/MRL.1994.v1.n5.a5](https://doi.org/10.4310/MRL.1994.v1.n5.a5) —
  bibliographic data verified through the Crossref-verified reference list of
  Paolini–Salvetti (below).
- **Large type: YES.** H. Hendriks, *Hyperplane complements of large type*,
  Invent. Math. 79 (1985), 375–381,
  [doi:10.1007/BF01388979](https://doi.org/10.1007/BF01388979); and independently
  K. Appel–P. Schupp, *Artin groups and infinite Coxeter groups*, Invent. Math. 72
  (1983), 201–220, doi:10.1007/BF01389320 — both verified through the
  Crossref-verified reference lists of Charney–Davis and Paolini–Salvetti.
- **FC type and 2-dimensional Artin groups: YES.** R. Charney–M. Davis, *The
  $K(\pi,1)$-problem for hyperplane complements associated to infinite reflection
  groups*, J. Amer. Math. Soc. 8 (1995), 597–627,
  [doi:10.1090/S0894-0347-1995-1303028-9](https://doi.org/10.1090/s0894-0347-1995-1303028-9)
  — verified via Crossref.
- **Affine type: YES.** G. Paolini–M. Salvetti, *Proof of the $K(\pi,1)$ conjecture
  for affine Artin groups*, Invent. Math. 224 (2021), 487–572,
  [doi:10.1007/s00222-020-01016-y](https://doi.org/10.1007/s00222-020-01016-y) —
  verified via Crossref (abstract confirms: "We prove the $K(\pi,1)$ conjecture for
  affine Artin groups").
- **Further recent progress:** J. Huang, *Cycles in spherical Deligne complexes and
  application to $K(\pi,1)$-conjecture for Artin groups*,
  [arXiv:2405.12068](https://arxiv.org/abs/2405.12068) (2024) proves the conjecture
  for all 3-dimensional hyperbolic-type Artin groups except one example, for
  quasi-Lannér hyperbolic types up to dimension 4, and for complete bipartite Coxeter
  diagrams — verified via arXiv.
- Surveys: L. Paris, *$K(\pi,1)$ conjecture for Artin groups*, Ann. Fac. Sci.
  Toulouse 23 (2014), 361–415, [doi:10.5802/afst.1411](https://www.numdam.org/item/10.5802/afst.1411.pdf);
  R. Boyd, *An introduction to the geometric and combinatorial group theory of Artin
  groups*, [arXiv](https://arxiv.org/html/2601.08658v1) (survey written January 2024).

## Work done

I verified the source wording directly against Bestvina's PDF, then verified each
key citation against Crossref records or arXiv. On the mathematical side, the
useful rigorous content I can contribute is a precise statement of the reduction
and why the question is hard:

1. **Van der Lek / Salvetti reduction.** By van der Lek's thesis (Nijmegen, 1983),
   every Artin group $A_\Gamma$ is the fundamental group of the quotient
   $X_\Gamma/W_\Gamma$ of the complement of the complexified Coxeter hyperplane
   arrangement of the associated Coxeter group $W_\Gamma$. Salvetti (1987, 1994)
   constructed an explicit **finite** CW complex $\mathrm{Sal}(\Gamma)$ (now called
   the Salvetti complex), with one $k$-cell per subset of $k$ generators whose
   parabolic Coxeter subgroup is finite, which is a homotopy model for
   $X_\Gamma/W_\Gamma$; in particular $\pi_1(\mathrm{Sal}(\Gamma)) \cong A_\Gamma$.

2. **Hence the following are equivalent / related:**
   - ($K(\pi,1)$ conjecture) $X_\Gamma/W_\Gamma$ is aspherical;
   - $\mathrm{Sal}(\Gamma)$ is aspherical, i.e. is itself a $K(A_\Gamma,1)$;
   - (Bestvina's Q 2.9) $A_\Gamma$ has *some* finite $K(G,1)$.
   The conjecture $\Rightarrow$ Q 2.9, since $\mathrm{Sal}(\Gamma)$ is finite.
   Whether Q 2.9 is strictly weaker is itself unknown; no Artin group is known to
   have a finite $K(G,1)$ without $\mathrm{Sal}(\Gamma)$ being aspherical, and no
   counterexample is known in either direction.

3. **Why the general case resists attack.** The obstructions are algebraic as much
   as topological: outside the Garside realm (finite/affine type, where the Artin
   monoid embeds in the group and yields finite classifying spaces via Bestvina's
   normal form complex — cf. Charney–Meier–Whittlesey, Geom. Dedicata 105 (2004),
   171–188) and the FC-type/Deligne-complex methods of Charney–Davis, there is no
   known contractible complex with a cocompact $A_\Gamma$-action. In particular,
   even the following weaker consequences of a positive answer are **open in
   general**: (a) every Artin group is torsion-free; (b) every Artin group has
   finite cohomological dimension. This shows Bestvina's question is genuinely at
   the frontier — it cannot currently be settled even in its weakest corollaries.

4. **Attempt at direct progress.** I considered whether one could attack Q 2.9
   without the full $K(\pi,1)$ conjecture, e.g. by exhibiting a finite-dimensional
   contractible complex with free cocompact $A_\Gamma$-action other than the
   universal cover of the Salvetti complex, or by an inductive scheme over parabolic
   subgroups (adding one generator at a time, using that amalgamated products over
   parabolic subgroups with finite $K(\pi,1)$'s have finite-dimensional classifying
   spaces). The obstruction is that $A_\Gamma$ is not known to decompose as such an
   amalgam along inclusions that induce $K(\pi,1)$-preserving pushouts: the
   required asphericity of the relevant pushout spaces is exactly the content of
   the $K(\pi,1)$ conjecture for $\Gamma$ (this is essentially the Charney–Davis
   "union of chambers" criterion, which needs the Deligne complex to be
   CAT(1)-like / the complexes of groups to be developable — unknown in general).
   So no unconditional progress beyond the known families seems available by these
   routes, consistent with the literature.

## Result

**OPEN-TRIAGE.** The question is open in general. It is answered affirmatively for
the following verified families of Artin groups: finite type (Deligne 1972),
right-angled (Salvetti 1987/1994), large type (Appel–Schupp 1983; Hendriks 1985),
FC type and 2-dimensional (Charney–Davis 1995), affine type (Paolini–Salvetti
2021), and various hyperbolic-type and bipartite-diagram classes (Huang 2024).
For a general Artin group, neither a finite $K(G,1)$ nor even torsion-freeness or
finite cohomological dimension is known. The question is implied by, and widely
regarded as essentially equivalent in difficulty to, the $K(\pi,1)$ conjecture
for Artin groups.

## What remains

- The general case: prove or disprove that every Artin group has a finite
  $K(G,1)$ — equivalently, decide asphericity of the Salvetti complex for an
  arbitrary Coxeter diagram, or find a counterexample.
- Even weaker open targets: torsion-freeness of all Artin groups; finite
  cohomological dimension of all Artin groups; whether Bestvina's question is
  strictly weaker than the $K(\pi,1)$ conjecture.
- The single remaining 3-dimensional hyperbolic-type exception in Huang's 2024
  result, and higher-dimensional hyperbolic types beyond the quasi-Lannér range.
- Verification caveat: the Appel–Schupp, Hendriks, Salvetti, and van der Lek items
  were confirmed via the Crossref-verified reference lists of Deligne- and
  Charney–Davis-level sources rather than by fetching each DOI record directly
  (budget constraint); their publication data quoted here comes from those records.