research/AMR-010-0209.md

metadata

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 (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: "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 — 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 — 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; 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 — 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 — 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 (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; R. Boyd, An introduction to the geometric and combinatorial group theory of Artin groups, arXiv (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.