Smallest nonabelian quotient conjecture for Artin groups

Let $A_\Gamma$ be an Artin group and let $W_\Gamma$ be its associated Coxeter group, obtained from $A_\Gamma$ by imposing $s_i^2=1$ for every standard generator $s_i$ . For a group $G$ admitting finite non-abelian quotients, let $m(G)=\min\{|Q|:Q\text{ is a finite non-abelian quotient of }G\}$ . The conjecture asserts that the isomorphism classes of finite non-abelian quotients $Q$ of $A_\Gamma$ with $|Q|=m(A_\Gamma)$ are exactly the isomorphism classes of finite non-abelian quotients $Q$ of $W_\Gamma$ with $|Q|=m(W_\Gamma)$ . Equivalently, a smallest finite non-abelian quotient of $A_\Gamma$ is isomorphic to a smallest finite non-abelian quotient of $W_\Gamma$ .

References

Primary source

arXiv

Additional references

Finite quotients of spherical Artin groups — arXiv — Sam Hughes, Thomas Ng, Kaitlin Ragosta, Nancy Scherich, Yvon Verberne

geometric-group-theory group-theory

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A September 2026 preprint claims substantial progress for spherical and affine Artin groups, but the conjecture remains unresolved for arbitrary Artin groups.

The conjecture predicts that the smallest nonabelian finite quotients of Artin groups are determined by their Coxeter quotients. The reported result addresses spherical and affine groups, not the full class of Artin groups.

September 2026 spherical and affine progress

Hughes, Ng, Ragosta, Scherich, and Verberne report a proof of the conjectured identification for spherical and affine Artin groups. A related preprint claims explicit smallest-quotient calculations for irreducible spherical groups and derives profinite rigidity among spherical-type Artin groups. These are claims in arXiv preprints, without independent verification or referee assessment in the retrieved sources.

Current status (as of September 2026): the conjectured identification is claimed, but unverified, for spherical and affine Artin groups, while the case of arbitrary Artin groups remains open.

Sources

No solutions have been posted yet.