Coxeter Quotients of Automorphism Groups of Coxeter Groups

Definitions and mathematical statement

A Coxeter system consists of a group $W$ with a generating set $S$ and a presentation \[ W=\left\langle S\ \middle|\ (st)^{m(s,t)}=1\text{ whenever }m(s,t)<\infty\right\rangle, \] where $m(s,s)=1$ and $m(s,t)=m(t,s)\in\{2,3,\ldots\}\cup\{\infty\}$ for distinct $s,t\in S$. The value $m(s,t)=\infty$ imposes no relation on $st$. A Coxeter group is a group admitting such a presentation. Its rank relative to this presentation is $|S|$. Write $\operatorname{Aut}(W)$ for its group of abstract automorphisms. A group $G$ virtually maps onto a group $V$ if there are a subgroup $H\leq G$ of finite index and a surjective homomorphism $H\twoheadrightarrow V$.

The supplied conjecture asks whether, for every infinite Coxeter group $W_\Gamma$, there exist an infinite Coxeter group $V$, a subgroup $H\leq\operatorname{Aut}(W_\Gamma)$, and a surjective homomorphism $\varphi:H\twoheadrightarrow V$ such that \[ [\operatorname{Aut}(W_\Gamma):H]<\infty. \] The group $V$, the subgroup $H$, and the homomorphism $\varphi$ may depend on $W_\Gamma$.

The source does not specify whether $\Gamma$ must be finite, or equivalently whether the displayed Coxeter generating set must be finite. Thus the intended rank restriction remains unresolved; the definition above accommodates both finite and infinite rank. The supplied label corrected_verified and the supplied open status are registry annotations, not independent mathematical verification. Moreover, the accompanying literature assessment concerns a different subject and cannot establish the status of this Coxeter-group conjecture.

Short English statement

Does the automorphism group of every infinite Coxeter group have a subgroup of finite index that surjects onto some infinite Coxeter group? The intended restriction on Coxeter rank is unspecified. The supplied open status requires review because the attached literature assessment discusses a different problem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004454$, title Coxeter Quotients of Automorphism Groups of Coxeter Groups, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2020; no individual proposer is supplied. The record labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. These are supplied registry claims.
• Supplied report citation: Geometric Structures in Group Theory (2021), Oberwolfach Reports, pp. 877--918, DOI 10.4171/owr/2020/16; also labeled Oberwolfach Report 16/2020. https://doi.org/10.4171/owr/2020/16. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied 2021 citation and 2020 report/proposal labels are retained without resolving their bibliographic relationship.
• Supplied literature reference: Hyde--Moore (2023). https://doi.org/10.4171/GGD/708. No title or further bibliographic details are supplied.
• Literature checked at: 2026-08-22. This is supplied dated curation, not independently verified source consultation or a new literature assessment. The supplied status is open. The assessment describes several conjectures about the subgroup quasiorder of PL-positive-I, including whether $F$ is a bottleneck and whether $F$-obstructions are complete; it says that later work proves strong obstruction theorems while leaving converse and classification questions open. This assessment is attributed to the supplied OWR curation and explicitly calls for independent verification of time-sensitive status and cited claims. Its relationship to the Coxeter-group statement is unexplained.