Profinite Rigidity of Coxeter Groups

Definitions and mathematical statement

A Coxeter system is specified by a set $S$ and a symmetric matrix $m_{st}$ with $m_{ss}=1$ and $m_{st}\in\{2,3,4,\ldots\}\cup\{\infty\}$ for distinct $s,t\in S$. Its Coxeter group is \[ W(S,m)=\left\langle S\ \middle|\ s^2=1\ (s\in S),\ (st)^{m_{st}}=1\ (s\ne t,\ m_{st}<\infty)\right\rangle. \] An entry $m_{st}=\infty$ imposes no relation on $st$. In the graph notation of the supplied excerpt, a nonedge has $m_{st}=2$, and an edge whose label is omitted has label $3$. Write $W_\Gamma$ for the group specified by such a labelled graph $\Gamma$.

For a group $G$, let \[ \mathcal N_G=\{N\trianglelefteq G:[G:N]<\infty\}. \] Its profinite completion is \[ \widehat G=\varprojlim_{N\in\mathcal N_G}G/N, \] where each finite quotient has the discrete topology and the inverse limit has the induced topology from their product. An isomorphism of profinite completions is understood here to be an isomorphism of topological groups.

Let $\mathcal W$ denote the class described in the supplied statement as all Coxeter groups. A group $W\in\mathcal W$ is $\mathcal W$-profinitely rigid if, for every $W'\in\mathcal W$, an isomorphism $\widehat W\cong\widehat{W'}$ of profinite groups implies an isomorphism $W\cong W'$ of abstract groups.

Question. Is every Coxeter group $\mathcal W$-profinitely rigid? Equivalently, is the following implication valid for every pair $W_\Gamma,W_\Lambda\in\mathcal W$? \[ \widehat{W_\Gamma}\cong\widehat{W_\Lambda} \quad\Longrightarrow\quad W_\Gamma\cong W_\Lambda. \] The conclusion concerns the groups, without requiring an isomorphism to preserve their specified Coxeter generators or labelled graphs.

Unresolved scope qualification. The supplied statement says all Coxeter groups but does not explicitly specify whether the Coxeter generating sets must be finite. The definition above permits arbitrary sets $S$, which gives the literal unrestricted reading. If the intended convention is finite rank, the question must instead quantify only over systems with finite generating sets. The supplied record does not resolve this distinction.

Short English statement

Does the profinite completion determine a Coxeter group up to group isomorphism among Coxeter groups? The supplied record labels the problem partially solved, but that status is not independently verified here. Whether the intended question is restricted to finite-rank Coxeter groups remains unclear.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30006199$, title Profinite Rigidity of Coxeter Groups, exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2025$; no proposer is named.
• Supplied primary report citation: Geometric Structures in Group Theory (2025), Oberwolfach Reports, pp. 275--338, DOI 10.4171/owr/2025/6. https://doi.org/10.4171/owr/2025/6. This report is also listed under the supplied literature-source label "OWR report."
• Supplied literature source labelled "Affine Coxeter rigidity": https://doi.org/10.1007/s00209-025-03799-3.
• Supplied literature source labelled "Profinite properties of Coxeter groups": https://arxiv.org/abs/2505.08701.
• Supplied literature assessment, checked on 2026-08-22: the general $\mathcal W$-profinite-rigidity question is described as open, with affine, right-angled, minimal non-spherical, triangle, and other subclasses described as proved rigid. The registry status is partially_solved. This is supplied dated curation derived from the OWR material, not an independently verified assessment; the precise hypotheses and comparison classes of those subclass results require checking.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied verification claims. The original extracted passage has damaged presentation notation and an incomplete description of profinite completion; it also describes the isomorphism problem among Coxeter groups as open. That historical description is preserved as a source claim, without independent verification.