Profinite Invariance among Finite-Rank Coxeter Groups

Definitions and mathematical statement

A finite-rank Coxeter group is a group admitting a presentation \[ W=\left\langle S\ \middle|\ s^2=1\ (s\in S),\ (st)^{m_{st}}=1\ (s\ne t,\ m_{st}<\infty)\right\rangle, \] where $S$ is finite and $m_{st}=m_{ts}\in\{2,3,\ldots\}\cup\{\infty\}$. When $m_{st}=\infty$, no relation on $st$ is imposed. Finite rank refers to the existence of such a finite generating set; it does not require $W$ to be finite.

For a group $G$, its profinite completion is the topological group \[ \widehat G=\varprojlim_{\substack{N\trianglelefteq G\\ [G:N]<\infty}}G/N, \] with its inverse-limit topology, each finite quotient being discrete. Isomorphisms of profinite completions below are isomorphisms of topological groups.

Let $\mathcal C$ denote the class of finite-rank Coxeter groups. For an invariant $I$ defined on $\mathcal C$, being determined by the profinite completion within $\mathcal C$ means that \[ \forall W_1,W_2\in\mathcal C,\qquad \widehat{W_1}\cong\widehat{W_2} \ \Longrightarrow\ I(W_1)\simeq I(W_2), \] where $\simeq$ denotes equality or the appropriate isomorphism relation for the values of $I$. For a property $P$, the corresponding condition is \[ \forall W_1,W_2\in\mathcal C,\qquad \widehat{W_1}\cong\widehat{W_2} \ \Longrightarrow\left(P(W_1)\Longleftrightarrow P(W_2)\right). \] No equality of the ranks of $W_1$ and $W_2$ is assumed here.

Problem (supplied clean formulation). Determine which homological and cohomological properties or invariants of finite-rank Coxeter groups are determined by their profinite completions.

Unresolved scope. This is a broad research program, not a specified assertion about a particular invariant. The supplied record does not specify coefficients, module actions, homological degrees, particular invariants, or the equivalence relation intended for their values. The displayed implications give a precise interpretation for comparison within finite-rank Coxeter groups; the supplied question does not explicitly settle whether comparison with a broader class of groups is also intended. The literal extracted statement contains only a problem label and a page marker, so it cannot itself resolve these qualifications.

Short English statement

The supplied question asks which homological and cohomological features must agree for finite-rank Coxeter groups whose profinite completions are isomorphic as topological groups. It leaves the particular features and coefficient choices unspecified. This is a statement-only draft of a broad program requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30005698$, title Profinite Invariance among Finite-Rank Coxeter Groups, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Homological aspects of TDLC-groups (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/55, pp. 3103--3138. https://doi.org/10.4171/owr/2023/55. The literal extracted text reads "We pose: Problem 10. Mini-Workshop: Homological aspects of TDLC-groups 3123" and supplies no mathematical formulation beyond that label.
• Supplied literature source: Oberwolfach report, https://ems.press/content/serial-article-files/48166. The literature assessment dated 2026-08-22 describes Problem 10 as a broad open program on profinite cohomological invariants of finite-rank Coxeter groups. This is supplied dated curation, not an independently verified assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The record labels the problem open and the clean statement corrected_verified, and reports that it was checked against the official report. These are supplied curation claims, not verification performed for this draft. The source citation gives 2024, whereas the proposed year, year tag, and DOI contain 2023; their relationship is not explained in the supplied record.