Euler Characteristic and Coherence of Random Few-Relator Groups

Definitions and mathematical statement

A group $H$ is coherent if every finitely generated subgroup of $H$ is finitely presented. It is incoherent if it has a finitely generated subgroup that is not finitely presented.

Write \[ G(n,m,\ell)=\langle x_1,\ldots,x_n\mid r_1,\ldots,r_m\rangle \] for the random presentation in the supplied few-relators model, with the numbers $n$ of generators and $m$ of relators held fixed as the length parameter $\ell$ tends to infinity. The supplied record does not specify the admissible parameter pairs, the distribution of the relators, or the precise meaning of the length parameter; these must be fixed before the probabilities below are fully defined.

Conjecture as supplied, with unresolved model qualifications. For every admissible fixed parameter pair $(n,m)$, let $\chi$ be the Euler characteristic intended by the source. The proposed limits are \[ \begin{aligned} \chi\leq 0&\quad\Longrightarrow\quad \lim_{\ell\to\infty}\Pr\bigl(G(n,m,\ell)\text{ is coherent}\bigr)=1,\\ \chi>0&\quad\Longrightarrow\quad \lim_{\ell\to\infty}\Pr\bigl(G(n,m,\ell)\text{ is incoherent}\bigr)=1. \end{aligned} \] In particular, the boundary case $\chi=0$ belongs to the coherence assertion.

The presentation complex, consisting of one vertex, $n$ edges, and $m$ two-dimensional cells, has Euler characteristic $1-n+m$. If it is a classifying space for the presented group, this is also the group Euler characteristic. The record does not specify whether $\chi$ means this presentation Euler characteristic or the group Euler characteristic, nor does it provide hypotheses ensuring their identification. Thus substituting $\chi=1-n+m$ into the conjecture requires clarification of that convention or justification of the relevant classifying-space hypothesis. The displayed limits express the intended target but remain qualified by these missing specifications.

Short English statement

The conjecture predicts coherence with probability tending to one for nonpositive Euler characteristic, including zero, and incoherence with probability tending to one for positive Euler characteristic, as the relator length parameter grows with the numbers of generators and relators fixed. The sampling model, admissible parameters, and Euler characteristic convention require clarification.

Sources

• Supplied local problem-registry record, permanent UnsolvedMath ID $30004453$, exact source problem code catalogue entry, title: Euler Characteristic and Coherence of Random Few-Relator Groups. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2020; no individual proposer is identified.
• Supplied source citation: Geometric Structures in Group Theory (2021), Oberwolfach Reports, pp. 877--918, DOI 10.4171/owr/2020/16. https://doi.org/10.4171/owr/2020/16. The literature-source list also identifies this reference as Oberwolfach Report 16/2020.
• Supplied literature reference: Kielak--Kropholler--Wilkes (2020). https://arxiv.org/abs/2003.06354. No paper title, theorem numbers, or detailed parameter hypotheses are supplied.
• Supplied literature assessment, checked on 2026-08-22: the registry labels the problem partially solved and reports coherence with high probability for negative Euler characteristic, several positive-probability coherence cases at zero Euler characteristic, and asymptotic incoherence for two-generator positive Euler characteristic. It describes the full all-parameters conjecture as remaining open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the status and cited claims require independent verification.