Hyperbolic Three-Manifold Groups in $\operatorname{SL}_3(\mathbb Z)$

Definitions and mathematical statement

Write \[ \operatorname{SL}_3({\mathbb{Z}})=\{A\in\operatorname{Mat}_{3\times3}({\mathbb{Z}}):\det A=1\}. \] Under the usual convention, a hyperbolic $3$-manifold is a connected, complete Riemannian $3$-manifold without boundary whose sectional curvature is constantly $-1$. Finite volume means $\operatorname{Vol}(M)<\infty$. For a base point $m_0\in M$, let $\pi_1(M,m_0)$ denote its fundamental group.

Does there exist a finite-volume hyperbolic $3$-manifold $M$ and a group homomorphism \[ \rho:\pi_1(M,m_0)\longrightarrow\operatorname{SL}_3({\mathbb{Z}}) \qquad\text{such that}\qquad \ker\rho=\{1\}? \] The existence of such a homomorphism is independent of the choice of base point. The supplied statement imposes no orientability or compactness restriction; both compact and noncompact finite-volume manifolds are within the stated scope. Completeness and absence of boundary are included here as the usual hyperbolic-manifold convention, but are not explicitly specified in the supplied wording and require confirmation against the report.

Short English statement

Does some finite-volume hyperbolic $3$-manifold have a fundamental group that embeds in the group of determinant-one $3\times3$ integer matrices? The supplied record labels this question open; that status is not independently verified here.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; supplied record with permanent UnsolvedMath ID $30003299$, source problem code catalogue entry. No individual proposer is identified; the supplied proposed year is 2016.
• Supplied report citation: Surface Bundles (2017), Oberwolfach Reports, pp. 3149--3195, DOI 10.4171/owr/2016/56. Literature-source label: OWR 56 (2017). https://doi.org/10.4171/owr/2016/56.
• Supplied literature reference: Long--Reid--Thistlethwaite, small subgroups of $\operatorname{SL}_3({\mathbb{Z}})$. https://web.math.ucsb.edu/~long/pubpdf/smallsubgroups2.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the existence question remains posed as a small-subgroups question, with no example or obstruction found in that review. This is supplied dated curation derived from the Oberwolfach material, not an independently verified assessment of the literature.
• The supplied statement status is corrected_verified; its verification note reports comparison with the official Oberwolfach report and preservation of its hypotheses, notation, and target. This is a supplied verification claim; no source was independently consulted for this draft.