Decidability of Subgroup Membership in SL3 of the Integers

Definitions and mathematical statement

Let \[ \operatorname{SL}(3,{\mathbb{Z}})=\{A\in M_3({\mathbb{Z}}):\det A=1\}. \] For a finite list $A_1,\ldots,A_k\in\operatorname{SL}(3,{\mathbb{Z}})$, let \[ H=\langle A_1,\ldots,A_k\rangle =\{A_{i_1}^{\varepsilon_1}\cdots A_{i_m}^{\varepsilon_m}:m\geq0,\ i_j\in\{1,\ldots,k\},\ \varepsilon_j\in\{-1,1\}\}. \] The empty product is the identity matrix; an empty generating list generates the trivial subgroup. Matrix entries are given exactly as integers.

Question. Does there exist a single algorithm which, for every finite generating list $A_1,\ldots,A_k\in\operatorname{SL}(3,{\mathbb{Z}})$ and every $B\in\operatorname{SL}(3,{\mathbb{Z}})$, terminates and correctly decides whether \[ B\in\langle A_1,\ldots,A_k\rangle? \] The restriction to finitely generated subgroups is explicit in the supplied background assessment, although absent from the short source question. The formulation above interprets decidability uniformly over generating lists. The supplied short question does not explicitly distinguish this uniform formulation from decidability separately for every fixed finitely generated subgroup; that distinction requires source review. The registry labels the problem open, but that label and its dated literature assessment are supplied curation, not independently verified conclusions.

Short English statement

Given finitely many determinant-one integer $3\times3$ matrices and another such matrix, can one algorithm always decide whether the latter belongs to the subgroup generated by the former? The intended uniformity should be confirmed against the source.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003188$, source problem code catalogue entry, title: Decidability of Subgroup Membership in SL3 of the Integers. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2016.
• Supplied source citation: Computational Group Theory (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/37, pp. 2123--2169. https://doi.org/10.4171/owr/2016/37. The citation's 2017 label and the registry's 2016 proposed year are retained without resolving their relationship.
• Supplied literature reference: Lohrey, subgroup membership. https://doi.org/10.1007/s00224-023-10122-2.
• Supplied literature reference: Detinko--Flannery--Hulpke. https://arxiv.org/abs/1706.04166.
• Supplied literature reference: Oberwolfach Report 37. https://doi.org/10.4171/owr/2016/37.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes decidability for arbitrary finitely generated subgroups of $\operatorname{SL}(3,{\mathbb{Z}})$ as a long-standing open problem. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment.
• The registry assigns the statement status corrected_verified and reports that the official problem-session summary was checked and directly states the decidability question. This records the supplied verification claim; no independent source consultation is asserted here.