Finding Free Subgroups Containing a Given Matrix

Definitions and mathematical statement

For notation, a linear group is a subgroup $G\leq \mathrm{GL}_n(K)$, where $K$ is a field and $n\geq 1$. A free basis of a subgroup $H\leq G$ is a subset $B\subseteq H$ that generates $H$ and for which every nonempty reduced word in the formal letters $b^{\pm1}$, $b\in B$, represents a nonidentity element. A subgroup is free if it has such a basis.

The supplied problem asks for an algorithm that, given a linear group $G$ and a matrix $M\in G$, decides whether \[ \exists H\leq G:\qquad M\in H,\qquad H\neq\{1\},\qquad H\text{ is free}, \] and, in an affirmative case, constructs a free basis $B$ of a witnessing subgroup $H$. Membership of $M$ in $H$ is required; the statement does not require $M$ to be a basis element or exclude $M=1$.

Unresolved qualifications. The displayed formulation reads "nontrivial free subgroup" literally, allowing free groups of rank one. If the intended requirement is a nonabelian free subgroup, the condition must instead require a free basis with at least two elements. The supplied record does not resolve this distinction. It also does not specify the coefficient field, the permitted dimensions, how the group and its matrix entries are presented effectively, or how a basis is to be output. In particular, finite generation of the input group and finiteness of the output basis are not stated. These missing specifications prevent a fully determined algorithmic formulation.

Short English statement

Given a linear group and a matrix belonging to it, decide whether some nontrivial free subgroup contains that matrix and, if so, produce a free basis of such a subgroup. The meaning of nontrivial and the effective input and output conventions require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30004768$, title "Finding Free Subgroups Containing a Given Matrix," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The recorded proposed year is 2021; no proposer is named.
• Supplied source citation: Computational Group Theory (2022), Oberwolfach Reports, pp. 2027--2087, DOI 10.4171/owr/2021/38. The literature-source label is "Oberwolfach Report 38/2021." https://doi.org/10.4171/owr/2021/38.
• Supplied literature source: Algorithms for the Tits Alternative and Related Problems. https://www.math.auckland.ac.nz/~obrien/research/Tits.pdf. No authors or publication details are supplied for this reference.
• Supplied dated literature curation, checked 2026-08-22: the record labels the problem open and states that known virtually-free rank-two cases do not resolve the stated matrix-input problem in dimensions at least three; it further attributes to the source report the statement that nothing appears to be known there. This assessment is supplied dated curation, not independently verified here, and its time-sensitive status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified and reports that the official report was checked directly and that the cleaned statement preserves the complete mathematical target while restoring definitions and removing extraction artifacts or source-dependent wording. These are supplied verification claims, not verification performed for this draft.