Free Subgroups Avoiding All Dehn-Surgery Kernels

Definitions and mathematical statement

Let $K\subset S^3$ be a hyperbolic knot, meaning that its complement admits a complete hyperbolic metric of finite volume. Let $X_K=S^3\setminus\operatorname{int}N(K)$ be its compact exterior, where $N(K)$ is a tubular neighborhood, and identify $G(K)=\pi_1(S^3\setminus K)$ with $\pi_1(X_K)$.

Use the standard meridian--preferred-longitude coordinates $(\mu,\lambda)$ on the boundary torus. For $r=p/q\in{\mathbb{Q}}$, with $p\in{\mathbb{Z}}$, $q\in{\mathbb{N}}$, and $\gcd(p,q)=1$, the slope $r$ is represented by the unoriented primitive class $p\mu+q\lambda$. Let $s_r\in G(K)$ be the image of a representative of this slope under boundary inclusion, after choosing a path to the basepoint. Define \[ N_r=\langle\!\langle s_r\rangle\!\rangle_{G(K)} =\bigcap\{N\trianglelefteq G(K):s_r\in N\}. \] This normal subgroup is independent of the conjugacy and inversion choices in the representative. Write $X_K(r)$ for the Dehn filling in which the slope $r$ bounds a meridian disk of the attached solid torus.

Interpret the hypothesis that $K$ has no finite surgery in its standard nonmeridional sense: \[ \left|\pi_1(X_K(r))\right|=\infty \qquad\text{for every }r\in{\mathbb{Q}}. \] The meridional slope $\infty=1/0$ is excluded. The supplied record does not explicitly state the slope coordinates or the exclusion implicit in the phrase "no finite surgery"; these conventions require source confirmation. Including meridional filling in that hypothesis would make it impossible, since meridional filling gives $S^3$.

Question. For every hyperbolic knot $K\subset S^3$ satisfying this no-finite-surgery hypothesis, does there exist a subgroup $H\leq G(K)$, isomorphic to the free group $F_2$ on two generators, such that \[ H\cap\left(\bigcup_{r\in{\mathbb{Q}}}N_r\right)=\{1\}? \] Equivalently, the required condition is $H\cap N_r=\{1\}$ for every rational slope $r$. The union here is a set-theoretic union; it is not asserted to be a subgroup.

Short English statement

For every hyperbolic knot with no finite nonmeridional Dehn surgery, the question asks for a free subgroup on two generators such that every nonidentity element survives every rational Dehn filling. The meridional filling is excluded. The supplied record labels the question open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath problem registry, supplied complete record, permanent ID $30004404$, source problem code catalogue entry, entitled "Free Subgroups Avoiding All Dehn-Surgery Kernels." The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2020; no individual proposer is supplied.
• Supplied source citation: Low-dimensional Topology (2021), Oberwolfach Reports, pp. 465--516, DOI 10.4171/owr/2020/8. https://doi.org/10.4171/owr/2020/8. The supplied title year is retained despite the report's 2020 designation.
• Supplied literature source: "Oberwolfach Report 8/2020," https://ems.press/content/serial-article-files/46844.
• Supplied literature source: "Oberwolfach Report 8/2020," https://doi.org/10.4171/owr/2020/8.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open and reports that existing results produce infinitely many cyclic subgroups with the required disjointness, while the rank-two free subgroup question remains open for hyperbolic knots without finite surgery. This is supplied dated curation, not an independently verified literature assessment. The record requests independent verification of time-sensitive status and cited claims.