A question in Oberwolfach Report 8/2020 asks whether every hyperbolic knot without finite surgery has a rank-two free subgroup that injects under every rational Dehn filling. The figure-eight knot gives a negative answer. Its zero-surgery group is the metabelian group Z^2 semidirect_A Z, where A has rows (1,1) and (1,2). More precisely, the zero-filling kernel equals the second derived subgroup of the knot group. Hence the nontrivial second derived subgroup of every rank-two free subgroup is killed by zero filling. An explicit length-eighteen word witnesses the obstruction.

The note combines standard figure-eight surgery facts with elementary group theory. It answers the universal formulation recorded as OWR-17471-010 in ulamai/UnsolvedMath version 1.6.0, not the separate existence question for some other hyperbolic knot or a variant excluding zero surgery. Classical inputs are attributed; no absolute priority claim is made.

Alper Ferudun, Mercury Software GmbH. English unrefereed preprint, with portable LaTeX source, written proof audit and executable exact diagnostics. AI tools assisted the author with research, checks and drafting. The work is self-audited, not independently reviewed or formally verified. CC BY 4.0.
