Futer–Purcell–Schleimer's purely cosmetic surgery conjecture for cusped hyperbolic manifolds
Futer–Purcell–Schleimer's purely cosmetic surgery conjecture for cusped hyperbolic manifolds
Let be a cusped hyperbolic manifold with at least one cusp, and let and be sets of slopes on the cusps. Futer–Purcell–Schleimer's conjecture. If the fillings and are purely cosmetic, then admits an orientation-preserving symmetry sending to . Here, cosmetic means homeomorphic, and the conjecture generalizes the one-cusped setting to manifolds with multiple cusps. The source presents it as proposed and then constructs purely cosmetic surgeries on hyperbolic manifolds with multiple cusps, so the conjecture is disproved.
Sources & referencesView supporting material
Primary source
Qiuyu Ren, “Families of cosmetic surgeries”, arXiv:2604.02672 (2026).
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