Futer–Purcell–Schleimer's purely cosmetic surgery conjecture for cusped hyperbolic manifolds

Let MM be a cusped hyperbolic manifold with at least one cusp, and let s\mathbf{s} and s\mathbf{s'} be sets of slopes on the cusps. Futer–Purcell–Schleimer's conjecture. If the fillings M(s)M(\mathbf{s}) and M(s)M(\mathbf{s'}) are purely cosmetic, then MM admits an orientation-preserving symmetry sending s\mathbf{s} to s\mathbf{s'}. 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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