Mathematical analysis reveals exact involution ranks across mapping class groups of closed orientable surfaces, resolving a longstanding problem of Farb.
We determine the involution rank of the mapping class group of every closed orientable surface of genus g ≥ 2, thereby resolving a problem of Farb. More precisely, irank MCG(Σ_g) = ∞ for g = 2, 4 for g = 3, and 3 for g ≥ 4. The three previously unresolved cases are settled by two different methods: a mod-2 symplectic obstruction in genus three, and explicit generating pairs reversed by half-turn involutions in genera four and five.
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Guo et al. (2026) studied this question.
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