Maximum orders of cyclic and abelian extendable actions on surfaces

A faithful action of a group G on the genus g > 1 orientable closed surface Σ _g is extendable (over the three-dimensional sphere S ³ ), with respect to an embedding e: Σ _g → S ³ , if there is an action of G on S ³ such that h ◦ e = e ◦ h for any h ∈ G. We show that the maximum order of extendable cyclic group actions on Σ _g is 4g + 4 when g is even, and 4g4 when g is odd; the maximum order of extendable abelian group actions on Σ _g is 4g + 4. We also give the maximum orders of cyclic and abelian group actions on handlebodies.