Extending finite group actions on surfaces over S3S3

Let OEgOEg (resp. CEgCEg and AEgAEg) and resp. OEgo be the maximum order of finite (resp. cyclic and abelian) groups G   acting on the closed orientable surfaces ΣgΣg which extend over (S3,Σg)(S3,Σg) among all embeddings Σg→S3Σg→S3 and resp. unknotted embeddings Σg→S3Σg→S3.It is known that OEgo⩽12(g−1), and we show that 12(g−1)12(g−1) is reached for an unknotted embedding Σg→S3Σg→S3 if and only if g=2,3,4,5,6,9,11,17,25,97,121,241,601g=2,3,4,5,6,9,11,17,25,97,121,241,601. Moreover AEgAEg is 2g+22g+2; and CEgCEg is 2g+22g+2 for even g  , and 2g−22g−2 for odd g.Efforts are made to see intuitively how these maximal symmetries are embedded into the symmetries of the 3-sphere.