Extending finite group actions on surfaces over $S^3$

Let $OE_g$ (resp. $CE_g$ and $AE_g$) and resp. $OE^o_g$ be the maximum order of finite (resp. cyclic and abelian) groups $G$ acting on the closed orientable surfaces $\Sigma_g$ which extend over $(S^3, \Sigma_g)$ among all embeddings $\Sigma_g\to S^3$ and resp. unknotted embeddings $\Sigma_g\to S^3$. It is known that $OE^o_g\le 12(g-1)$, and we show that $12(g-1)$ is reached for an unknotted embedding $\Sigma_g \to S^3$ if and only if $g = 2$, 3, 4, 5, 6, 9, 11, 17, 25, 97, 121, 241, 601. Moreover $AE_g$ is $2g+2$; and $CE_g$ is $2g+2$ for even $g$, and $2g-2$ for odd $g$. Efforts are made to see intuitively how these maximal symmetries are embedded into the symmetries of the 3-sphere.

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