Extending finite group actions on surfaces over S~3

摘要

Let OE_g (resp. CE_g and AE_g) and resp. OE_g~o be the maximum order of finite (resp. cyclic and abelian) groups G acting on the closed orientable surfaces ∑_g which extend over (S~3,∑_g) among all embeddings ∑_g → S~3 and resp. unknotted embeddings ∑_g→ S~3. It is known that OE_g~o is contained in 12(g - 1), and we show that 12(g - 1) is reached for an unknotted embedding ∑_g → 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.