JournalsggdVol. 9, No. 4pp. 1001–1045

Embedding surfaces into S3S^3 with maximum symmetry

  • Chao Wang

    Peking University, Beijing, China
  • Shicheng Wang

    Peking University, Beijing, China
  • Yimu Zhang

    Peking University, Beijing, China
  • Bruno Zimmermann

    Università degli Studi di Trieste, Italy
Embedding surfaces into $S^3$ with maximum symmetry cover
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Abstract

We restrict our discussion to the orientable category. For g>1g > 1, let OEg\mathrm {OE}_g be the maximum order of a finite group GG acting on the closed surface Σg\Sigma_g of genus gg which extends over (S3,Σg)(S^3, \Sigma_g), for all possible embeddings ΣgS3\Sigma_g\hookrightarrow S^3. We will determine OEg\mathrm {OE}_g for each gg, indeed the action realizing OEg\operatorname{OE}_g.

In particular, with 23 exceptions, OEg\operatorname{OE}_g is 4(g+1)4(g+1) if gk2g\ne k^2 or 4(g+1)24(\sqrt{g}+1)^2 if g=k2g=k^2,and moreover OEg\operatorname{OE}_g can be realized by unknotted embeddings for all gg except for g=21g=21 and 481481.

Cite this article

Chao Wang, Shicheng Wang, Yimu Zhang, Bruno Zimmermann, Embedding surfaces into S3S^3 with maximum symmetry. Groups Geom. Dyn. 9 (2015), no. 4, pp. 1001–1045

DOI 10.4171/GGD/334