Groups, Geometry, and Dynamics


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Volume 9, Issue 4, 2015, pp. 1001–1045
DOI: 10.4171/GGD/334

Published online: 2015-11-16

Embedding surfaces into $S^3$ with maximum symmetry

Chao Wang[1], Shicheng Wang[2], Yimu Zhang[3] and Bruno Zimmermann[4]

(1) Peking University, Beijing, China
(2) Peking University, Beijing, China
(3) Peking University, Beijing, China
(4) Università degli Studi di Trieste, Italy

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

In particular, with 23 exceptions, $\operatorname{OE}_g$ is $4(g+1)$ if $g\ne k^2$ or $4(\sqrt{g}+1)^2$ if $g=k^2$,and moreover $\operatorname{OE}_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$.

Keywords: Surface symmetry, extendable action, 3-orbifolds, maximal order

Wang Chao, Wang Shicheng, Zhang Yimu, Zimmermann Bruno: Embedding surfaces into $S^3$ with maximum symmetry. Groups Geom. Dyn. 9 (2015), 1001-1045. doi: 10.4171/GGD/334