- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:01
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
9
2015
4
Embedding surfaces into $S^3$ 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, TRIESTE, ITALY
Surface symmetry, extendable action, 3-orbifolds, maximal order
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 $481$.
Manifolds and cell complexes
Combinatorics
Group theory and generalizations
1001
1045
10.4171/GGD/334
http://www.ems-ph.org/doi/10.4171/GGD/334