Title: |
Embedding surfaces into ${S}^{3}$ with maximum symmetry |

Speaker: |
Shicheng Wang (Peking University) |

Abstract: |
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
${\mathrm{OE}}_{g}$.
In particular, with 23 exceptions, ${\mathrm{OE}}_{g}$ is $4(g+1)$ if $g\ne {k}^{2}$ or $4{(\sqrt{g}+1)}^{2}$ if $g={k}^{2}$, and moreover ${\mathrm{OE}}_{g}$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and 481. This is a joint work with Chao Wang, Yimu Zhang and Bruno Zimmermann |