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 $\left({S}^{3},{\Sigma }_{g}\right)$, for all possible embeddings ${\Sigma }_{g}↪{S}^{3}$. 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 $4\left(g+1\right)$ if $g\ne {k}^{2}$ or $4{\left(\sqrt{g}+1\right)}^{2}$ 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