The most symmetric surfaces in the 3-torus

Suppose an orientation preserving action of a finite group $G$ on the closed surface $Σ_g$ of genus $g>1$ extends over the 3-torus $T^3$ for some embedding $Σ_g\subset T^3$. Then $|G|\le 12(g-1)$, and this upper bound $12(g-1)$ can be achieved for $g=n^2+1, 3n^2+1, 2n^3+1, 4n^3+1, 8n^3+1, n\in \mathbb{Z}_+$. Those surfaces in $T^3$ realizing the maximum symmetries can be either unknotted or knotted. Similar problems in non-orientable category is also discussed. Connection with minimal surfaces in $T^3$ is addressed and when the maximum symmetric surfaces above can be realized by minimal surfaces is identified.

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