The most symmetric surfaces in the 3-torus

Abstract

Suppose an orientation preserving action of a finite group $G$ on the closed surface ${\Sigma }_g$ of genus $g>1$ extends over the 3-torus $T^3$ for some embedding ${\Sigma }_g\subset T^3$. 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.

19 pages, 12 figures