The maximally symmetric surfaces in the 3-torus

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 _gsubset 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, nin {mathbb {Z}}_+). The surfaces in (T^3) realizing a maximal symmetry can be either unknotted or knotted. Similar problems in the non-orientable category are also discussed. The connection with minimal surfaces in (T^3) is addressed and the situation when the maximally symmetric surfaces above can be realized by minimal surfaces is identified.