The maximally symmetric surfaces in the 3-torus

Abstract

Suppose an orientation-preserving action of a finite group G on the closed surface g of genus g > 1 extends over the 3-torus T³ for some embedding Σg ⊂ T³. Then |G| ≤ 12(g − 1), and this upper bound 12(g − 1) can be achieved for g = n² + 1, 3n² + 1, 2n³ + 1, 4n³ + 1, 8n³ + 1, n ∈ Z+. The surfaces in T³ 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 T³ is addressed and the situation when the maximally symmetric surfaces above can be realized by minimal surfaces is identified. Show more