The most symmetric surfaces in the 3-torus

TL;DR

This paper investigates the maximum symmetry groups of surfaces embedded in the 3-torus, establishing an upper bound on group order and characterizing surfaces that attain this bound, including their knottedness and relation to minimal surfaces.

Contribution

It provides a sharp upper bound on the symmetry group size for surfaces in the 3-torus and characterizes the surfaces achieving this bound, including their knottedness and minimal surface realizations.

Findings

Maximum symmetry group order is 12(g-1).

Surfaces achieving maximum symmetry can be knotted or unknotted.

Identifies conditions under which these surfaces are minimal.

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.