A classification of maximally symmetric surfaces in the 3-dimensional torus

TL;DR

This paper classifies all maximally symmetric surfaces in the 3-torus under orientation-preserving symmetries, identifying those with the largest possible symmetry group order and relating unknotted cases to minimal surfaces.

Contribution

It provides a complete classification of maximally symmetric surfaces in the 3-torus, including explicit descriptions of group actions achieving maximal symmetry.

Findings

Maximal order of symmetry group is 12(g-1).

Classification of all such symmetric surfaces up to conjugation.

Unknotted surfaces correspond to equivariant minimal surfaces.

Abstract

If a finite group of orientation-preserving diffeomorphisms of the 3-dimensional torus leaves invariant an oriented, closed, embedded surface of genus g>1 and preserves the orientation of the surface, then its order is bounded from above by 12(g-1). In the present paper we classify (up to conjugation) all such group actions and surfaces for which the maximal possible order 12(g-1) is achieved, and note that the unknotted surfaces can be realized by equivariant minimal surfaces in a 3-torus.