Bai, ShengRobins, VanessaWang, ChaoWang, Shicheng2017-04-100046-5755http://hdl.handle.net/1885/114523Suppose 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.Chao Wang is supported by Grant No. 11501534 of NSFC and the last author is supported by Grant No. 11371034 of NSFC. Vanessa Robins is supported by ARC fellowship FT140100604.17 pagesapplication/pdf© 2017 Springer Science+Business Media Dordrecht.Maximal surface symmetryTriply periodic minimal surfacesKnotted surfaces in the 3-torus2017-02-2210.1007/s10711-017-0218-0