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 \(g=n^2+1, 3n^2+1, 2n^3+1, 4n^3+1, 8n^3+1, n\in {\mathbb {Z}}_+\). 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.
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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.
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Bai, S., Robins, V., Wang, C. et al. The maximally symmetric surfaces in the 3-torus. Geom Dedicata 189, 79–95 (2017). https://doi.org/10.1007/s10711-017-0218-0
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DOI: https://doi.org/10.1007/s10711-017-0218-0