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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brakke, K.: Triply periodic minimal surfaces. http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html. Accessed 18 Jan 2016
Frohman, C., Hass, J.: Unstable minimal surfaces and Heegaard splittings. Invent. Math. 95(3), 529–540 (1989)
Hahn, T. (ed.): International Tables for Crystallography. D. Reidel Publishing Company, Dordrecht (2005)
Hempel, J.: 3-Manifolds. Princeton University Press, Princeton (1976)
Hurwitz, A.: Über algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41(3), 403–442 (1892)
Hyde, S.T., Blum, Z., Landh, T., Lidin, S., Ninham, B.W., Andersson, S., Larsson, K.: The Language of Shape: The Role of Curvature in Condensed Matter. Elsevier, New York (1996)
Hyde, S.T., O’Keeffe, M., Proserpio, D.M.: A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics. Angew. Chem. Int. Ed. 47, 79968000 (2008)
May, C.L.: Automorphisms of compact Klein surfaces with boundary. Pac. J. Math. 59, 199–210 (1975)
May, C.L.: Large automorphism groups of compact Klein surfaces with boundary. Glasg. Math. J. 1, 1–10 (1977)
Meeks, W.H.: The conformal structure and geometry of triply periodic minimal surfaces in \(R^3\). Bull. Am. Math. Soc. 83(1), 134–136 (1977)
Meeks, W.H., Yau, S.T.: The equivariant Dehn’s lemma and loop theorem. Comment. Math. Helv. 56(1), 225–239 (1981)
O’Keeffe, M., Peskov, M.A., Ramsden, S.J., Yaghi, O.M.: The reticular chemistry structure resource (RCSR) database of, and symbols for crystal nets. Acc. Chem. Res. 41, 1782–1789 (2008)
Schwarz, H.A.: Gesammelte Mathematische Abhandlungen. Bd 1. Springer, Berlin (1890)
Schoen, A.H.: NASA Tech. Note No. D-5541 (1970)
Wang, C., Wang, S.C., Zhang, Y.M., Zimmermann, B.: Extending finite group actions on surfaces over \(S^3\). Topol. Appl. 160(16), 2088–2103 (2013)
Wang, C., Wang, S.C., Zhang, Y.M., Zimmerman, B.: Maximal orders of extendable finite group actions on surfaces. Groups Geom. Dyn. 9(4), 1001–1045 (2015)
Wang, C., Wang, S.C., Zhang, Y.M.: Maximal orders of extendable actions on surfaces. Acta Math. Sin. (Engl. Ser.) 32(1), 54–68 (2016)
Zimmermann, B.: Uber Homomorphismen \(n\)-dimensionaler Henkelkrper und endliche Erweiterungen von Schottky-Gruppen. Comment. Math. Helv. 56, 474–486 (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-017-0218-0