The number of groups of a given genus
HTML articles powered by AMS MathViewer
- by T. W. Tucker
- Trans. Amer. Math. Soc. 258 (1980), 167-179
- DOI: https://doi.org/10.1090/S0002-9947-1980-0554326-3
- PDF | Request permission
Abstract:
It is shown that the number of groups with a given genus greater than one is finite. The proof depends heavily on V. K. Proulx’s classification of groups of genus one. The key observation is that as the number of vertices of a graph imbedded on a given surface increases, the average face size of the imbedding approaches the average face size of a toroidal imbedding. The result appears to be related to Hurwitz’s theorem bounding the order of a group of conformal automorphisms on a Riemann surface of genus g.References
- R. P. Baker, Cayley Diagrams on the Anchor Ring, Amer. J. Math. 53 (1931), no. 3, 645–669. MR 1506843, DOI 10.2307/2371172
- Jonathan L. Gross, Voltage graphs, Discrete Math. 9 (1974), 239–246. MR 347651, DOI 10.1016/0012-365X(74)90006-5
- Jonathan L. Gross, Every connected regular graph of even degree is a Schreier coset graph, J. Combinatorial Theory Ser. B 22 (1977), no. 3, 227–232. MR 450121, DOI 10.1016/0095-8956(77)90068-5
- Henry Levinson and Bernard Maskit, Special embeddings of Cayley diagrams, J. Combinatorial Theory Ser. B 18 (1975), 12–17. MR 384598, DOI 10.1093/comjnl/18.3.287
- H. Maschke, The Representation of Finite Groups, Especially of the Rotation Groups of the Regular Bodies of Three-and Four-Dimensional Space, by Cayley’s Color Diagrams, Amer. J. Math. 18 (1896), no. 2, 156–194. MR 1505708, DOI 10.2307/2369680
- Viera Krňanová Proulx, Classification of the toroidal groups, J. Graph Theory 2 (1978), no. 3, 269–273. MR 480167, DOI 10.1002/jgt.3190020312
- Viera Krňanová Proulx, Classification of the toroidal groups, J. Graph Theory 2 (1978), no. 3, 269–273. MR 480167, DOI 10.1002/jgt.3190020312
- C. L. Siegel, Topics in complex function theory. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Automorphic functions and abelian integrals; Translated from the German by A. Shenitzer and M. Tretkoff; With a preface by Wilhelm Magnus; Reprint of the 1971 edition; A Wiley-Interscience Publication. MR 1008931 A. T. White, Graphs, groups, and surfaces, North-Holland, Amsterdam, 1973.
- A. T. White, Graphs of groups on surfaces, Combinatorial surveys (Proc. Sixth British Combinatorial Conf., Royal Holloway Coll., Egham, 1977) Academic Press, London, 1977, pp. 165–197. MR 0491290
Bibliographic Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 258 (1980), 167-179
- MSC: Primary 05C10; Secondary 05C25
- DOI: https://doi.org/10.1090/S0002-9947-1980-0554326-3
- MathSciNet review: 554326