Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The number of groups of a given genus


Author: T. W. Tucker
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 05C10, 05C25

Retrieve articles in all journals with MSC: 05C10, 05C25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0554326-3
Keywords: Cayley graph, genus of a group, Euler characteristic, Riemann surface
Article copyright: © Copyright 1980 American Mathematical Society