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On a Result of A. M. Macbeath on Normal Subgroups of a Fuchsian Group

Published online by Cambridge University Press:  20 November 2018

W. Jonsson
W. Jonsson*
Affiliation:
McGill University, Montreal, Quebec
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A. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.

Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.

This theorem will follow from a consideration of the exact sequence

*

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 15 - 16
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Cartan, H., and Eilenberg, S., Homological Algebra, Princeton, 1956.Google Scholar
2. Stallings, J., Homology and central series of groups, Journal of Algebra 2, No. 2, 170-181.Google Scholar
3. Stammbach, U., Anwendung der homologietheorie auf zentralreihen und invarianten von pr?sentierungen, Math. Z. 94 (1966), 157-177.Google Scholar
4. Sah, C. H., Groups related to compact Rieman surfaces, Acta Math. 123 (1969), 13-42.Google Scholar