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Generalized triangle groups

Published online by Cambridge University Press:  24 October 2008

Gilbert Baumslag ,
John W. Morgan  and
Peter B. Shalen
Gilbert Baumslag
Affiliation:
Department of Mathematics, City College, New York, NY 10031, U.S.A.
John W. Morgan
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, U.S.A.
Peter B. Shalen
Affiliation:
Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60680, U.S.A.
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Extract

A group G is called a triangle group if it can be presented in the form

It is well-known that G is isomorphic to a subgroup of PSL2(ℂ), that a is of order l, b is of order m and ab is of order n. If

then G contains the fundamental group of a positive genus orientable surface as a subgroup of finite index whenever s(G) ≤ 1; in particular G is infinite. Furthermore, if s(G) < 1, the genus of the surface is greater than 1 and consequently G contains a free group of rank 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 1 , July 1987 , pp. 25 - 31
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1] Baumslag, B. and Pride, S. J.. Groups with two more generators than relators. J. London Math. Soc. (2) 17 (1978), 425–426.CrossRefGoogle Scholar
[2] Boyer, S.. On proper powers in free products and Dehn surgery. Preprint (1986).Google Scholar
[3] Culler, M. and Shalen, P. B.. Varieties of group representations and splittings of 3-manifolds. Annals of Math. 117 (1983), 109–146.CrossRefGoogle Scholar
[4] González-Acuña, F. and Short, H.. Knot surgery and primeness. Math. Proc. Cambridge Philos. Soc. 99 (1986), 89–102.CrossRefGoogle Scholar
[5] McA. Gordon, C. and Luecke, J.. Only integral Dehn surgeries can yield reducible manifolds. Math. Proc. Cambridge Philos. Soc. 102 (1987), 97–101.CrossRefGoogle Scholar
[6] Horowitz, R.. Characters of free groups represented in the two-dimensional linear group. Comm. Pure & Appl. Math. 25 (1972), 635–649.CrossRefGoogle Scholar
[7] Kurosh, A. G.. Die Untergruppen der freien Produkte von beliebigen Gruppen. Math. Ann. 109 (1934), 647–660.CrossRefGoogle Scholar
[8] Lubotzky, A. and Magid, A.. Varieties of representations of finitely generated groups. To appear.Google Scholar
[9] Mal'cev, A. I.. On faithful representations of infinite groups of matrices. American Math. Soc. Translations (2) 45 (1965), 1–18.CrossRefGoogle Scholar
[10] Ree, R. and Mendelsohn, N. S.. Free subgroups of groups with a single defining relation. Arch. Math. 19 (1968), 577–580.CrossRefGoogle Scholar
[11] Wall, C. T. C.. Rational Euler characteristics. Proc. Cambridge Philos. Soc. 57 (1961), 182–184.CrossRefGoogle Scholar