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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

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