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Comments On a Discreteness Condition for Subgroups of SL(2, C)

Published online by Cambridge University Press:  20 November 2018

Troels Jørgensen*
Affiliation:
Harvard University, Cambridge, Massachusetts
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SL(2,C) is the group of all complex unimodular 2 × 2 matrices. A subgroup of SL(2, C) is said to be discrete if it does not contain any convergent sequence of distinct elements. A subgroup is said to be elementary if the commutator of any two elements of infinite order has trace 2. The discreteness condition which this note relates to is the following:

PROPOSITION 1. If two complex, unimodular 2 × 2 matrices X and Y generatea non-elementary, discrete group, then

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Jorgensen, T., On discrete groups of Môbius transformations, Amer. J. Math. 98 (1976), 739749.Google Scholar
2. Jorgensen, T. and Kiikka, M., Some extreme discrete groups, Ann. Acad. Sci. Fenn., Ser. A. I. Math. 1 (1975), 245248.Google Scholar
3. Jorgensen, T., Compact 3-manifolds of constant negative curvature fibering over the circle, Ann. of Math.. 106 (1977), 6172.Google Scholar
4. Jorgensen, T., A note on subgroups of SL (2, C), Quart. J. Math. Oxford, (2), 28 (1977), 209212.Google Scholar
5. Wielenberg, N., On the fundamental polyhedra of discrete Môbius groups, Amer. J. Math.. 99 (1977), 861877.Google Scholar