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Inequalities in Discrete Subgroups of PSL(2, R)

Published online by Cambridge University Press:  20 November 2018

Jane Gilman
Jane Gilman*
Affiliation:
Rutgers University, Newark, New Jersey
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Conditions for a subgroup, F, of PSL(2, R) to be discrete have been investigated by a number of authors. Jørgensen's inequality [5] gives an elegant necessary condition for discreteness for subgroups of PSL(2, C). Purzitsky, Rosenberger, Matelski, Knapp, and Van Vleck, among others [12, 13, 14, 9, 16, 17, 18, 19, 20, 7, 21] studied two generator discrete subgroups of PSL(2, R) in a long series of papers. The complete classification of two generator subgroups was surprisingly complicated and elusive. The most complete result appears in [20].

In this paper we use the results of [20] to prove that a nonelementary subgroup F of PSL(2, R) is discrete if and only if every non-elementary subgroup, G, generated by two hyperbolics is discrete (Theorem 5.2) and that F contains no elliptics if and only if each such G is free (Theorem 5.1). Thus, we produce necessary and sufficient conditions for a non-elementary subgroup F of PSL(2, R) to be a discrete group without elliptic elements (Theorem 6.1) or a discrete group containing only hyperbolic elements (Theorem 7.1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 1 , 01 February 1988 , pp. 115 - 130
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Beardon, A F., The geometry of discrete groups, Graduate Texts in Math. 91 (Springer, 1983).CrossRefGoogle Scholar
2. Doyle, C. and James, D., Discreteness criteria and high order generators for subgroups of SL(2, R), Ill. J. 25 (1981), 191–200.Google Scholar
3. Ford, , Automorphic functions (McGraw-Hill, 1929).Google Scholar
4. Gilman, J., On characterizing finite subgroups of the mapping-class group, Proc. Alta Conference, Annals of Math. Studies (1987), 433–442.Google Scholar
5. Jørgensen, T., On discrete groups of Mobi us transformation, Amer. J. Math. 98 (1976), 739–749.Google Scholar
6. Jørgensen, T., A note on subgroups of SL(2, C), Quart J. Math. Oxford Ser. II, 28 (1977), 209–212.Google Scholar
7. Knapp, AW., Doubly generated Fuchsian groups, Mich. Math. J.. 75 (1968), 289–304.Google Scholar
8. Lehner, J., Discontinuous groups and automorphic functions, A. M. S. Surveys. 8 (Providence, R. I., 1964).CrossRefGoogle Scholar
9. Matelski, J. P., The classification of discrete 2-generator subgroups of PSL(2, R), Israel J. Math.. 42 (1982), 309–317.Google Scholar
10. Magnus, Karass and Solitar, Combinatorial group theory (Wiley and Sons, N. Y., 1966).Google Scholar
11. Pommerenke, Ch. and Purzitsky, N., On some universal bounds for Fuchsian groups, Studies in Pure Mathematics, 561–575.Google Scholar
12. Purzitski, N., Two generator discrete free products, Math. Z.. 126 (1972), 209–223.Google Scholar
13. Purzitski, N., Real two-dimensional representation of two-generator free groups, Math. Z. 127 (1972), 95–104.Google Scholar
14. Purzitski, N., All two-generator Fuchsian groups, Math. Z. 147 (1976), 87–92.Google Scholar
15. Purzitsky, N. and Rosenberger, G., Two generator Fuchsian groups of genus one, Math. Z.. 128 (1972), 245–251. Correction: Math. Z. 132 (1973), 261–262.Google Scholar
16. Rosenberger, G., Fuchssche Gruppen, diefreies Produkt zweier zyklischer Gruppen sind, und die Gleichung x2 + y2 + z2 = xyz, Math. Ann. 199 (1972), 213–228.Google Scholar
17. Rosenberger, G., Von Untergruppen der Triangel Gruppen, Ill. J. Math. 22 (1978), 404–413.Google Scholar
18. Rosenberger, G., Fine Bemerkung zu einer Arbeit von T. Jørgensen, Math. Z. 165 (1979), 261–265.Google Scholar
19. Rosenberger, G., Some remarks on a paper of C. Doyle and D. James on subgroups of SL(2, R), Ill. J. 28 (1984), 348–351.Google Scholar
20. Rosenberger, G., All generating pairs of all two-generator Fuchsian groups, Arch. Math. 46 (1986), 198–204.Google Scholar
21. Van Vleck, E., On the combination of non-loxodromic subsituations, Trans. A. M. S.. 21 (1919), 299–312.Google Scholar