Inequalities in Discrete Subgroups of PSL(2, R)

Gilman, Jane

Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 115-130

Résumé

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).

Détail

DOI : 10.4153/CJM-1988-005-x

Bibliographie

[1] 1. Beardon, A F., The geometry of discrete groups, Graduate Texts in Math. 91 (Springer, 1983). Google Scholar | DOI

[2] 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] 3. Ford, , Automorphic functions (McGraw-Hill, 1929). Google Scholar

[4] 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] 5. Jørgensen, T., On discrete groups of Mobi us transformation, Amer. J. Math. 98 (1976), 739–749. Google Scholar

[6] 6. Jørgensen, T., A note on subgroups of SL(2, C), Quart J. Math. Oxford Ser. II, 28 (1977), 209–212. Google Scholar

[7] 7. Knapp, AW., Doubly generated Fuchsian groups, Mich. Math. J.. 75 (1968), 289–304. Google Scholar

[8] 8. Lehner, J., Discontinuous groups and automorphic functions, A. M. S. Surveys. 8 (Providence, R. I., 1964). Google Scholar | DOI

[9] 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] 10. Magnus, Karass and Solitar, Combinatorial group theory (Wiley and Sons, N. Y., 1966). Google Scholar

[11] 11. Pommerenke, Ch. and Purzitsky, N., On some universal bounds for Fuchsian groups, Studies in Pure Mathematics, 561–575. Google Scholar

[12] 12. Purzitski, N., Two generator discrete free products, Math. Z.. 126 (1972), 209–223. Google Scholar

[13] 13. Purzitski, N., Real two-dimensional representation of two-generator free groups, Math. Z. 127 (1972), 95–104. Google Scholar

[14] 14. Purzitski, N., All two-generator Fuchsian groups, Math. Z. 147 (1976), 87–92. Google Scholar

[15] 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] 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] 17. Rosenberger, G., Von Untergruppen der Triangel Gruppen, Ill. J. Math. 22 (1978), 404–413. Google Scholar

[18] 18. Rosenberger, G., Fine Bemerkung zu einer Arbeit von T. Jørgensen, Math. Z. 165 (1979), 261–265. Google Scholar

[19] 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] 20. Rosenberger, G., All generating pairs of all two-generator Fuchsian groups, Arch. Math. 46 (1986), 198–204. Google Scholar

[21] 21. Van Vleck, E., On the combination of non-loxodromic subsituations, Trans. A. M. S.. 21 (1919), 299–312. Google Scholar