Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-24T22:48:18.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-generator arithmetic Fuchsian groups

Published online by Cambridge University Press:  24 October 2008

C. Maclachlan  and
G. Rosenberger
C. Maclachlan
Affiliation:
Aberdeen University and Universität Dortmund
G. Rosenberger
Affiliation:
Aberdeen University and Universität Dortmund
Get access Rights & Permissions [Opens in a new window]

Extract

In (12), Takeuchi characterized those Fuchsian groups which are arithmetic in terms of the traces of the matrices involved in the group. Further, he used these criteria (13) to show that there are only finitely many arithmetic Fuchsian triangle groups and indeed to determine the signatures of all such groups. We note that, if two Fuchsian triangle groups have the same signature, then they are conjugate in PGL(2, ) and hence either all Fuchsian triangle groups of a given signature are arithmetic or none of them are. This will not, in general, be the case for non-triangle Fuchsian groups and in this paper, we examine Fuchsian groups with signature of one of the forms (1; n; 0), n ≽ 2 or (0; 2,2, 2, n; 0), n ≽ 3 and odd in which case the space of conjugacy classes has dimension two. Our principal results state that for each n, there are only finitely many conjugacy classes of arithmetic Fuchsian groups of given signature (1; n; 0) or (0; 2, 2, 2, n; 0) and for large enough n there are no arithmetic Fuchsian groups of that given signature (Theorems 3 and 4). Together with the results of (13) and known results for non-cocompact Fuchsian groups, these results show that there are only finitely many conjugacy classes of two-generator arithmetic Fuchsian groups (see §7).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 383 - 391
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Helling, H.Bestimmung der Kommensurabilitätklasse der Hilbertschen Modulgruppe. Math. Z. 92 (1966), 269–280.CrossRefGoogle Scholar
(2)Kern-Isberner, G. and Rosenberger, G. Einige Bemerkungen über Untergruppen der PSL(2, ). Resultate der Math.Google Scholar
(3)Maclachlan, C.Groups of units of zero ternary quadratic forms. Proc. Roy. Soc. Edinburgh. Sect. A 88 (1981), 141–157.Google Scholar
(4)Purzitsky, N. and Rosenberger, G.Two-generator Fuchsian groups of genus one. Math. Z. 128 (1972), 245–251. Correction Math. Z. 132 (1973), 261–2.CrossRefGoogle Scholar
(5)Rosenberger, G.Fuchssche Gruppen, die freies Product zweir zyklischer Gruppen sind. und die Gleichung x 2 + y 2 + z 2 = xyz. Math. Ann. 199 (1972), 213–227.CrossRefGoogle Scholar
(6)Rosenberger, G.On discrete free subgroups of linear groups. J. London Math. Soc. 17 (1978), 79–85.CrossRefGoogle Scholar
(7)Rosenberger., G.Über die Diophantische Gleichung ax 2 + by* + cz 2 = dxyz. J. Reine Angew. Math. 305 (1979), 122–125.Google Scholar
(8)Rosenberger, G. and Kalia., R. N.Automorphisms of the Fuchsian groups of type (0; 2, 2, 2, q; 0). Comm. Alg. 6 (11), (1978) 1115–1129.Google Scholar
(9)Salem, R.Algebraic numbers and Fourier analysis (Heath, Boston, 1963.)Google Scholar
(10)Schur, I.Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 1 (1918), 377–402.CrossRefGoogle Scholar
(11)Takeuchi, K.On some discrete subgroups of SL2(). J. Fac. Sci. Univ. Tokyo Sect. I 16 (1967), 97–100.Google Scholar
(12)Takeuchi, K.A characterization of arithmetic Fuchsian groups. J. Math. Soc. Japan 27 (1975), 600–612.Google Scholar
(13)Takeuchi, K.Arithmetic triangle groups. J. Math. Soc. Japan 29 (1977), 91–106.Google Scholar
(14)Takeuchi, K.Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo Sect. I A 24 (1977), 201–222.Google Scholar
(15)Vigneras, M-F. Arithmétique des algebres de quaternions. Lecture Notes in Math. no. 800 (Springer-Verlag, 1980).Google Scholar
(16)Weiss., E.Algebraic number theory (McGraw-Hill, New York, 1963.)Google Scholar