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Counting cusps of subgroups of $ \mathrm{PSL}_2(\mathcal{O}_K)$


Author: Kathleen L. Petersen
Journal: Proc. Amer. Math. Soc. 136 (2008), 2387-2393
MSC (2000): Primary 11F23, 22E40, 11A07
DOI: https://doi.org/10.1090/S0002-9939-08-09262-9
Published electronically: March 14, 2008
MathSciNet review: 2390505
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Abstract: Let $ K$ be a number field with $ r$ real places and $ s$ complex places, and let $ \mathcal{O}_K$ be the ring of integers of $ K$. The quotient $ [\mathbb{H}^2]^r\times [\mathbb{H}^3]^s/\mathrm{PSL}_2(\mathcal{O}_K)$ has $ h_K$ cusps, where $ h_K$ is the class number of $ K$. We show that under the assumption of the generalized Riemann hypothesis that if $ K$ is not $ \mathbb{Q}$ or an imaginary quadratic field and if $ i \not \in K$, then $ \mathrm{PSL}_2(\mathcal{O}_K)$ has infinitely many maximal subgroups with $ h_K $ cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.


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  • 1. M. D. Baker and A. W. Reid, Arithmetic knots in closed 3-manifolds, J. Knot Theory Ramifications 11 (2002), no. 6, 903-920, Knots 2000 Korea, Vol. 3 (Yongpyong). MR 1936242 (2004b:57009)
  • 2. M. Harper and M. R. Murty, Euclidean rings of algebraic integers, Canad. J. Math. 56 (2004), no. 1, 71-76. MR 2031123 (2005h:11261)
  • 3. C. Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209-220. MR 0207630 (34:7445)
  • 4. C. J. Leininger, Compressing totally geodesic surfaces, Topology Appl. 118 (2002), no. 3, 309-328. MR 1874553 (2002j:57017)
  • 5. H. W. Lenstra, Jr., On Artin's conjecture and Euclid's algorithm in global fields, Invent. Math. 42 (1977), 201-224. MR 0480413 (58:576)
  • 6. K. L. Petersen, One-cusped congruence subgroups of $ \mathrm{PSL}_2(\mathcal{O}_k)$, University of Texas at Austin, 2005, Doctoral Thesis.
  • 7. -, One-cusped congruence subgroups of Bianchi groups, Math. Ann. 338 (2007), no. 2, 249-282. MR 2302062
  • 8. H. Petersson, Über einen einfachen Typus von Untergruppen der Modulgruppe, Arch. Math. 4 (1953), 308-315. MR 0057910 (15,291a)
  • 9. -, Über die Konstruktion zykloider Kongruenzgruppen in der rationalen Modulgruppe, J. Reine Angew. Math. 250 (1971), 182-212. MR 0294255 (45:3324)
  • 10. A. W. Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991), no. 1, 171-184. MR 1099096 (92a:57011)
  • 11. J.-P. Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489-527. MR 0272790 (42:7671)
  • 12. M. Suzuki, Group theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 247, Springer-Verlag, Berlin, 1982, Translated from the Japanese by the author. MR 648772 (82k:20001c)
  • 13. G. van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR 930101 (89c:11073)
  • 14. P. J. Weinberger, On Euclidean rings of algebraic integers, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, R. I., 1973, pp. 321-332. MR 0337902 (49:2671)

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Additional Information

Kathleen L. Petersen
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
Email: petersen@mast.queensu.ca

DOI: https://doi.org/10.1090/S0002-9939-08-09262-9
Received by editor(s): June 5, 2006
Received by editor(s) in revised form: July 12, 2006, November 28, 2006, and June 11, 2007
Published electronically: March 14, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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