Fuchsian groups and algebraic number fields

Authors:
P. L. Waterman and C. Maclachlan

Journal:
Trans. Amer. Math. Soc. **287** (1985), 353-364

MSC:
Primary 20H10; Secondary 11F06, 32G15

MathSciNet review:
766224

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given the signature of a finitely-generated Fuchsian group, we find the minimal extension of the rationals for which there is a Fuchsian group having the required signature, whose matrix entries lie in this field.

**[1]**Lipman Bers,*On boundaries of Teichmüller spaces and on Kleinian groups. I*, Ann. of Math. (2)**91**(1970), 570–600. MR**0297992****[2]**Allan L. Edmonds, John H. Ewing, and Ravi S. Kulkarni,*Torsion free subgroups of Fuchsian groups and tessellations of surfaces*, Invent. Math.**69**(1982), no. 3, 331–346. MR**679761**, 10.1007/BF01389358**[3]**Allan L. Edmonds, John H. Ewing, and Ravi S. Kulkarni,*Regular tessellations of surfaces and (𝑝,𝑞,2)-triangle groups*, Ann. of Math. (2)**116**(1982), no. 1, 113–132. MR**662119**, 10.2307/2007049**[4]**Allan L. Edmonds, Ravi S. Kulkarni, and Robert E. Stong,*Realizability of branched coverings of surfaces*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 773–790. MR**732119**, 10.1090/S0002-9947-1984-0732119-5**[5]**Leon Greenberg,*Homomorphisms of triangle groups into 𝑃𝑆𝐿(2,𝐶)*, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 167–181. MR**624813****[6]**-,*Homomorphisms of triangle groups into*, Modular Functions in Analysis and Number Theory, (T. Metzer, ed.), Univ. of Pittsburgh, 1983.**[7]**-,*Homomorphisms and quadratic differentials of triangle groups*(unpublished).**[8]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, Oxford, at the Clarendon Press, 1954. 3rd ed. MR**0067125****[9]**A. W. Knapp,*Doubly generated Fuchsian groups*, Michigan Math. J.**15**(1969), 289–304. MR**0248231****[10]**Ravi S. Kulkarni,*An extension of a theorem of Kurosh and applications to Fuchsian groups*, Michigan Math. J.**30**(1983), no. 3, 259–272. MR**725780****[11]**J. Lehner and M. Newman,*Real two-dimensional representations of the modular group and related groups*, Amer. J. Math.**87**(1965), 945–954. MR**0188298****[12]**J. Lehner and M. Newman,*Real two-dimensional representations of the free product of two finite cyclic groups*, Proc. Cambridge Philos. Soc.**62**(1966), 135–141. MR**0193159****[13]**W. Magnus,*Rational representations of Fuchsian groups and non-parabolic subgroups of the modular group*, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II (1973), 179–189. MR**0346069****[14]**Jens Mennicke,*A note on regular coverings of closed orientable surfaces*, Proc. Glasgow Math. Assoc.**5**(1961), 49–66 (1961). MR**0161910****[15]**M. H. Millington,*Subgroups of the classical modular group*, J. London Math. Soc. (2)**1**(1969), 351–357. MR**0244160****[16]**Kisao Takeuchi,*Fuchsian groups contained in 𝑆𝐿₂(𝑄)*, J. Math. Soc. Japan**23**(1971), 82–94. MR**0271342****[17]**P. L. Waterman,*Fuchsian groups and algebraic number fields*, Thesis, Aberdeen, 1982.**[18]**André Weil,*On discrete subgroups of Lie groups*, Ann. of Math. (2)**72**(1960), 369–384. MR**0137792**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
20H10,
11F06,
32G15

Retrieve articles in all journals with MSC: 20H10, 11F06, 32G15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766224-5

Article copyright:
© Copyright 1985
American Mathematical Society