Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Farey polytopes and continued fractions associated with discrete hyperbolic groups


Author: L. Ya. Vulakh
Journal: Trans. Amer. Math. Soc. 351 (1999), 2295-2323
MSC (1991): Primary 11J99
DOI: https://doi.org/10.1090/S0002-9947-99-02151-0
Published electronically: February 5, 1999
MathSciNet review: 1467477
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The known definitions of Farey polytopes and continued fractions are generalized and applied to diophantine approximation in $n$-dimensional euclidean spaces. A generalized Remak-Rogers isolation theorem is proved and applied to show that certain Hurwitz constants for discrete groups acting in a hyperbolic space are isolated. The approximation constant for the imaginary quadratic field of discriminant $-15$ is found.


References [Enhancements On Off] (What's this?)

  • 1. B.N. Apanasov, Discrete Groups in Space and Uniformization Problems, Kluwer Academic Publishers, Dordrecht/Boston/London 1991. MR 93h:57026
  • 2. A.F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, New York, 1983. MR 85d:22026
  • 3. A.F. Beardon and B. Maskit, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1-12. MR 48:11489
  • 4. C.T.Benson and L.C. Grove, Finite Reflection Groups, Springer-Verlag, 2nd. ed., 1985. MR 85m:20007
  • 5. L. Bianchi, Sui gruppi de sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari, Math. Ann. 40 (1892), 332-412.
  • 6. É. Borel, Contribution à l'analyse arithmétique du continu, J. Math Pures Appl. (5) 9 (1903), 329-375.
  • 7. A.M. Brunner, Y.W. Lee, N.J. Wielenberg, Polyhedral groups and graph amalgamation products, Topology and its Applications, 20 (1985), 289-304. MR 87a:57049
  • 8. J.W.S. Cassels, An introduction to Diophantine approximation, Cambridge Univ. Press, 1957. MR 19:396h
  • 9. J.H Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, New York/ Berlin/ Heidelberg/ London/ Paris/ Tokyo, 1988. MR 89a:11067
  • 10. H.S.M. Coxeter, Regular Polytopes, Dover, 1973. MR 51:6554
  • 11. T.W. Cusick and M.E. Flahive, Markoff and Lagrange spectra, Math. Surveys and Monos., vol. 30, Amer. Math. Soc., Providence, R.I., 1989. MR 90i:11069
  • 12. L.R. Ford, A geometric proof of a theorem of Hurwitz, Proc. Edinburgh Math. Soc., 35 (1917), 59-65.
  • 13. L.R. Ford, On the closeness of approach of complex rational fraction to a complex irrational number, Trans. Amer. Math. Soc., 27 (1925), 146-154.
  • 14. F. Grunewald, A.C. Gushoff, und J. Mennicke, Komplex-quadratische Zahlk[??]'orper kleiner discriminante und pflasterungen des dreidimensionalen hyperbolischen raumes, Geometriae Dedicata 12 (1982), 227-237. MR 83h:51033
  • 15. A. Haas and C. Series, The Hurwitz constant and Diophantine approximation on Hecke groups, J. London Math. Soc. (2) 34 (1986), 219-234. MR 87m:11060
  • 16. G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, Oxford, 1960.MR 16:673c
  • 17. A. Hurwitz, Über die angenaherte Darstellungen der Irrationalzahlen durch rationale Brüche, Math. Ann. 39 (1891), 279-284.
  • 18. C. Maclachlan, P.L. Waterman, and N.J. Weilenberg, Higher dimensional analogues of the modular and Picard groups, Trans. Amer. Math. Soc. 312 (1989), 739-753. MR 90b:11039
  • 19. P.J. Nicholls, The Ergodic Theory of Discrete Groups, London Math. Soc. Lecture Note Ser. 143, Cambridge University Press, 1989. MR 91i:58104
  • 20. D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J. 21 (1954), 549-562. MR 16:458d
  • 21. Asmus L. Schmidt, Farey triangles and Farey quadrangles in the complex plane, Math. Scand. 21 (1967), 241-295. MR 39:6831
  • 22. Asmus L. Schmidt, Farey simplices in the space of quaternions, Math. Scand. 24 (1969), 31-65. MR 40:7206
  • 23. Asmus L. Schmidt, Diophantine approximation of complex numbers, Acta Math. 134 (1975), 1-85. MR 54:10160
  • 24. Asmus L. Schmidt, Diophantine approximation in the field $\textbf{Q}(i(11^{1/2}))$, J. Number Theory 10 (1978), 151-176. MR 58:5537
  • 25. Asmus L. Schmidt, Diophantine approximation in the Eisensteinian field, J. Number Theory 16 (1983), 169-204. MR 84m:10024
  • 26. C. Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), 69-80. MR 87c:58094
  • 27. M.K. Shaiheev, Reflection subgroups in Bianchi groups, Selecta Mathematica Sovietica 9 (1990), 315-322. MR 91j:20118
  • 28. R.G. Swan, Generators and relations for certain special linear groups, Adv. in Math. 6 (1971), 1-77.MR 44:1741
  • 29. K. Th. Vahlen, Über Näherungswerte und Kettenbrüche, J. Reine Angew. Math. 115 (1895), 221-233.
  • 30. L.Ya. Vulakh, The Markov spectrum of imaginary quadratic field $\mathbf{Q}(i\sqrt{D})$, where $D \not \equiv 3 \pmod{4}$ (Russian), Vestnik Moskov. Univ. Ser. 1 Math. Meh. 26 (1971), no. 6, 32-41. MR 45:1847
  • 31. L.Ya. Vulakh, The Markov spectrum of the Gaussian field (Russian), Izv. Vys[??]s. U[??]cebn. Zaved. Matematika, 129 (1973), no. 2, 26-40. MR 47:6616
  • 32. L.Ya. Vulakh, The accumulation point of the Markov spectrum of the field $\mathbf{Q}(i\sqrt{D})$ (Russian), Vestnik Moskov. Univ. Ser. 1 Math. Meh. 30 (1975), no. 1, 9-11; 31 (1976), no. 5, 125. MR 51:12732; MR 55:5550
  • 33. L.Ya. Vulakh, Maximal Fuchsian Subgroups of Extended Bianchi Groups, in Number Theory with an Emphasis on the Markoff Spectrum (Provo, UT, 1991), 297-310, Lecture Notes in Pure and Appl. Math., 147, Dekker, New York, 1993. MR 94g:11028
  • 34. L.Ya. Vulakh, On Hurwitz constants for Fuchsian groups, Canad. J. Math. 49 (1997), 405-416. MR 98a:11087
  • 35. L.Ya. Vulakh, Diophantine approximation on Bianchi groups, J. Number Theory, 54 (1995), 73-80. MR 96g:11076
  • 36. L.Ya. Vulakh, Diophantine approximation in $\mathbf{R}^{n}$, Trans. Amer. Math. Soc. 347 (1995), 573-585. MR 95e:11076

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11J99

Retrieve articles in all journals with MSC (1991): 11J99


Additional Information

L. Ya. Vulakh
Affiliation: Department of Mathematics, The Cooper Union, 51 Astor Place, New York, New York 10003
Email: vulakh@cooper.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02151-0
Keywords: Diophantine approximation, Clifford algebra, hyperbolic geometry
Received by editor(s): February 26, 1996
Received by editor(s) in revised form: May 19, 1997
Published electronically: February 5, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society