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The Markov spectra for Fuchsian groups
Author:
L. Ya. Vulakh
Journal:
Trans. Amer. Math. Soc. 352 (2000), 4067-4094
MSC (2000):
Primary 11J06, 11F06
Posted:
April 17, 2000
MathSciNet review:
1650046
Full-text PDF Free Access
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Abstract: Applying the Klein model  of the hyperbolic plane and identifying the geodesics in  with their poles in the projective plane, the author develops a method of determining infinite binary trees in the Markov spectrum for a Fuchsian group. The method is applied to a maximal group commensurable with the modular group and other Fuchsian groups.
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- 2.
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- 3.
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- A. Haas, Diophantine approximation on hyperbolic orbifolds, Duke Math. J. 56 (1988), 531-547. MR 90c:11031
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 Arise from the Markoff Spectrum, Bull. Amer. Math. Soc. 11 (1984), 359-362. MR 86b:11033
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- A. A. Markoff, Sur les formes binaires indéfinies, Math. Ann. 15 (1879), 381-409.
- 21.
- P.J. Nicholls, Diophantine approximation via the modular group, J. London Math. Soc. (2) 17 (1978), 11-17. MR 58:16551
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- G. Rosenberger, Uber die Diophantische Gleichung
 , J. Reine Angen. Math. 305 (1979), 122-125. MR 80d:10030
- 23.
- Asmus L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. I, J. Reine Angew. Math. 286/287 (1976), 341-368. MR 56:15566
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- Asmus L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. II, J. Reine Angew. Math. 292 (1977), 109-114. MR 56:15567
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- Asmus L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. III, (preprint).
- 26.
- C. Series, The Markoff spectrum in the Hecke group
 , Proc. London Math. Soc. 57 (1988), 151-181. MR 89g:11056
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- L.Ya. Vulakh, The Markov spectrum of imaginary quadratic field
 , where  (Russian), Vestnik Moskov. Univ. Ser. 1 Math. Meh. 26 (1971), no. 6, 32-41. MR 45:1847
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- L.Ya. Vulakh, The Diophantine equation and the Markov spectrum (Russian), Trudy Moskov. Ins. Radiotehn., Electron. i Avtomat., 57 (1972), 54-58.
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- L.Ya. Vulakh, The Diophantine equation
 and the Markov spectrum (Russian), Trudy Moskov. Ins. Radiotehn., Electron. i Avtomat., 67 (1973), 105-112. MR 58:21957
- 31.
- L.Ya. Vulakh, On Markov spectra on sublattices related to Diophantine equations (Russian), Trudy XXV Nauchn.-tehn. Konfer. Moskov. Ins. Radiotehn., Electron. i Avtomat., Sec. Mat. (1976), 16-21.
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- L.Ya. Vulakh, Classification of Fuchsian subgroups of some Bianchi groups, Canad. Math. Bull. (3) 34 (1991), 417-422. MR 92i:11047
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- 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
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- L.Ya. Vulakh, Higher dimensional analogues of Fuchsian subgroups of
 , Trans. Amer. Math. Soc. 337 (1993), 947-963. MR 93h:20055
- 35.
- L.Ya. Vulakh, Diophantine approximation on Bianchi groups, J. Number Theory, 54 (1995), 73-80. MR 96g:11076
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- L.Ya. Vulakh, Diophantine approximation in
 , Trans. Amer. Math. Soc. 347 (1995), 573-585. MR 95e:11076
- 37.
- L.Ya. Vulakh, On Hurwitz constants for Fuchsian groups, Canad. J. Math., 49, (1997), 405-416. MR 98a:11087
- 38.
- L.Ya. Vulakh, Farey polytopes and continued fractions associated with discrete hyperbolic groups, Trans. Amer. Math. Soc. 351 (1999), 2295-2323. MR 99i:11054
- 39.
- L.Ya. Vulakh, The Markov spectrum for triangle groups, J. Number Theory, (67), (1997), 11-28. MR 99e:11093
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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:
http://dx.doi.org/10.1090/S0002-9947-00-02455-7
PII:
S 0002-9947(00)02455-7
Keywords:
Diophantine approximation,
projective geometry,
hyperbolic geometry
Received by editor(s):
September 17, 1997
Received by editor(s) in revised form:
August 25, 1998
Posted:
April 17, 2000
Article copyright:
© Copyright 2000 American Mathematical Society
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