The Markov spectra for Fuchsian groups

Vulakh, L.

doi:10.1090/S0002-9947-00-02455-7

The Markov spectra for Fuchsian groups
HTML articles powered by AMS MathViewer

by L. Ya. Vulakh PDF

Trans. Amer. Math. Soc. 352 (2000), 4067-4094 Request permission

Abstract:

Applying the Klein model $D^2$ of the hyperbolic plane and identifying the geodesics in $D^2$ 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.

References

Boris N. Apanasov, Discrete groups in space and uniformization problems, Mathematics and its Applications (Soviet Series), vol. 40, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated and revised from the 1983 Russian original. MR 1191903
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
A. F. Beardon, J. Lehner, and M. Sheingorn, Closed geodesics on a Riemann surface with application to the Markov spectrum, Trans. Amer. Math. Soc. 295 (1986), no. 2, 635–647. MR 833700, DOI 10.1090/S0002-9947-1986-0833700-7
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Harvey Cohn, Representation of Markoff’s binary quadratic forms by geodesics on a perforated torus, Acta Arith. 18 (1971), 125–136. MR 288079, DOI 10.4064/aa-18-1-125-136
H. S. M. Coxeter, The real projective plane, Cambridge University Press, New York, 1960. 2nd. ed. MR 0112071
Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, Mathematical Surveys and Monographs, vol. 30, American Mathematical Society, Providence, RI, 1989. MR 1010419, DOI 10.1090/surv/030
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
J. Elstrodt, F. Grunewald, and J. Mennicke, Arithmetic applications of the hyperbolic lattice point theorem, Proc. London Math. Soc. (3) 57 (1988), no. 2, 239–283. MR 950591, DOI 10.1112/plms/s3-57.2.239
D. S. Gorškov, Lobačevskiĭ geometry in connection with some problems in arithmetic, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 67 (1977), 39–85, 225 (Russian). Studies in number theory (LOMI), 4. MR 0563093
Andrew Haas, Diophantine approximation on hyperbolic Riemann surfaces, Acta Math. 156 (1986), no. 1-2, 33–82. MR 822330, DOI 10.1007/BF02399200
Andrew Haas, Length spectra as moduli for hyperbolic surfaces, Duke Math. J. 52 (1985), no. 4, 923–934. MR 816393, DOI 10.1215/S0012-7094-85-05249-4
Andrew Haas, Diophantine approximation on hyperbolic orbifolds, Duke Math. J. 56 (1988), no. 3, 531–547. MR 948532, DOI 10.1215/S0012-7094-88-05622-0
Heinz Helling, On the commensurability class of the rational modular group, J. London Math. Soc. (2) 2 (1970), 67–72. MR 277620, DOI 10.1112/jlms/s2-2.1.67
C. G. Lekkerkerker, Geometry of numbers, Bibliotheca Mathematica, Vol. VIII, Wolters-Noordhoff Publishing, Groningen; North-Holland Publishing Co., Amsterdam-London, 1969. MR 0271032
J. Lehner and M. Sheingorn, Simple closed geodesics on $H^{+}/\Gamma (3)$ arise from the Markov spectrum, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 359–362. MR 752798, DOI 10.1090/S0273-0979-1984-15307-2
A. V. Malyšev, Markov and Lagrange spectra (a survey of the literature), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 67 (1977), 5–38, 225 (Russian). Studies in number theory (LOMI), 4. MR 0441876
A. A. Markoff, Sur les formes binaires indéfinies, Math. Ann. 15 (1879), 381-409.
Peter J. Nicholls, Diophantine approximation via the modular group, J. London Math. Soc. (2) 17 (1978), no. 1, 11–17. MR 498427, DOI 10.1112/jlms/s2-17.1.11
Gerhard Rosenberger, Über die diophantische Gleichung $ax^{2}+by^{2}+cz^{2}=dxyz$, J. Reine Angew. Math. 305 (1979), 122–125 (German). MR 518854, DOI 10.1515/crll.1979.305.122
Asmus L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. I, J. Reine Angew. Math. 286(287) (1976), 341–368. MR 457358, DOI 10.1515/crll.1976.286-287.341
Asmus L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. II, J. Reine Angew. Math. 292 (1977), 109–114. MR 457359, DOI 10.1515/crll.1977.292.109
Asmus L. Schmidt, Minimum of quadratic forms with respect to Fuchsian groups. III, (preprint).
Caroline Series, The Markoff spectrum in the Hecke group $G_5$, Proc. London Math. Soc. (3) 57 (1988), no. 1, 151–181. MR 940433, DOI 10.1112/plms/s3-57.1.151
L. Ja. Vulah, The Markov spectrum of imaginary quadratic fields $Q(i\surd D)$, where $D\not \equiv 3(\textrm {mod}\ 4)$, Vestnik Moskov. Univ. Ser. I Mat. Meh. 26 (1971), no. 6, 32–41 (Russian, with English summary). MR 0292765
L. Ja. Vulah, Diophantine closed sets, Trudy Moskov. Inst. Radiotehn. Èlektron. i Avtomat. 52 Mat. (1971), 27–31 (Russian). MR 0563108
L.Ya. Vulakh, The Diophantine equation $p_{1}^{2} + p_{2}^{2} + 5q^{2} = 5p_{1}p_{2}q$ and the Markov spectrum (Russian), Trudy Moskov. Ins. Radiotehn., Electron. i Avtomat., 57 (1972), 54-58.
L. Ja. Vulah, The Diophantine equation $p^{2}+2q^{2}+3r^{2}=6pqr$, and the Markov spectrum, Trudy Moskov. Inst. Radiotehn. Èlektron. i Avtomat. Vyp. 67 Mat. (1973), 105–112, 152 (Russian). MR 0506051
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.
L. Ya. Vulakh, Classification of maximal Fuchsian subgroups of some Bianchi groups, Canad. Math. Bull. 34 (1991), no. 3, 417–422. MR 1127768, DOI 10.4153/CMB-1991-067-5
L. Ya. Vulakh, Maximal Fuchsian subgroups of extended Bianchi groups, Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991) Lecture Notes in Pure and Appl. Math., vol. 147, Dekker, New York, 1993, pp. 297–310. MR 1219343
L. Ya. Vulakh, Higher-dimensional analogues of Fuchsian subgroups of $\textrm {PSL}(2,\mathfrak {o})$, Trans. Amer. Math. Soc. 337 (1993), no. 2, 947–963. MR 1145965, DOI 10.1090/S0002-9947-1993-1145965-9
L. Ya. Vulakh, Diophantine approximation on Bianchi groups, J. Number Theory 54 (1995), no. 1, 73–80. MR 1352637, DOI 10.1006/jnth.1995.1102
L. Ya. Vulakh, Diophantine approximation in $\textbf {R}^n$, Trans. Amer. Math. Soc. 347 (1995), no. 2, 573–585. MR 1276937, DOI 10.1090/S0002-9947-1995-1276937-3
L. Ya. Vulakh, On Hurwitz constants for Fuchsian groups, Canad. J. Math. 49 (1997), no. 2, 405–416. MR 1447498, DOI 10.4153/CJM-1997-020-x
L. Ya. Vulakh, Farey polytopes and continued fractions associated with discrete hyperbolic groups, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2295–2323. MR 1467477, DOI 10.1090/S0002-9947-99-02151-0
L. Ya. Vulakh, The Markov spectra for triangle groups, J. Number Theory 67 (1997), no. 1, 11–28. MR 1485425, DOI 10.1006/jnth.1997.2181