Geometry and Markoff’s spectrum for $\mathbb {Q}(i)$, I
HTML articles powered by AMS MathViewer
- by Ryuji Abe and Iain R. Aitchison PDF
- Trans. Amer. Math. Soc. 365 (2013), 6065-6102 Request permission
Abstract:
We develop a study of the relationship between geometry of geodesics and Markoff’s spectrum for $\mathbb {Q}(i)$. There exists a particular immersed totally geodesic twice punctured torus in the Borromean rings complement, which is a double cover of the once punctured torus having Fricke coordinates $(2\sqrt {2}, 2\sqrt {2}, 4)$. The set of the simple closed geodesics on this once punctured torus is decomposed into two subsets. The discrete part of Markoff’s spectrum for $\mathbb {Q}(i)$ (except for one) is given by the maximal Euclidean height of the lifts of the simple closed geodesics composing one of the subsets.References
- Ryuji Abe, On correspondences between once punctured tori and closed tori: Fricke groups and real lattices, Tokyo J. Math. 23 (2000), no. 2, 269–293. MR 1806465, DOI 10.3836/tjm/1255958671
- R. Abe and B. Rittaud, Combinatorics on words associated to Markoff spectra, preprint.
- I. R. Aitchison, E. Lumsden, and J. H. Rubinstein, Cusp structures of alternating links, Invent. Math. 109 (1992), no. 3, 473–494. MR 1176199, DOI 10.1007/BF01232034
- I. R. Aitchison and J. H. Rubinstein, Combinatorial cubings, cusps, and the dodecahedral knots, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 17–26. MR 1184399
- I. R. Aitchison and J. H. Rubinstein, Canonical surgery on alternating link diagrams, Knots 90 (Osaka, 1990) de Gruyter, Berlin, 1992, pp. 543–558. MR 1177446
- Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada, and Yasushi Yamashita, Punctured torus groups and 2-bridge knot groups. I, Lecture Notes in Mathematics, vol. 1909, Springer, Berlin, 2007. MR 2330319
- 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
- B. H. Bowditch, A proof of McShane’s identity via Markoff triples, Bull. London Math. Soc. 28 (1996), no. 1, 73–78. MR 1356829, DOI 10.1112/blms/28.1.73
- B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3) 77 (1998), no. 3, 697–736. MR 1643429, DOI 10.1112/S0024611598000604
- Harvey Cohn, Approach to Markoff’s minimal forms through modular functions, Ann. of Math. (2) 61 (1955), 1–12. MR 67935, DOI 10.2307/1969618
- 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
- Harvey Cohn, Markoff forms and primitive words, Math. Ann. 196 (1972), 8–22. MR 297847, DOI 10.1007/BF01419427
- Harvey Cohn, Growth types of Fibonacci and Markoff, Fibonacci Quart. 17 (1979), no. 2, 178–183. MR 536967
- Harvey Cohn, Minimal geodesics on Fricke’s torus-covering, 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. 73–85. MR 624806
- Harvey Cohn, Remarks on the cyclotomic Fricke groups, Kleinian groups and related topics (Oaxtepec, 1981) Lecture Notes in Math., vol. 971, Springer, Berlin-New York, 1983, pp. 15–23. MR 690274
- 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
- Benjamin Fine, The structure of $\textrm {PSL}_{2}(R)$; $R$, the ring of integers in a Euclidean quadratic imaginary number field, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 145–170. MR 0352289
- Lester R. Ford, On the closeness of approach of complex rational fractions to a complex irrational number, Trans. Amer. Math. Soc. 27 (1925), no. 2, 146–154. MR 1501304, DOI 10.1090/S0002-9947-1925-1501304-X
- R. Fricke, Über die Theorie der automorphen Modulgruppen, Nachr. Akad. Wiss. Göttingen (1896), 91-101.
- 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, The geometry of Markoff forms, Number theory (New York, 1984–1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 135–144. MR 894509, DOI 10.1007/BFb0072978
- Linda Keen, Intrinsic moduli on Riemann surfaces, Ann. of Math. (2) 84 (1966), 404–420. MR 203000, DOI 10.2307/1970454
- 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
- W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Interscience Publishers, New York, 1966.
- A. V. Malyshev, Markov and Lagrange spectra, Zap. Nauch. Sem. Leningrad. Otdel. Math. Inst. Steklov. 67 (1977), 3-38. [transl., J. Soviet Math. 16 (1981), 767-788]
- A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann. 15 (1879), 381-406.
- A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann. 17 (1880), no. 3, 379–399 (French). MR 1510073, DOI 10.1007/BF01446234
- Toshihiro Nakanishi and Marjatta Näätänen, The Teichmüller space of a punctured surface represented as a real algebraic space, Michigan Math. J. 42 (1995), no. 2, 235–258. MR 1342488, DOI 10.1307/mmj/1029005226
- John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730, DOI 10.1007/978-1-4757-4013-4
- Asmus L. Schmidt, Diophantine approximation of complex numbers, Acta Math. 134 (1975), 1–85. MR 422168, DOI 10.1007/BF02392098
- Asmus L. Schmidt, On $C$-minimal forms, Math. Ann. 215 (1975), 203–214. MR 376530, DOI 10.1007/BF01343890
- 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
- Paul Schmutz, Systoles of arithmetic surfaces and the Markoff spectrum, Math. Ann. 305 (1996), no. 1, 191–203. MR 1386112, DOI 10.1007/BF01444218
- Caroline Series, The geometry of Markoff numbers, Math. Intelligencer 7 (1985), no. 3, 20–29. MR 795536, DOI 10.1007/BF03025802
- W.P. Thurston, The geometry and topology of 3-manifolds. Princeton University Lecture Notes 1978.
- 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. 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
- L. Ya. Vulakh, The Markov spectra for Fuchsian groups, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4067–4094. MR 1650046, DOI 10.1090/S0002-9947-00-02455-7
- Norbert Wielenberg, The structure of certain subgroups of the Picard group, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 427–436. MR 503003, DOI 10.1017/S0305004100055250
Additional Information
- Ryuji Abe
- Affiliation: Department of Mathematics, Tokyo Polytechnic University, Atsughi, Kanagawa 243-0297, Japan
- Email: ryu2abe@email.plala.or.jp
- Iain R. Aitchison
- Affiliation: Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
- Email: I.Aitchison@ms.unimelb.edu.au
- Received by editor(s): October 22, 2011
- Received by editor(s) in revised form: March 30, 2012
- Published electronically: August 1, 2013
- Additional Notes: The first author was partially supported by Université de Tours (LMPT) and Université de Caen (LMNO)
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 6065-6102
- MSC (2010): Primary 57M50, 20H10, 53C22, 11J06
- DOI: https://doi.org/10.1090/S0002-9947-2013-05850-3
- MathSciNet review: 3091276