250 References [17] E. B. Burger: Exploring the Number Jungle: A Journey into Diophantine Analysis. American Mathematical Society [18] J. O. But

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2 250 References [17] E. B. Burger: Exploring the Number Jungle: A Journey into Diophantine Analysis. American Mathematical Society [18] J. O. Button: The uniqueness of the prime Markoff numbers. J. London Math. Soc. 58(2) (1998), [19] J. O. Button: Markoff numbers, principal ideals and continued fraction expansions. Journal of Number Theory 87 (2001), [20] J. W. S. Cassels: The Markoff chain. Annals of Math. 50 (1949), [21] J. W. S. Cassels: An Introduction to Diophantine Approximation. Cambridge University Press [22] E. B. Christoffel: Observatio arithmetica. Annali di Matematica 6 (1875), [23] E. B. Christoffel: Lehrsätze über arithmetische Eigenschaften der Irrationalzahlen. Annali di Matematica Pura ed Applicata, Series II 15 (1888), [24] D. Clemens: Von der diophantischen Approximation zu Matchings in Schlangengraphen Eigenschaften und Anwendungen von Markoffzahlen und Cohnmatrizen. Diplomarbeit (Master s thesis), Freie Universität Berlin [25] H. Cohen: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 238, 4th edition, Springer [26] M. Cohen, W. Metzler, and A. Zimmermann: What does a basis of F(a, b) look like? Mathematische Annalen 257 (1981), [27] H. Cohn: Approach to Markoff s minimal forms through modular functions. Annals of Math. 61 (1955), [28] H. Cohn: Representation of Markoff s binary quadratic forms by geodesics on a perforated torus. Acta Arithmetica 18 (1971), [29] H. Cohn: Markoff forms and primitive words. Mathematische Annalen 196 (1972), [30] H. Cohn: Growth types of Fibonacci and Markoff. Fibonacci Quarterly 17(2) (1979), [31] H. Cohn: Minimal geodesics on Fricke s torus-covering. In: Proceedings of the Conference on Riemann Surfaces and Related Topics (Stony Brook, 1978, I. Kra and B. Maskit, eds.), Ann. Math. Studies 97, Princeton Univ. Press (1981), [32] D. A. Cox : Primes of the Form x 2 + ny 2. Wiley Interscience [33] H. S. M. Coxeter and W. O. J. Moser: Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 14, 4th edition, Springer [34] T. W. Cusick: The connection between the Lagrange and Markoff spectra. Duke Math. Journal 42 (1975), [35] T. W. Cusick and M. E. Flahive: The Markoff and Lagrange Spectra. Mathematical Surveys and Monographs 30, AMS [36] L. E. Dickson: Studies in the Theory of Numbers. Chelsea Publishing Co. 1957, reprint of the 1930 first edition.

3 References 251 [37] A. Dujella and A. Pethő: Generalization of a theorem of Baker and Davenport. Oxford Quarterly Journal of Mathematics 49 (1998), [38] N. J. Fine and H. S. Wilf: Uniqueness theorem for periodic functions. Proc. Amer. Math. Soc. 16 (1965), [39] A. S. Fraenkel, M. Mushkin, and U. Tassa: Determination of [nθ] by its sequence of differences. Canad. Math. Bull. 21 (1978), [40] G. A. Freiman: Noncoincidence of the Markoff and Lagrange spectra. Mat. Zametki 3 (1968), , English translation: Math. Notes 3 (1968), [41] G. A. Freiman: Diophantine approximation and geometry of numbers (the Markoff spectrum). Kalininskii Gosudarstvennyi Universitet, Moscow [42] R. Fricke: Über die Theorie der automorphen Modulgruppen. Nachrichten Ges. Wiss. Göttingen (1896), [43] G. Frobenius: Über die Markoffschen Zahlen. Gesammelte Abhandlungen, Band III, Springer (1968), [44] A. M. W. Glass: The ubiquity of free groups. Math. Intelligencer 14(3) (1992), [45] A. Glen, A. Lauve, and F. V. Saliola: A note on the Markoff condition and central words. Information Processing Letters 105 (2008), [46] F. González-Acuña and A. Ramirez: A composition formula in the rank two free group. Proc. Amer. Math. Soc. 127 (1999), [47] D. S. Gorshkov: Geometry of Lobachevskii in connection with certain questions of arithmetic. Zap. Nauch. Sem. Lenin. Otd. Math. Inst. V. A. Steklova AN SSSR 67 (1977), 39 85, English translation in J. Soviet Math. 16 (1981), [48] C. Gurwood: Diophantine approximation and the Markoff chain. Thesis, New York University [49] R. Guy: Don t try to solve these problems. Amer. Math. Monthly 90 (1983), [50] R. Guy: Unsolved Problems in Number Theory, third edition. Springer [51] A. Haas: Diophantine approximation on hyperbolic Riemann surfaces. Acta Mathematica 156 (1986), [52] M. Hall, Jr.: On the sum and product of continued fractions. Annals of Math. 48 (2) (1947), [53] G. H. Hardy and E. M. Wright: An Introduction to the Theory of Numbers, 6th edition. Oxford University Press [54] F. Hirzebruch and D. Zagier: The Atiyah Singer theorem and elementary number theory. Publish or Perish 1974, Chapter 8. [55] A. Hurwitz: Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Mathematische Annalen 39 (1891), [56] A. Hurwitz: Über eine Aufgabe der unbestimmten Analysis. Arch. Math. Phys. 11 (3) (1907), [57] B. W. Jones: The Arithmetic Theory of Quadratic Forms, 3rd edition. Carus Mathematical Monographs 10, Mathematical Association of America 1967.

4 252 References [58] C. Kassel and C. Reutenauer: Sturmian morphisms, the braid group B 4, Christoffel words and bases of F 2. Annali di Matematica Pura ed Applicata 186 (2007), [59] M. A. Korkine and G. Zolotareff: Sur les formes quadratiques. Mathe-matische Annalen 6 (1873), [60] E. H. Kuo: Applications of graphical condensation for enumerating matchings and tilings. Theoretical Computer Science 319 (2004), [61] M. L. Lang and S. P. Tan: A simple proof of the Markoff conjecture for prime powers. arxiv: math/ v1 (2005), 1 5. [62] J. Lehner and M. Sheingorn: Simple closed geodesics on H + /Γ (3) arise from the Markov spectrum. Bull. Amer. Math. Soc. 11(2) (1984), [63] M. Lothaire: Algebraic Combinatorics on Words. Cambridge University Press [64] A. de Luca: Sturmian words: structure, combinatorics, and their arithmetics. Theoretical Computer Science 183(1) (1997), [65] A. de Luca and F. Mignosi: On some combinatorial properties of Sturmian words. Theoretical Computer Science 136(2) (1994), [66] R. C. Lyndon and P. E. Schupp: Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer [67] W. Magnus, A. Karrass, and D. Solitar: Combinatorial Group Theory. Presentation of Groups in Terms of Generators and Relations. Dover Publications 2004, reprint of the 1976 second edition. [68] A. V. Malyshev: Markov and Lagrange spectra [Survey of the literature]. Zap. Nauch. Sem. Lenin. Otd. Math. Inst. V. A. Steklova AN SSSR 67 (1977), 5 38, English translation in J. Soviet Math. 16 (1981), [69] A. A. Markoff: Sur les formes quadratiques binaires indéfinies. Mathematische Annalen 15 (1879), [70] A. A. Markoff: Sur les formes quadratiques binaires indéfinies II. Mathematische Annalen 17 (1880), [71] A. A. Markoff: Sur une question de Jean Bernoulli. Mathematische Annalen 19 (1882), [72] E. M. Matveev: An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II. Izvestiya Rossiiskaya Akad. Nauk Ser. Matem. 64 (2000), , English translation om Izv. Math. 64 (2000), [73] J. Meier: Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups (London Mathematical Society Student Texts 73). Cambridge University Press [74] R. A. Mollin: Quadratics. CRC Press [75] R. A. Mollin: Fundamental Number Theory with Applications. CRC Press [76] R. A. Mollin: Algebraic Number Theory. CRC Press [77] M. Morse and G. A. Hedlund: Symbolic dynamics II: Sturmian trajectories. American Journal of Mathematics 62 (1940), 1 42.

5 References 253 [78] B. H. Neumann: Die Automorphismen der freien Gruppen. Mathematische Annalen 107 (1933), [79] J. Nielsen: Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Mathematische Annalen 78 (1918), [80] I. Niven: Irrational Numbers. Carus Math. Monographs 11, Mathematical Association of America, Wiley [81] I. Niven, H. S. Zuckerman, and H. L. Montgomery: An Introduction to the Theory of Numbers, 5th edition. Wiley [82] R. P. Osborne and H. Zieschang: Primitives in the free group on two generators. Inventiones mathematicae 63 (1981), [83] S. Perrine: Sur une généralisation de la théorie de Markoff. Journal of Number Theory 37 (1991), [84] S. Perrine: L interprétation matricielle de la théorie de Markoff classique. International Journal of Mathematical Sciences 32 (2002), [85] S. Perrine: La théorie de Markoff et ses développements. Tessier & Ashpool [86] S. Perrine: De Frobenius à Riedel: analyse des solutions de l équation de Markoff. archives ouvertes 2009, 145 pp. [87] O. Perron: Über die Approximation irrationaler Zahlen durch rationale. Sitzungsberichte Heidelberg Akad. Wiss., Abh. 4 (1921), 17 pp. [88] O. Perron: Über die Approximation irrationaler Zahlen durch rationale II. Sitzungsberichte Heidelberg Akad. Wiss., Abh. 8 (1921), 12 pp. [89] O. Perron: Die Lehre von den Kettenbrüchen. Teubner [90] J. Propp: The combinatorics of frieze patterns and Markoff numbers. arxiv: math/ v4 (2008), [91] R. A. Rankin: Modular Forms and Functions. Cambridge University Press [92] G. Rauzy: Mots infinis en arithmétique. In: Automata on Infinite Words (M. Nivat and D. Perrin, eds.). Lecture Notes Computer Science, Vol. 192 (1985), , Springer. [93] R. Remak: Über indefinite binäre quadratische Minimalformen. Mathematische Annalen 92 (1924), [94] C. Reutenauer: On Markoff s property and Sturmian words. Mathematische Annalen 336 (2006), [95] C. Reutenauer: Christoffel words and Markoff triples. Integers 9 (2009), [96] N. Riedel: Markoff equation and nilpotent matrices. arxiv: math/ v5 (2009), [97] D. Rosen and G. S. Patterson, Jr.: Some numerical evidence concerning the uniqueness of the Markov numbers. Mathematics of Computation 25 (1971), [98] G. Rosenberger: The uniqueness of the Markoff numbers. Mathematics of Computation 30 (1976),

6 254 References [99] K. E. Roth: Rational approximations to algebraic numbers. Mathematika 2 (1955), [100] A. N. Rudakov: Markov numbers and exceptional bundles on P 2. Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 1, , English translation in Math. USSR Izv. 32 (1989), [101] H. Schecker: Über die Menge der Zahlen, die als Minima quadratischer Formen auftreten. Journal of Number Theory 9 (1977), [102] P. Schmutz: Systoles of arithmetic surfaces and the Markoff spectrum. Mathematische Annalen 305 (1996), [103] V. Senkel: Markoffzahlen. Diplomarbeit (Master s thesis), Universität Bielefeld [104] C. Series: The geometry of Markoff numbers. Math. Intelligencer 7(3) (1985), [105] C. Series: The modular surface and continued fractions. J. London Math. Soc. 31 (2) (1985), [106] J.-P. Serre: A Course in Arithmetic. Graduate Texts in Mathematics 7, Springer [107] J. H. Silverman: The Markoff equation x 2 + y 2 + z 2 = axyz over quadratic imaginary fields. Journal of Number Theory 37 (1990), [108] N. J. A. Sloane: Online Encyclopedia of Integer Sequences. Sequence A [109] H. J. S. Smith: Note on continued fractions. Messenger Math. 6 (1876), [110] A. Srinivasan: A really simple proof of the Markoff conjecture for prime powers, [111] A. Srinivasan: Markoff numbers and ambiguous classes. Journal de Théorie des Nombres de Bordeaux 21 (3) (2009), [112] K. B. Stolarsky: Beatty sequences, continued fractions, and certain shift operators. Canad. Math. Bull. 19 (1976), [113] L. Tornheim: Asymmetric minima of quadratic forms and asymmetric Diophantine approximation. Duke Math. J. 22 (1955), [114] M. Waldschmidt: Open Diophantine problems. Moscow Math. Journal 4(1) (2004), [115] K. Weller: Christoffelwörter, Matchings und Matrizen Markoffzahlen in verschiedenen Kontexten. Diplomarbeit (Master s thesis), Freie Universität Berlin [116] S.-I. Yasutomi: The continued fraction expansion of α with μ(α) = 3. Acta Arithmetica 84 (1998), [117] D. Zagier: On the number of Markoff numbers below a given bound. Mathematics of Computation 39 (1982), [118] Y. Zhang: An elementary proof of Markoff conjecture for prime powers. arxiv: math/ v1 (2006), 1 7. [119] Y. Zhang: An elementary proof of uniqueness of Markoff numbers which are prime powers. arxiv:math/ v2 (2007), [120] Y. Zhang: Congruence and uniqueness of certain Markoff numbers. Acta Arithmetica 128 (2007),

7 Index X freely generates G, 113 Δ-equivalent, 102 Q 0,1 rationals between 0 and 1, 53 m-chain, 236 nth Farey row, 51 nth convergent, 13 abelianization, 124 admissible sequence, 190 algebraic integer, 239 algebraic number of degree d, 5 aperiodic word, 160 approximation to order t, 5 associated elements in ring, 234 associated sequence, 187 associated word, 194 automorphism group AutF(a, b), 124 automorphism group AutZ(a, b), 123 axis of map, 97 badly approximable numbers, 27 balanced set of words, 161 balanced word, 150, 159 basis of Z 2, 122 basis of F 2, 124 basis of free group, 114 block of sequence, 192 central word, 179, 221 chain C t, 226 characteristic number, 37, 56, 71 characteristic sequence, 196 characteristic word c α, 173 Chinese remainder theorem, 59 Christoffel word ch p/q, 152 class group of R, 241 closed geodesic, 105 Cohn matrix, 66, 119 Cohn matrix C t (a), 70 Cohn tree T C,68 Cohn triple, 66, 119 Cohn word, 120, 197 commutator, 89 commutator subgroup SL(2, Z), 117 commutator subgroup G,90 conjugacy class, 82 conjugate group elements, 82 conjugate ideal, 238 conjugate quadratic irrational, 15 conjugate ring element, 234 constant sequence, 193 continued fraction expansion, 9 convergent sequence of words, 160 corresponding element in R, 236 cusp, 96 cut u,vof sequence188 defining set of relations, 115 definite quadratic form, 36 degenerate sequence, 193 Diophantine equations, 31 directive sequence, 174 discriminant, 36 distance in hyperbolic plane, 109 domino graph D( p q ), 142 doubly infinite sequence, 187 doubly infinite word, 181, 194 elliptic map, 95 equivalent geodesics, 109 equivalent numbers, 24, 93, 200 Euclid tree, 211 Euclidean number, 211 Euler totient function, 211 factor of an ideal, 240 factor of word, 150 factorial set of words, 161 factorization into prime ideals, 242 Farey index, 54 M. Aigner, Markov s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings, DOI / , Springer International Publishing Switzerland

8 256 Index Farey neighbors, 51 Farey table, 51, 208 Farey tree T F,53 Farey triple, 51 Fibonacci branch, 49, 73, 232 Fibonacci numbers, 11, 49, 148, 208 Fibonacci word f, 160 Fibonacci words W 1, 122 n finite word, 113 fixed points of a map, 95 formal inverse, 113 fractional R-ideal, 240 fractional linear map, 92 free abelian group Z 2 of rank 2, 116 free group F n of rank n, 115 free group with basis X, 113 freely reduced word, 113 fundamental domain, 102 fundamental unit, 235 general linear group GL(2, Z), 63 general weight of word, 224 group generated by X, 81 harmonic number, 214 height h(w) of word, 150 hitting word h α,ρ, 171 homogeneous modular group, 94 hyperbolic lines, 96 hyperbolic map, 95 hyperbolic plane H, 96 hyperbolic points, 96 ideal, 238 identity of Fricke, 64 indefinite quadratic form, 36 index of Cohn matrix, 70 infinite word, 159 inhomogeneous modular group, 94 inner automorphism, 125 integral ideal, 240 intercept of line, 163 invertible ideal, 240 juxtaposition of words, 114 Lagrange number, 8 Lagrange number of sequence, 189 Lagrange spectrum L, 8 Lagrange spectrum L (d), 189 lattice path L p q, 136 lattice path l p q, 136 Lemma of Baker Davenport, 230 length of word, 113 lexicographic order, 154 limit point of the pair sequence (α n+1,β n ), 187 list of Markov numbers, 219 lower Christoffel word ch p q, 137 lower mechanical word s α,ρ, 164 Markov condition for words, 181, 194 Markov form, 37 Markov number, 31 Markov number of matrix, 66 Markov spectrum, 37 Markov tree, 207 Markov tree T M,47 Markov triple, 31 Markov word, 195, 196 Markov s equation, 31 Markov s theorem, 35, 185, 204 matching number, 144 mediant, 51 Möbius function, 214 Möbius inversion formula, 215 Möbius series, 215 Möbius transformation, 97 modular group SL(2, Z), 63 narrow class group, 241 nonsimple closed geodesic, 105 nonsingular Markov triple, 45 norm of element, 234 norm of ideal, 241 order of group element, 82 palindrome, 138 parabolic map, 95 Pell branch, 50, 73, 232 Pell equation, 20 Pell numbers, 12, 49, 148, 208 Pell words W n 1, 122 n perfect matching, 144 period of word, 151 periodic infinite continued fraction, 14 point at infinity, 93 positive automorphism, 128, 197

9 Index 257 prefix, 150 presentation of group, 113, 116 prime ideal, 240 primitive element in Z(a, b), 122 primitive element in F(a, b), 124 primitive ideal, 238 primitive map, 109 primitive matrix, 97 principal congruence group of level n, 100 principal ideal, 238 product of ideals, 240 projection π(a T ), 105 projection from F(a, b) to Z(a, b), 124 proper prefix, 150 proper suffix, 150 purely periodic continued fraction, 15 purely periodic word, 159 quadratic field, 234 quadratic form, 36 quadratic irrational, 15 quadratic residue, 55 reduced fraction, 51 reflection matrix, 92 regular sequence, 193 regular word, 195, 196 relation of generators, 115 reverse word, 138 Riemann ζ-function, 215 Riemann sphere, 93 Riemann surface, 102 right special factor, 161 ring O K, 239 ring R, 234 section of word, 195 semicircle, 96 set D of doubly infinite sequences, 189 set F n (x) of factors of length n, 159 set of generators, 81 shifted sequence, 188 simple closed geodesic, 105 simple continued fraction, 11 simple infinite continued fraction, 13 singular Markov triple, 45 slope σ (u) of word u, 165 slope of line, 163 snake graph S( p q ), 141 special linear group SL(2, Z), 63 stabilizer St(A T ),97 standard pair, 178 standard sequence of words, 174 standard tree T st, 179 standard weight, 224 standard word, 174 Stern Brocot tree, 54 strong Markov word, 195 strongly admissible sequence, 190 Sturmian word, 161 suffix, 150 trace, 64 transcendental number, 5 translation, 95 transpose, 64 tree-standard word, 178 ultimately periodic word, 159 unique Markov number, 58 uniqueness conjecture, 39, 48, 54, 68, 110, 148, 204, 227, 228 uniqueness property, 58 unit in ring, 234 upper Christoffel word Ch p q, 137 upper mechanical word S α,ρ, 164 weight of word, 221 word tree T W, 120