Abstract
We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F 2) of automorphisms of the rank two free group F 2 and show that it can be realized as a monoid in the group B 4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F 2 lifting any given basis of the free abelian group Z 2. We further give an algorithm allowing to decide whether two elements of F 2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Allouche, J.-P., Shallit, J.: Automatic Sequences. Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
Bernoulli, J.: Sur une nouvelle espèce de calcul. In: Recueil pour les astronomes, t. 1. pp. 255–284. Springer, Berlin Heidelberg New York (1772)
Berstel, J., de Luca, A.: Sturmian words, Lyndon words and trees. Theoret. Comput. Sci. 178, 171–203 (1997)
Birman, J.S.: Automorphisms of the fundamental group of a closed, orientable 2-manifold. Proc. Amer. Math. Soc. 21, 351–354 (1969)
Birman, J.S.: Braids, links and map** class groups. Annals of Math. Studies, No. 82, Princeton University Press, Princeton (1975)
Birman, J.S., Hilden, H.M.: On the map** class groups of closed surfaces as covering spaces. In: Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies, vol. 66, pp. 81–115. Princeton University Press, Princeton (1971)
Borel, J.-P., Laubie, F.: Quelques mots sur la droite projective réelle. J. Théorie des Nombres de Bordeaux 5, 23–51 (1993)
Bourbaki, N.: Groupes et algèbres de Lie, chapter IV–VI. Hermann, Paris (1968)
Christoffel, E.B.: Observatio arithmetica. Ann. Mat. Pura Appl. 6, 148–152 (1875)
Cohn, H.: Markoff forms and primitive words. Math. Ann. 196, 8–22 (1972)
Coxeter, H.S.M., Moser, W.O.J.: Generators and relations for discrete groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 14, 4th edn. Springer, Berlin Heidelberg New York (1980)
Droubay, X., Pirillo, G.: Palindromes and Sturmian words. Theoret. Comput. Sci. 223, 73–85 (1999)
Dyer, J.L., Formanek, E., Grossman, E.K.: On the linearity of automorphism groups of free groups. Arch. Math. 38, 404–409 (1982)
Dyer, J.L., Grossman, E.K.: The automorphism groups of the braid groups. Amer. J. Math. 103, 1151–1169 (1981)
Fenchel, W.: Jakob Nielsen in memoriam. Acta Math. 103, VII–XIX (1960)
Gassner, B.J.: On braid groups. Abh. Math. Sem. Univ. Hamburg 25, 10–22 (1962)
González-Acuña, F., Ramí rez, A.: A composition formula in the rank two free group. Proc. Am. Math. Soc. 127, 2779–2782 (1999)
Gorin, E.A., Lin, V.Ya.: Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids. Mat. Sbornik 78(120), 579–610 (1969) (English translation: Math. USSR-Sbornik 7(4), 569–596 (1969))
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, Oxford (1979)
Helling, H.: A note on the automorphism group of the rank two free group. J. Algebra 223, 610–614 (2000)
Karrass, A., Pietrowski, A., Solitar, D.: Some remarks on braid groups. In: Contributions to braid groups, Contemp. Math. 33, 341–352. Providence: Amer. Math. Soc. (1984)
Lothaire, M.: Algebraic Combinatorics on Words, pp. 45–110. Cambridge University Press, Cambridge (2002)
de Luca, A., Mignosi, F.: On some combinatorial properties of Sturmian words. Theoret. Comput. Sci. 136, 361–385 (1994)
Markoff, A.: Sur une question de Jean Bernouilli. Math. Ann. 19, 27–36 (1882)
Mignosi, F., Séébold, P., Morphismes sturmiens et règles de Rauzy. J. Théorie des Nombres de Bordeaux 5, 221–233 (1993)
Nielsen, J.: Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Math. Ann. 78, 385–397 (1918)
Osborne, R.P., Zieschang, H.: Primitives in the free group on two generators. Invent. Math. 63, 17–24 (1981)
Pirillo, G.: A new characteristic property of the palindrome prefixes of a standard Sturmian word. Sém. Lothar. Combin., art. B43f, 4pp. (2000) (electronic)
Séébold, P.: On the conjugation of standard morphisms. Theoret. Comput. Sci. 195, 91–109 (1998)
Wen, Z.-X., Wen, Z.-Y.: Local isomorphisms of invertible substitutions. C.R. Acad. Sci., Paris 318(série I), 299–304 (1994)
Wollmershäuser, F.R.:Das Mathematische Seminar der Universität Strass-burg. In: Butzer, P.L., Fehér, F., (eds.). E.B. Christoffel, The Influence of His Work on Mathematics and the Physical Sciences, pp. 52–70. Birkhäuser Verlag, Basel, Boston, Stuttgart(1981)
Wynn, P.: The work of E.B. Christoffel on the theory of continued fractions. In: Butzer, P.L., Fehér, F., (eds.). E.B. Christoffel, The Influence of his Work on Mathematics and the Physical Sciences, pp. 190–202. Birkhäuser Verlag, Basel, Boston, Stuttgart (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15
Rights and permissions
About this article
Cite this article
Kassel, C., Reutenauer, C. Sturmian morphisms, the braid group B 4, Christoffel words and bases of F 2 . Annali di Matematica 186, 317–339 (2007). https://doi.org/10.1007/s10231-006-0008-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-006-0008-z