Group theory, Number theory
New sequences of non-free rational points
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 983-989.

We exhibit some new infinite families of rational values of τ, some of them squares of rationals, for which the group or even the semigroup generated by the matrices (1101) and  (10τ1) is not free.

Received:
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Accepted:
Published online:
DOI: 10.5802/crmath.230
Smilga, Ilia 1

1 Institut des Hautes Études Scientifiques et CNRS, 35 route de Chartres, 91440 Bures-sur-Yvette, France.
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Smilga, Ilia. New sequences of non-free rational points. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 983-989. doi : 10.5802/crmath.230. http://www.numdam.org/articles/10.5802/crmath.230/

[1] Bamberg, John Non-free points for groups generated by a pair of 2×2 matrices, J. Lond. Math. Soc., Volume 62 (2000) no. 3, pp. 795-801 | DOI | MR | Zbl

[2] Beardon, Alan F. Pell’s equation and two generator free Möbius groups, Bull. Lond. Math. Soc., Volume 25 (1993) no. 6, pp. 527-532 | DOI | Zbl

[3] Brenner, Joël L. Quelques groupes libres de matrices, C. R. Math. Acad. Sci. Paris, Volume 241 (1955), pp. 1689-1691 | MR | Zbl

[4] Brenner, Joël L.; Charnow, A. Free semigroups of 2×2 matrices, Pac. J. Math., Volume 77 (1978) no. 1, pp. 57-69 | DOI | MR | Zbl

[5] Chang, Bomshik; Jennings, Stephen A.; Ree, Rimhak On certain pairs of matrices which generate free groups, Can. J. Math., Volume 10 (1958), pp. 279-284 | DOI | MR | Zbl

[6] Gilman, Jane The structure of two-parabolic space: parabolic dust and iteration, Geom. Dedicata, Volume 131 (2008), pp. 27-48 | DOI | MR | Zbl

[7] Ignatov, Yu. A. Rational nonfree points of the complex plane, Algorithmic problems in the theory of groups and semigroups (Russian), Tul’skij Gosudarstvennyj Pedagogicheskij Institut, Tula, 1986, pp. 72-80 | MR

[8] Keen, Linda; Series, Caroline The Riley slice of Schottky space, Proc. Lond. Math. Soc., Volume 69 (1994) no. 1, pp. 72-90 | DOI | MR | Zbl

[9] Kim, Sang-Hyun; Koberda, Thomas Non-freeness of groups generated by two parabolic elements with small rational parameters (2020) (https://arxiv.org/abs/1901.06375, To appear in the Michigan Mathematical Journal, 2022)

[10] Lyndon, Roger C.; Ullman, Joseph L. Groups generated by two parabolic linear fractional transformations, Can. J. Math., Volume 21 (1969), pp. 1388-1403 | DOI | MR | Zbl

[11] Ree, Rimhak On certain pairs of matrices which do not generate a free group, Can. Math. Bull., Volume 4 (1961), pp. 49-52 | MR | Zbl

[12] Sanov, Ivan N. A property of a representation of a free group, Dokl. Akad. Nauk SSSR, n. Ser., Volume 57 (1947), pp. 657-659 | MR | Zbl

[13] Słanina, P. On some free semigroups, generated by matrices, Czech. Math. J., Volume 65(140) (2015) no. 2, pp. 289-299 | DOI | MR | Zbl

[14] Tan, Eng-Chye; Tan, Ser-Peow Quadratic Diophantine equations and two generator Möbius groups, J. Aust. Math. Soc., Ser. A, Volume 61 (1996) no. 3, pp. 360-368 | Zbl

[15] The Sage Developers SageMath, the Sage Mathematics Software System (Version 8.1), 2017 (http://www.sagemath.org)

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