Previous |  Up |  Next

Article

Keywords:
free semigroup; semigroup of matrices
Summary:
Let $$ A=\left [ \begin {matrix} 1 & 2 \\ 0 & 1 \end {matrix} \right ],\quad B_{\lambda }=\left [ \begin {matrix} 1 & 0 \\ \lambda & 1 \end {matrix} \right ]. $$ We call a complex number $\lambda $ “semigroup free“ if the semigroup generated by $A$ and $B_{\lambda }$ is free and “free” if the group generated by $A$ and $B_{\lambda }$ is free. First families of semigroup free $\lambda $'s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free $\lambda $'s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common picture.
References:
[1] Bamberg, J.: Non-free points for groups generated by a pair of {$2\times 2$} matrices. J. Lond. Math. Soc., II. Ser. 62 (2000), 795-801. DOI 10.1112/S0024610700001630 | MR 1794285 | Zbl 1019.20014
[2] Brenner, J. L.: Quelques groupes libres de matrices. C. R. Acad. Sci., Paris 241 French (1955), 1689-1691. MR 0075952 | Zbl 0065.25402
[3] Brenner, J. L., Charnow, A.: Free semigroups of {$2\times 2$} matrices. Pac. J. Math. 77 (1978), 57-69. DOI 10.2140/pjm.1978.77.57 | MR 0507620 | Zbl 0382.20044
[4] Chang, B., Jennings, S. A., Ree, R.: On certain pairs of matrices which generate free groups. Can. J. Math. 10 (1958), 279-284. DOI 10.4153/CJM-1958-029-2 | MR 0094388 | Zbl 0081.25902
[5] Harpe, P. de la: Topics in Geometric Group Theory. Chicago Lectures in Mathematics The University of Chicago Press, Chicago (2000). MR 1786869 | Zbl 0965.20025
[6] Fuchs-Rabinowitsch, D. J.: On a certain representation of a free group. Leningrad State Univ. Annals Math. Ser. 10 Russian (1940), 154-157. MR 0003414
[7] Ignatov, J. A.: Free and nonfree subgroups of $ PSL_{2}(\mathbb{C})$ that are generated by two parabolic elements. Mat. Sb., Nov. Ser. 106 Russian (1978), 372-379. MR 0505107
[8] Ignatov, Y. A., Evtikhova, A. V.: Free groups of linear-fractional transformations. Chebyshevskiĭ Sb. 3 Russian (2002), 78-81. MR 2023624 | Zbl 1112.20045
[9] Ignatov, Y. A., Gruzdeva, T. N., Sviridova, I. A.: Free groups of linear-fractional transformations. Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform. 5 (1999), 116-120. MR 1749349
[10] Keen, L., Series, C.: The Riley slice of Schottky space. Proc. Lond. Math. Soc. (3) 69 (1994), 72-90. MR 1272421 | Zbl 0807.30031
[11] Lyndon, R. C., Ullman, J. L.: Groups generated by two parabolic linear fractional transformations. Can. J. Math. 21 (1969), 1388-1403. DOI 10.4153/CJM-1969-153-1 | MR 0258975 | Zbl 0191.01904
[12] Robinson, D. J. S.: A Course in the Theory of Groups. Graduate Texts in Mathematics 80 Springer, New York (1996). MR 1357169
[13] Saks, S., Zygmund, A.: Analytic Functions. PWN-Polish Scientific Publishers Warsaw (1965). MR 0180658 | Zbl 0136.37301
[14] Sanov, I. N.: A property of a representation of a free group. Dokl. Akad. Nauk SSSR, N. Ser. 57 Russian (1947), 657-659. MR 0022557
[15] Słanina, P.: On some free groups, generated by matrices. Demonstr. Math. (electronic only) 37 (2004), 55-61. Zbl 1050.20016
Partner of
EuDML logo