On some free semigroups, generated by matrices

Piotr Słanina

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 289-299
  • ISSN: 0011-4642

Abstract

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Let A = 1 2 0 1 , B λ = 1 0 λ 1 . We call a complex number λ “semigroup free“ if the semigroup generated by A and B λ is free and “free” if the group generated by A and B λ is free. First families of semigroup free λ ’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free λ ’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.

How to cite

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Słanina, Piotr. "On some free semigroups, generated by matrices." Czechoslovak Mathematical Journal 65.2 (2015): 289-299. <http://eudml.org/doc/270096>.

@article{Słanina2015,
abstract = {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.},
author = {Słanina, Piotr},
journal = {Czechoslovak Mathematical Journal},
keywords = {free semigroup; semigroup of matrices},
language = {eng},
number = {2},
pages = {289-299},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some free semigroups, generated by matrices},
url = {http://eudml.org/doc/270096},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Słanina, Piotr
TI - On some free semigroups, generated by matrices
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 289
EP - 299
AB - 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.
LA - eng
KW - free semigroup; semigroup of matrices
UR - http://eudml.org/doc/270096
ER -

References

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  1. Bamberg, J., 10.1112/S0024610700001630, J. Lond. Math. Soc., II. Ser. 62 (2000), 795-801. (2000) Zbl1019.20014MR1794285DOI10.1112/S0024610700001630
  2. Brenner, J. L., Quelques groupes libres de matrices, C. R. Acad. Sci., Paris 241 French (1955), 1689-1691. (1955) Zbl0065.25402MR0075952
  3. Brenner, J. L., Charnow, A., 10.2140/pjm.1978.77.57, Pac. J. Math. 77 (1978), 57-69. (1978) Zbl0382.20044MR0507620DOI10.2140/pjm.1978.77.57
  4. Chang, B., Jennings, S. A., Ree, R., 10.4153/CJM-1958-029-2, Can. J. Math. 10 (1958), 279-284. (1958) Zbl0081.25902MR0094388DOI10.4153/CJM-1958-029-2
  5. Harpe, P. de la, Topics in Geometric Group Theory, Chicago Lectures in Mathematics The University of Chicago Press, Chicago (2000). (2000) Zbl0965.20025MR1786869
  6. Fuchs-Rabinowitsch, D. J., On a certain representation of a free group, Leningrad State Univ. Annals Math. Ser. 10 Russian (1940), 154-157. (1940) MR0003414
  7. Ignatov, J. A., Free and nonfree subgroups of P S L 2 ( ℂ ) that are generated by two parabolic elements, Mat. Sb., Nov. Ser. 106 Russian (1978), 372-379. (1978) MR0505107
  8. Ignatov, Y. A., Evtikhova, A. V., Free groups of linear-fractional transformations, Chebyshevskiĭ Sb. 3 Russian (2002), 78-81. (2002) Zbl1112.20045MR2023624
  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. (1999) MR1749349
  10. Keen, L., Series, C., The Riley slice of Schottky space, Proc. Lond. Math. Soc. (3) 69 (1994), 72-90. (1994) Zbl0807.30031MR1272421
  11. Lyndon, R. C., Ullman, J. L., 10.4153/CJM-1969-153-1, Can. J. Math. 21 (1969), 1388-1403. (1969) Zbl0191.01904MR0258975DOI10.4153/CJM-1969-153-1
  12. Robinson, D. J. S., A Course in the Theory of Groups, Graduate Texts in Mathematics 80 Springer, New York (1996). (1996) MR1357169
  13. Saks, S., Zygmund, A., Analytic Functions, PWN-Polish Scientific Publishers Warsaw (1965). (1965) Zbl0136.37301MR0180658
  14. Sanov, I. N., A property of a representation of a free group, Dokl. Akad. Nauk SSSR, N. Ser. 57 Russian (1947), 657-659. (1947) MR0022557
  15. Słanina, P., On some free groups, generated by matrices, Demonstr. Math. (electronic only) 37 (2004), 55-61. (2004) Zbl1050.20016MR2053102

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