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Free Groups Generated by Two Heisenberg Translations

Published online by Cambridge University Press:  20 November 2018

BaoHua Xie ,
JieYan Wang  and
YuePing Jiang
BaoHua Xie
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China e-mail: [email protected]@[email protected]
JieYan Wang
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China e-mail: [email protected]@[email protected]
YuePing Jiang
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China e-mail: [email protected]@[email protected]
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Abstract.

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In this paper, we will discuss the groups generated by two Heisenberg translations of $\text{PU}\left( 2,1 \right)$ and determine when they are free.

Keywords

30F4022E4020H10free groupHeisenberg groupcomplex triangle group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 881 - 889
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Brenner, J. L., Quelques groupes libres de matrices. C. R. Acad. Sci. Pari. 241 (1955), 1689–1691.Google Scholar
[2] Chang, B., Jenning, S. A., and Ree, R., On certain pairs of matrices which generate free groups. Canad. J. Math. 10 (1958), 279–284. http://dx.doi.org/10.4153/CJM-1958-029-2 Google Scholar
[3] Goldman, W. M., Complex hyperbolic geometry. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999.Google Scholar
[4] Goldman, W. M. and Parker, J. R., Complex hyperbolic ideal triangle groups. J. Reine Angew. Math. 425 (1992), 71–86.Google Scholar
[5] Lyndon, R. C. and Ullman, J. L., Groups generated by two parabolic linear fractional transformations. Canad. J. Math. 21 (1969), 1388–1403. http://dx.doi.org/10.4153/CJM-1969-153-1 Google Scholar
[6] Massey, W. S., Algebraic topology: an introduction. Graduate Texts in Mathematics, 56, Springer-Verlag, New York-Heidelberg, 1977.Google Scholar
[7] Roger, R. C. and Schupp, P. E., Combinatorial group theory. Classics in Mathematics, Springer-Verlag, Berlin, 2001.Google Scholar
[8] Sanov, L. N., A property of a representation of a free group. (Russian) Doklady Akad. Nauk SSS. 57 (1947), 657–659.Google Scholar
[9] Schwartz, R. E., Ideal triangle groups, dented tori, and numerical analysis. Ann. of Math. 153 (2001), no. 3, 533–598. http://dx.doi.org/10.2307/2661362 Google Scholar
[10] Xie, B. and Jiang, Y.,Groups generated by two elliptic elements in PU(2; 1). Linear Algebra Appl. 433 (2010), no. 11–12, 2168–2177. http://dx.doi.org/10.1016/j.laa.2010.07.028 Google Scholar