| Title:
|
Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$ (English) |
| Author:
|
Sahin, R. |
| Author:
|
Bizim, O. |
| Author:
|
Cangul, I. N. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
253-259 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Hecke groups $H(\lambda _q)$ are the discrete subgroups of ${\mathrm PSL}(2,\mathbb{R})$ generated by $S(z)=-(z+\lambda _q)^{-1}$ and $T(z)=-\frac{1}{z} $. The commutator subgroup of $H$($\lambda _q)$, denoted by $H^{\prime }(\lambda _q)$, is studied in [2]. It was shown that $H^{\prime }(\lambda _q)$ is a free group of rank $q-1$. Here the extended Hecke groups $\bar{H}(\lambda _q)$, obtained by adjoining $R_1(z)=1/\bar{z}$ to the generators of $H(\lambda _q)$, are considered. The commutator subgroup of $\bar{H}(\lambda _q)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H(\lambda _q)$ case, the index of $H^{\prime }(\lambda _q)$ is changed by $q$, in the case of $\bar{H}(\lambda _q)$, this number is either 4 for $q$ odd or 8 for $q$ even. (English) |
| Keyword:
|
Hecke group |
| Keyword:
|
extended Hecke group |
| Keyword:
|
commutator subgroup |
| MSC:
|
11F06 |
| MSC:
|
20H05 |
| MSC:
|
20H10 |
| idZBL:
|
Zbl 1053.11038 |
| idMR:
|
MR2040237 |
| . |
| Date available:
|
2009-09-24T11:11:59Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127882 |
| . |
| Reference:
|
[1] R. B. J. T. Allenby: Rings, Fields and Groups. Second Edition.Edward Arnold, London-New York-Melbourne-Auckland, 1991. MR 1144518 |
| Reference:
|
[2] I. N. Cangül and D. Singerman: Normal subgroups of Hecke groups and regular maps.Math. Proc. Camb. Phil. Soc. 123 (1998), 59–74. MR 1474865, 10.1017/S0305004197002004 |
| Reference:
|
[3] H. S. M. Coxeter and W. O. J. Moser: Generators and Relations for Discrete Groups.Springer, Berlin, 1957. MR 0088489 |
| Reference:
|
[4] E. Hecke: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen.Math. Ann. 112 (1936), 664–699. MR 1513069, 10.1007/BF01565437 |
| Reference:
|
[5] D. L. Johnson: Topics in the Theory of Group Presentations. L.M.S. Lecture Note Series 42.Cambridge Univ. Press, Cambridge, 1980. MR 0695161 |
| Reference:
|
[6] G. A. Jones and J. S. Thornton: Automorphisms and congruence subgroups of the extended modular group.J. London Math. Soc. 34 (1986), 26–40. MR 0859146, 10.1112/jlms/s2-34.1.26 |
| . |
