Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$.

Title:

Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$.

Creator:

Şahin, Recep , Bizim, Osman , and Cangul, I. N.

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:3735f916-f5a6-4f1e-bde4-6371ef17eec5
uuid:3735f916-f5a6-4f1e-bde4-6371ef17eec5

Subject:

Hecke group, extended Hecke group, and commutator subgroup

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

253-259

Source:

Czechoslovak Mathematical Journal | 2004 Volume:54 | Number:1

Harvested from:

CDK

Metadata only:

false