The normal subgroup structure of the extended Hecke groups

Abstract.

We consider the extended Hecke groups $\ifmmode\expandafter\bar\else\expandafter\=\fi{H}{\left( \lambda \right)}$ generated by T(z) = −1/z, S(z) = −1/(z + λ) and R(z) = 1/z with λ ≥ 2. In this paper, firstly, we study the fundamental region of the extended Hecke groups $\ifmmode\expandafter\bar\else\expandafter\=\fi{H}{\left( \lambda \right)}$. Then, we determine the abstract group structure of the commutator subgroups ${\ifmmode\expandafter\bar\else\expandafter\=\fi{H}}\ifmmode{'}\else$'$\fi{\left( \lambda \right)}$, the even subgroup $\ifmmode\expandafter\bar\else\expandafter\=\fi{H}_{e} {\left( \lambda \right)}$, and the power subgroups $\ifmmode\expandafter\bar\else\expandafter\=\fi{H}^{m} {\left( \lambda \right)}$ of the extended Hecke groups $\ifmmode\expandafter\bar\else\expandafter\=\fi{H}{\left( \lambda \right)}$. Also, finally, we give some relations between them.