On the subgroups of the Picard group. (Q1403506)

The Picard group \(P=\text{PSL}(2,\mathbb Z[i])\) is a non-Fuchsian discrete subgroup of \(\text{PSL}(2,\mathbb C)\). The well-known modular group \(M=\text{PSL}(2,\mathbb Z)\) is a Fuchsian subgroup of the Picard group. The author obtains some results about some subgroups of the Picard group. Firstly, the author determines the group structure of the normal closure of \(M\) in \(P\) as \(M_1*_MM_2\) where \(M_1\cong M_2\cong S_3*_{\mathbb Z_3}A_4\). This is one of the three normal subgroups of \(P\) given by \textit{B. Fine} and \textit{M. Newman} [Trans. Am. Math. Soc. 302, 769-786 (1987; Zbl 0624.20031)] of index 2. Also, the author proves that the power subgroups \(P^{6n}\) of the Picard group are HNN groups. Furthermore, they are of infinite index in the Picard group.