Free subgroups of the parametrized modular group

We study free subgroups of index four of the parametrized modular group Π, the subgroup of SL(2; Z[ξ]) generated by (10 ξ1) and (01 -10). There are eight free subgroups, four of which are normal and four are non-normal. Then we study the intersections of the normal subgroups. We give canonical presentations in terms of generators and relations. At the end of the paper we study connections between and the Bianchi groups, the two-parabolic group and a group from relativity theory.