On a family of groups generated by parabolic matrices

We study various aspects of the family of groups generated by the parabolic matrices A(t1 ζ), ... , A(tm ζ) where A(z) = ( 1 z0 1 ) and by the elliptic matrix ( 0 -1  1 0 ). The elements of the matrices W in such groups can be computed by a recursion formula. These groups are special cases of the generalized parametrized modular groups introduced in [16].We study the sets {z : tr W(z) ∈ [-2; +2]} [13] and their critical points and geometry, furthermore some finite index subgroups and the discretness of subgroups.