  We study a generalization of the Fuchsian triangle groups to the hyperbolic
3-space, namely, the groups generated by half-turns in three hyperbolic lines.
The role of the hyperbolic triangles is now played by the right-angled
hexagons. This class of groups is very close to the arbitrary 2-generator
Kleinian groups but appear to be much more geometric by the construction. The
main idea of our research is to make use of the geometric properties of the
generalized triangle groups to study 2-generator arithmetic Kleinian groups. In
this paper we start with a parameterization for the generalized triangle groups
by three complex numbers. Then we give a matrix representation and introduce an
arithmeticity test for the group in terms of the given parameters. Finally, we
apply the arithmeticity test to prove the finiteness of the number of the
arithmetic groups in a partial case when all three parameters are equal. For
this partial case we also describe the parameters of the non cocompact
arithmetic triangle groups and present the singular sets of the corresponding
factor-orbifolds.
