The Picard group and the modular group

1. The classical modular group M is PSL 2 (ℤ) the 2 x 2 projective special linear group with integral entries. The modular group has been one of the most extensively studied single groups (see references). The reason for this is that M arises in so many different contexts - number theory, group theory, automorphic function theory, Riemann surfaces and elsewhere. Each of these disciplines claims the modular group for its own and looks at it in a slightly different manner. Related to M is its cousin the Pioard group T which is PSL 2 (ℤ[i]), the projective special linear group with Gaussian integer entries. Г has also been extensively studied (references) although the work more recent. Group theoretically r is quite similar to H [10]. However Г and M differ greatly in their action on the complex plane. Whereas M is a Fuchsian group, r is nowhere discontinuous in C and therefore has no Fuchsian subgroups of finite index [23]. What we will do in this survey is compare and contrast Г and M in four different areas - group theoretical structure, general subgroup structure, congruence subgroups and Fuchsian subgroups. As a general rule of thumb group theoretical properties of M will have close analogs in Г. As might be expected given the number theoretical similarities of ℤ and ℤ[i], the closer the group theoretical property reflects the underlying number theory (i.e., congruence subgroups) the closer the analog. Properties involving discontinuity however must be considerably revised.