On the Universal Group PSL(2, C)

The group Γ = PSL(2, C) arises in a wide variety of contexts; hyperbolic geometry, automorphic function theory, number theory and group theory. Much of combinatorial group theory arose out of the study of discrete subgroups of Γ = PSL(2, C), in particular Fuchsian Groups and Kleinian groups. From the Poincaré polygon theorem surface groups can be faithfully represented in PSL(2, C). Extending this, most cyclically pinched one-relator groups can also be embedded in Γ . Recent results of Fine and Rosenberger ([61],[62]) show that all finitely generated fully residually free groups, the so called limit groups, can also be faithfully represented in this group. In this paper we survey the tremendous impact this single group has had on combinatorial group theory in particular and infinite group theory in general.