On the Universal Group PSL(2, C)

The group Gamma = 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 Gamma = PSL(2, C), in particular Fuchsian Groups and Kleinian groups. From the Poincare 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 Gamma. 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.