Groups and embeddings in SL(2, C)

Abstract

We give results on when a finitely generated torsion free group does or does not embed in SL(2, C). For instance if we glue two copies of the figure 8 knot along its torus boundary then the fundamental group of the resulting closed 3-manifold sometimes embeds in SL(2, C) and sometimes does not, depending on the identification, whereas a graph of free groups with maximal cyclic edge groups need not embed, even if it is word hyperbolic.