Quantifying shape and position of discreteness loci in PSL(2,C)

A hyperbolic metric on a 3-manifold defines a discrete, faithful representation of the fundamental group into PSL(2,C). The deformation space of this representation in the character variety is essentially a Teichmueller space, but the way that this Teichmueller space is embedded is very complicated (the boundary of the embedding is fractal and in general not locally connected). We describe some semi-algebraic approximations of the deformation spaces which give certificates of discreteness for certain subgroups of PSL(2,C). As well as having applications to the enumeration of arithmetic groups and the theory of hyperbolic 2-bridge links, these certificates have surprising utility in the study of q-rational numbers.