UC Santa Barbara

Many well known representations of the braid groups are parameterized by a complex parameter, such as the Burau, Jones and BMW representations. This dissertation develops a construction for choosing specializations of the parameters so the images of the representations are discrete groups. This construction requires not only a parameterized representation, but the representations need to be sesquilinear. Squier showed that the Burau representation is sesquilinear. This dissertation extends Squier's result to all of the Jones and BMW representations, and finds discrete specializations of these representations.