On the universal deformations for SL₂-representations of knot groups

Based on the analogies between knot theory and number theory, we study a deformation theory for SL2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL₂-representations, we prove the existence of the universal deformation of a given SL₂-representation of a finitely generated group π over a perfect field k whose characteristic is not 2. We then show its connection with the character scheme for SL₂-representations of π when k is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.