Inverse Deformation Problems for Special Linear and Symplectic groups

The principal result of this thesis is an affirmative answer to the inverse deformation problem which asks: Does a given complete noetherian local ring have a realisation as the unrestricted universal deformation ring of any residual representation? This is proved in two ways: firstly a complete answer is given using the family of special linear groups over complete noetherian local rings and secondly, if the finite field does not have 3 elements or does not have characteristic 2, it is answered using the family of symplectic groups. Of central importance to the result in the symplectic case is the establishment of a structure theorem for subgroups of special linear groups which surject onto symplectic groups over finite fields.