Under mild conditions on p, closed subgroups of G(R) with full residual image G(F) are conjugates of G(A) for local subrings A, surjections arise from ring homomorphisms, and the identity is the universal deformation.
The inverse problem for universal deformation rings and the special linear group
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Deformation rings and images of Galois representations
Under mild conditions on p, closed subgroups of G(R) with full residual image G(F) are conjugates of G(A) for local subrings A, surjections arise from ring homomorphisms, and the identity is the universal deformation.