A structure theorem for subgroups of GLn over complete local Noetherian rings with large residual image

Given a complete local Noetherian ring (A, mA) with finite residue field and a subfield k of A/mA, we show that every closed subgroup G of GLn(A) such that G mod mA ⊇ SLn(k) contains a conjugate of SLn(W(k)A) under some small restrictions on k. Here W(k)A is the closed subring of A generated by the Teichm¨uller lifts of elements of the subfield k.