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

Abstract

Given a complete local Noetherian ring $(A,\m_A)$ with finite residue field and a subfield $\pmb{k}$ of $A/\m_A$, we show that every closed subgroup $G$ of $GL_n(A)$ such that $G\mod{\m_A}\supseteq SL_n(\pmb{k})$ contains a conjugate of $SL_n(W(\pmb{k})_A)$ under some small restrictions on $\pmb{k}$. Here $W(\pmb{k})_A$ is the closed subring of $A$ generated by the Teichm\"{u}ller lifts of elements of the subfield $\pmb{k}$.