A structure theorem for subgroups of $G{L}_n$ 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 $k$ of $A/\m_A$, we show that every closed subgroup $G$ of $G{L}_n(A)$ such that $G\mod{\m_A}\supseteq SL_n(\pmb{k})$ contains a conjugate of $S{L}_n(W(k{)}_A)$ under some small restrictions on $k$. Here $W(k{)}_A$ is the closed subring of $A$ generated by the Teichmüller lifts of elements of the subfield $k$.

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