Deformation rings and images of Galois representations

Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with residue field $\mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $\mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $\mathcal{G}(R)$ with full residual image $\mathcal{G}(\mathbb{F})$ is a conjugate of a group $\mathcal{G}(A)$ for $A\subset R$ a closed subring that is local and has residue field $\mathbb{F}$ . (2) Every surjective homomorphism $\mathcal{G}(R)\to\mathcal{G}(R')$ is, up to conjugation, induced from a ring homomorphism $R\to R'$. (3) The identity map on $\mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $\mathcal{G}(R)$ given by the reduction map $\mathcal{G}(R)\to\mathcal{G}(\mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $\mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $\mathcal{G}$, and we study in the case at hand in great detail what conditions on $\mathbb{F}$ or on $p$ in relation to $\mathcal{G}$ are necessary for the above results to hold.