Around the Lie correspondence for complete Kac-Moody groups and Gabber-Kac simplicity

Let $k$ be a field and $A$ be a generalised Cartan matrix, and let ${\mathfrak G}_A(k)$ be the corresponding minimal Kac-Moody group of simply connected type over $k$. Consider the completion ${\mathfrak G}_A^{pma}(k)$ of ${\mathfrak G}_A(k)$ introduced by O. Mathieu and G. Rousseau, and let ${\mathfrak U}_A^{ma+}(k)$ denote the unipotent radical of the positive Borel subgroup of ${\mathfrak G}_A^{pma}(k)$. In this paper, we exhibit some functoriality dependence of the groups ${\mathfrak U}_A^{ma+}(k)$ and ${\mathfrak G}_A^{pma}(k)$ on their Lie algebra. We also produce a large class of examples of minimal Kac-Moody groups ${\mathfrak G}_A(k)$ that are not dense in their Mathieu-Rousseau completion ${\mathfrak G}_A^{pma}(k)$. Finally, we explain how the problematic of providing a unified theory of complete Kac-Moody groups is related to the conjecture of Gabber-Kac simplicity of ${\mathfrak G}_A^{pma}(k)$, stating that every normal subgroup of ${\mathfrak G}_A^{pma}(k)$ that is contained in ${\mathfrak U}_A^{ma+}(k)$ must be trivial. We present several motivations for the study of this conjecture, as well as several applications of our functoriality theorem, with contributions to the question of (non-)linearity of ${\mathfrak U}_A^{ma+}(k)$, and to the isomorphism problem for complete Kac-Moody groups over finite fields. For $k$ finite, we also make some observations on the structure of ${\mathfrak U}_A^{ma+}(k)$ in the light of some important concepts from the theory of pro-$p$ groups.