On some pro- $$p$$ p groups from infinite-dimensional Lie theory

Abstract

We study some pro-\(p\)-groups arising from infinite-dimensional Lie theory. The starting point is incomplete Kac–Moody groups over finite fields. There are various completion procedures always providing locally pro-\(p\) groups. We show topological finite generation for their pro-\(p\) Sylow subgroups in most cases, whatever the (algebraic, geometric or representation-theoretic) completion. This implies abstract simplicity for complete Kac–Moody groups and provides identifications of the pro-\(p\) groups obtained from the same incomplete group. We also discuss the question of (non-)linearity of these pro-\(p\) groups.