On some pro- groups from infinite-dimensional Lie theory

We study some pro--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- groups. We show topological finite generation for their pro- 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- groups obtained from the same incomplete group. We also discuss the question of (non-)linearity of these pro- groups.