Residual profiles and bounded generation in pro-$p$ completions of amalgamated products

Abstract

This paper completely classifies when the pro-$p$ completion of an amalgamated product has bounded generation, assuming its starting vertex groups are already boundedly generated. We first prove that the pro-$p$ completion of any abstract amalgam always naturally forms a proper pro-$p$ amalgam. Within this framework, we show that bounded generation is extremely rare. Specifically, the completed group is boundedly generated if and only if one of the vertex groups completely collapses into the edge group (meaning it has an index of 1), or in the highly specific case where $p=2$ and both vertex groups have an index of exactly 2 over the edge intersection.Additionally, we develop a new set of criteria to determine when the original vertex and edge groups map into the completion without collapsing. Finally, we provide a cautionary counterexample: for any given prime $p$, we construct an amalgam whose pro-$p$ completion is boundedly generated, but its full profinite completion is not. This demonstrates that possessing bounded generation at a single prime does not guarantee that the full profinite group will share this property.