Overview
This paper presents a complete classification of the conditions under which the pro-$p$ completion of an amalgamated product exhibits bounded generation. The investigation proceeds under the assumption that the initial vertex groups of the amalgamated product are themselves already boundedly generated. The study also develops new criteria to ascertain when original vertex and edge groups map into their completion without undergoing collapse. A counterexample is constructed to illustrate that bounded generation in a pro-$p$ completion at a single prime $p$ does not ensure this property for the full profinite completion.
Research Context
The research is situated within the study of pro-$p$ completions and amalgamated products, specifically focusing on the property of bounded generation. Bounded generation refers to a group property where every element can be expressed as a product of a bounded number of elements from a fixed set of generators. The concept of an amalgamated product involves constructing a new group from two groups by identifying a common subgroup. Pro-$p$ completions represent a specific type of completion for groups, where the completion is formed with respect to the family of normal subgroups whose indices are powers of a prime $p$. The study builds upon the understanding of how these different group-theoretic constructs interact, particularly concerning the preservation or emergence of bounded generation.
Approach
The approach involves two main phases. Initially, the researchers establish that the pro-$p$ completion of any abstract amalgam naturally forms a proper pro-$p$ amalgam. This foundational step provides a framework for analyzing the bounded generation property within this specific completed structure. Subsequent analysis within this framework focuses on identifying the precise conditions under which bounded generation occurs. This involves examining the indices of vertex groups relative to the edge group. The research also includes the development of a set of criteria to determine the faithful embedding of original vertex and edge groups into their pro-$p$ completion. A final component of the approach is the construction of a specific counterexample for any given prime $p$. This counterexample demonstrates a scenario where the pro-$p$ completion possesses bounded generation, but the full profinite completion of the same amalgam does not. This construction serves to highlight a distinction between pro-$p$ and full profinite properties.
Findings
- The pro-$p$ completion of any abstract amalgam naturally forms a proper pro-$p$ amalgam.
- Bounded generation in the completed group is an exceptionally rare occurrence.
- The completed group is boundedly generated if and only if one of two specific conditions is met:
- One of the vertex groups completely collapses into the edge group, implying it has an index of 1 over the edge group.
- The highly specific case where $p=2$ and both vertex groups have an index of exactly 2 over the edge intersection.
- A new set of criteria was developed for determining when original vertex and edge groups map into the completion without collapsing.
- A cautionary counterexample was constructed: for any given prime $p$, an amalgam exists whose pro-$p$ completion is boundedly generated, but its full profinite completion is not.
- This counterexample demonstrates that bounded generation at a single prime does not guarantee that the full profinite group will share this property.
Why This Matters
The findings clarify the structural properties of pro-$p$ completions of amalgamated products, specifically regarding bounded generation. The precise classification of conditions for bounded generation in these complex group structures contributes to the theoretical understanding of infinite groups and their completions. The counterexample illustrating the divergence between pro-$p$ and full profinite bounded generation highlights a nuance in group theory that warrants careful consideration in future research concerning group completions.