[Paper Review] On the Theory of Ends of a pro-p Group

Kay Wingberg|arXiv (Cornell University)|May 3, 2013

Advanced Topology and Set Theory3 references3 citations

TL;DR

This paper establishes a pro-p analog of Stallings' decomposition theorem for finitely generated pro-p groups by introducing a cohomological invariant e(G) = 1 + h₁(G), where h₁(G) = dimFₚ H¹(G, Fₚ[[G]]). It proves that a finitely generated pro-p group G is freely decomposable if and only if its augmentation ideal IG is decomposable as a left Fₚ[[G]]-module, and for torsion-free groups, this is equivalent to h₁(G) = ∞. The key contribution is a complete cohomological characterization of free decomposability in the pro-p setting, extending classical results to the profinite context.

ABSTRACT

We study the group of ends of a pro-p group G and prove a pro-p analog of Stallings' decomposition theorem.

Motivation & Objective

• To develop a pro-p analog of Stallings’ decomposition theorem for abstract groups in the context of profinite groups.
• To define and study the number of ends e(G) for pro-p groups using continuous cohomology H¹(G, Fₚ[[G]]).
• To characterize free decomposability of finitely generated pro-p groups via module-theoretic properties of the augmentation ideal IG.
• To establish equivalences between cohomological invariants, module decomposability, and structural properties like duality and indecomposability.
• To extend classical results on abstract groups to the pro-p setting, particularly for torsion-free and duality groups.

Proposed method

• Define the number of ends e(G) = 1 + h₁(G), where h₁(G) = dimFₚ H¹(G, Fₚ[[G]]), generalizing the abstract group notion to pro-p groups.
• Use the completed group ring Fₚ[[G]] and its augmentation ideal IG to study cohomological invariants via continuous group cohomology.
• Apply the Krull-Schmidt-Azumaya theorem for finitely generated ΛG-modules to analyze decomposability of IG as a left Fₚ[[G]]-module.
• Establish equivalence between free decomposability of G and decomposability of IG as a ΛG-module.
• Use Frobenius reciprocity and induced module techniques to compute H¹(G, Fₚ[[G]]) as a right Fₚ[[G]]-module.
• Relate the invariant f(G) = dimFₚ H¹(G, Fₚ[[G]])^G to the number of freely indecomposable factors s(G), showing s(G) = f(G) or f(G)+1 depending on freeness.

Experimental results

Research questions

• RQ1 Can Stallings’ decomposition theorem for abstract groups be extended to the pro-p setting using cohomological invariants?
• RQ2 What is the pro-p analog of the number of ends e(G), and how does it behave for infinite pro-p groups?
• RQ3 Under what conditions is a finitely generated pro-p group freely decomposable, and how can this be detected cohomologically?
• RQ4 How does the structure of H¹(G, Fₚ[[G]]) as a right Fₚ[[G]]-module relate to the free decomposition type of G?
• RQ5 What is the relationship between the cohomological dimension, duality, and free decomposability in pro-p groups?

Key findings

• For a finitely generated pro-p group G, e(G) = 1 + h₁(G), with e(G) ∈ {0, 1, 2, ∞}, and e(G) = 0 if and only if G is finite.
• G is freely decomposable if and only if its augmentation ideal IG is decomposable as a left Fₚ[[G]]-module.
• For torsion-free G, h₁(G) = ∞ if and only if G is freely decomposable, and s(G) = f(G) + 1, where s(G) is the number of freely indecomposable factors and f(G) = dimFₚ H¹(G, Fₚ[[G]])^G.
• If G is free, then H¹(G, Fₚ[[G]]) fits into an exact sequence 0 → Fₚ[[G]] → Fₚ[[G]]^{f(G)} → H¹(G, Fₚ[[G]]) → 0.
• For pro-p groups of cohomological dimension ≤2, G is a duality group of dimension 2 if and only if it is freely indecomposable or IG is indecomposable as a module.
• The number of freely indecomposable factors satisfies s(H) = (G:H)(s(G)−1)+1 for open subgroups H, showing invariance under finite index subgroups.