Stallings Theorem About Ends of Groups - Formal Statement of Stallings' Theorem

Formal Statement of Stallings' Theorem

Let G be a finitely generated group.

Then e(G) > 1 if and only if one of the following holds:

• The group G admits a splitting G=H∗_CK as a free product with amalgamation where C is a finite group such that C ≠ H and C ≠ K.
• The group G admits a splitting is an HNN-extension where and C₁, C₂ are isomorphic finite subgroups of H.

In the language of Bass-Serre theory this result can be restated as follows: For a finitely generated group G we have e(G) > 1 if and only if G admits a nontrivial (that is, without a global fixed vertex) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.

For the case where G is a torsion-free finitely generated group, Stallings' theorem implies that e(G) = ∞ if and only if G admits a proper free product decomposition G = A∗B with both A and B nontrivial.