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.

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