**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*∗_{C}*K*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*C*_{1},*C*_{2}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.

Read more about this topic: Stallings Theorem About Ends Of Groups

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