**Cuts and Splittings Over Finite Groups**

If *G* = *H*∗*K* where *H* and *K* are nontrivial finitely generated groups then the Cayley graph of *G* has at least one essential cut and hence *e*(*G*) > 1. Indeed, let *X* and *Y* be finite generating sets for *H* and *K* accordingly so that *S* = *X* ∪ *Y* is a finite generating set for *G* and let Γ=Γ(*G*,*S*) be the Cayley graph of *G* with respect to *S*. Let *A* consist of the trivial element and all the elements of *G* whose normal form expressions for *G* = *H*∗*K* starts with a nontrivial element of *H*. Thus *A*∗ consists of all elements of *G* whose normal form expressions for *G* = *H*∗*K* starts with a nontrivial element of *K*. It is not hard to see that (*A*,*A*∗) is an essential cut in Γ so that *e*(*G*) > 1.

A more precise version of this argument shows that for a finitely generated group *G*:

- If
*G*=*H*∗_{C}*K*is a free product with amalgamation where*C*is a finite group such that*C*≠*H*and*C*≠*K*then*H*and*K*are finitely generated and*e*(*G*) > 1 . - If is an HNN-extension where
*C*_{1},*C*_{2}are isomorphic finite subgroups of*H*then*G*is a finitely generated group and*e*(*G*) > 1.

Stallings' theorem shows that the converse is also true.

Read more about this topic: Stallings Theorem About Ends Of Groups, Ends of Groups, Cuts and Almost Invariant Sets

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