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∗CK 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 C1, C2 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.
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