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.
Read more about this topic: Stallings Theorem About Ends Of Groups
Famous quotes containing the words theorem, formal and/or statement:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (19131960)
“True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
—Marcel Proust (18711922)
“The most distinct and beautiful statement of any truth must take at last the mathematical form.”
—Henry David Thoreau (18171862)