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 formal, statement and/or theorem:
“The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
—David Elkind (20th century)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)
“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)