In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group G has more than one end if and only if G admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.
The theorem was proved by John R. Stallings, first in the torsion-free case (1968) and then in the general case (1971).
Read more about Stallings Theorem About Ends Of Groups: Ends of Graphs, Ends of Groups, Formal Statement of Stallings' Theorem, Applications and Generalizations, See Also
Famous quotes containing the words stallings, theorem, ends and/or groups:
“I declare Billy. I like you so much personally I wish I could vote for you. But bein’ a member of the Grand Army of the Republic, I just as leave cut my throat as to vote for a Democrat.”
—Laurence Stallings (1894–1968)
“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 (1913–1960)
“In a valley late bees with whining gold
Thread summer to the loose ends of sleep....”
—Allen Tate (1899–1979)
“Some of the greatest and most lasting effects of genuine oratory have gone forth from secluded lecture desks into the hearts of quiet groups of students.”
—Woodrow Wilson (1856–1924)