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:
“Is that the Craig Jurgesen that Teddy Roosevelt gave you?... And you used it at San Juan Hill defending liberty. Now you want to destroy it.”
—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)
“No matter which word it is, when I pronounce repeatedly, it ends up sounding utterly ridiculous and meaningless to me.”
—Franz Grillparzer (1791–1872)
“The awareness of the all-surpassing importance of social groups is now general property in America.”
—Johan Huizinga (1872–1945)