In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group has more than one end if and only if the group 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 has more than one end if and only if 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)[1] and then in the general case (1971).[2]
- ^ John R. Stallings. On torsion-free groups with infinitely many ends. Annals of Mathematics (2), vol. 88 (1968), pp. 312–334
- ^ John Stallings. Group theory and three-dimensional manifolds. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.-London, 1971.