Stallings theorem about ends of groups

In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group ${\displaystyle G}$ has more than one end if and only if the group ${\displaystyle G}$ 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 ${\displaystyle G}$ has more than one end if and only if ${\displaystyle G}$ 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)^[1] and then in the general case (1971).^[2]