In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated 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
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).
be a connected graph where the degree of every vertex is finite.
as a topological space by giving it the natural structure of a one-dimensional cell complex.
are the ends of this topological space.
A more explicit definition of the number of ends of a graph is presented below for completeness.
infinite connected components.
is the smallest nonnegative integer
is called the number of ends of
is the number of "connected components at infinity" of
infinite connected components.
infinite connected components.
A basic fact in the theory of ends of groups says that
does not depend on the choice of a finite generating set
Hans Freudenthal[3] and independently Heinz Hopf[4] established in the 1940s the following two facts: Charles T. C. Wall proved in 1967 the following complementary fact:[5] Let
consists of all (topological) edges of
is called essential if both the sets
A simple but important observation states: If
are nontrivial finitely generated groups then the Cayley graph of
whose normal form expressions for
whose normal form expressions for
A more precise version of this argument shows that for a finitely generated group
: Stallings' theorem shows that the converse is also true.
if and only if one of the following holds: In the language of Bass–Serre theory this result can be restated as follows: For a finitely generated group
admits a nontrivial (that is, without a global fixed vertex) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.
is a torsion-free finitely generated group, Stallings' theorem implies that
admits a proper free product decomposition