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).

Ends of graphs

Let \Gamma be a connected graph where the degree of every vertex is finite. One can view \Gamma as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of \Gamma are the ends of this topological space. A more explicit definition of the number of ends of a graph is presented below for completeness. Let be a non-negative integer. The graph \Gamma is said to satisfy if for every finite collection F of edges of \Gamma the graph \Gamma - F has at most n infinite connected components. By definition, if and if for every the statement is false. Thus if m is the smallest nonnegative integer n such that. If there does not exist an integer such that, put. The number e(\Gamma) is called the number of ends of \Gamma. Informally, e(\Gamma) is the number of "connected components at infinity" of \Gamma. If, then for any finite set F of edges of \Gamma there exists a finite set K of edges of \Gamma with such that \Gamma - F has exactly m infinite connected components. If, then for any finite set F of edges of \Gamma and for any integer there exists a finite set K of edges of \Gamma with such that \Gamma - K has at least n infinite connected components.

Ends of groups

Let G be a finitely generated group. Let be a finite generating set of G and let \Gamma(G,S) be the Cayley graph of G with respect to S. The number of ends of G is defined as. A basic fact in the theory of ends of groups says that does not depend on the choice of a finite generating set S of G, so that e(G) is well-defined.

Basic facts and examples

Freudenthal-Hopf theorems

Hans Freudenthal and independently Heinz Hopf established in the 1940s the following two facts: Charles T. C. Wall proved in 1967 the following complementary fact:

Cuts and almost invariant sets

Let G be a finitely generated group, be a finite generating set of G and let be the Cayley graph of G with respect to S. For a subset denote by A^* the complement G-A of A in G. For a subset, the edge boundary or the co-boundary \delta A of A consists of all (topological) edges of \Gamma connecting a vertex from A with a vertex from A^. Note that by definition. An ordered pair (A,A^) is called a cut in \Gamma if \delta A is finite. A cut (A,A^) is called essential if both the sets A and A^ are infinite. A subset is called almost invariant if for every g \in G the symmetric difference between A and Ag is finite. It is easy to see that (A,A^) is a cut if and only if the sets A and A^ are almost invariant (equivalently, if and only if the set A is almost invariant).

Cuts and ends

A simple but important observation states:

Cuts and splittings over finite groups

If G = HK where H and K are nontrivial finitely generated groups then the Cayley graph of G has at least one essential cut and hence e(G) > 1. Indeed, let X and Y be finite generating sets for H and K accordingly so that is a finite generating set for G and let be the Cayley graph of G with respect to S. Let A consist of the trivial element and all the elements of G whose normal form expressions for G = HK starts with a nontrivial element of H. Thus A^* consists of all elements of G whose normal form expressions for G = HK starts with a nontrivial element of K. It is not hard to see that (A,A^) is an essential cut in Γ so that e(G) > 1. A more precise version of this argument shows that for a finitely generated group G: Stallings' theorem shows that the converse is also true.

Formal statement of Stallings' theorem

Let G be a finitely generated group. Then e(G) > 1 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 G we have e(G) > 1 if and only if G admits a nontrivial (that is, without a global fixed vertex) action on a simplicial tree with finite edge-stabilizers and without edge-inversions. For the case where G is a torsion-free finitely generated group, Stallings' theorem implies that if and only if G admits a proper free product decomposition G = A* B with both A and B nontrivial.

Applications and generalizations