# Stallings theorem about ends of groups

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 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)[1] and then in the general case (1971).[2] Contents 1 Ends of graphs 2 Ends of groups 2.1 Basic facts and examples 2.2 Freudenthal-Hopf theorems 2.3 Cuts and almost invariant sets 2.3.1 Cuts and ends 2.3.2 Cuts and splittings over finite groups 3 Formal statement of Stallings' theorem 4 Applications and generalizations 5 See also 6 Notes Ends of graphs Main article: End (graph theory) Let Γ be a connected graph where the degree of every vertex is finite. One can view Γ as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of Γ 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 n ≥ 0 be a non-negative integer. The graph Γ is said to satisfy e(Γ) ≤ n if for every finite collection F of edges of Γ the graph Γ − F has at most n infinite connected components. By definition, e(Γ) = m if e(Γ) ≤ m and if for every 0 ≤ n < m the statement e(Γ) ≤ n is false. Thus e(Γ) = m if m is the smallest nonnegative integer n such that e(Γ) ≤ n. If there does not exist an integer n ≥ 0 such that e(Γ) ≤ n, put e(Γ) = ∞. The number e(Γ) is called the number of ends of Γ. Informally, e(Γ) is the number of "connected components at infinity" of Γ. If e(Γ) = m < ∞, then for any finite set F of edges of Γ there exists a finite set K of edges of Γ with F ⊆ K such that Γ − F has exactly m infinite connected components. If e(Γ) = ∞, then for any finite set F of edges of Γ and for any integer n ≥ 0 there exists a finite set K of edges of Γ with F ⊆ K such that Γ − K has at least n infinite connected components. Ends of groups Let G be a finitely generated group. Let S ⊆ G be a finite generating set of G and let Γ(G, S) be the Cayley graph of G with respect to S. The number of ends of G is defined as e(G) = e(Γ(G, S)). A basic fact in the theory of ends of groups says that e(Γ(G, S)) 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 For a finitely generated group G we have e(G) = 0 if and only if G is finite. For the infinite cyclic group {displaystyle mathbb {Z} } we have {displaystyle e(mathbb {Z} )=2.} For the free abelian group of rank two {displaystyle mathbb {Z} ^{2}} we have {displaystyle e(mathbb {Z} ^{2})=1.} For a free group F(X) where 1 < |X| < ∞ we have e(F(X)) = ∞ Freudenthal-Hopf theorems Hans Freudenthal[3] and independently Heinz Hopf[4] established in the 1940s the following two facts: For any finitely generated group G we have e(G) ∈ {0, 1, 2, ∞}. For any finitely generated group G we have e(G) = 2 if and only if G is virtually infinite cyclic (that is, G contains an infinite cyclic subgroup of finite index). Charles T. C. Wall proved in 1967 the following complementary fact:[5] A group G is virtually infinite cyclic if and only if it has a finite normal subgroup W such that G/W is either infinite cyclic or infinite dihedral. Cuts and almost invariant sets Let G be a finitely generated group, S ⊆ G be a finite generating set of G and let Γ = Γ(G, S) be the Cayley graph of G with respect to S. For a subset A ⊆ G denote by A∗ the complement G − A of A in G. For a subset A ⊆ G, the edge boundary or the co-boundary δA of A consists of all (topological) edges of Γ connecting a vertex from A with a vertex from A∗. Note that by definition δA = δA∗. An ordered pair (A, A∗) is called a cut in Γ if δA is finite. A cut (A,A∗) is called essential if both the sets A and A∗ are infinite. A subset A ⊆ G is called almost invariant if for every g∈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: e(G) > 1 if and only if there exists at least one essential cut (A,A∗) in Γ. Cuts and splittings over finite groups If G = H∗K 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 S = X ∪ Y is a finite generating set for G and let Γ=Γ(G,S) 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 = H∗K starts with a nontrivial element of H. Thus A∗ consists of all elements of G whose normal form expressions for G = H∗K 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: If G = H∗CK is a free product with amalgamation where C is a finite group such that C ≠ H and C ≠ K then H and K are finitely generated and e(G) > 1 . If {displaystyle scriptstyle G=langle H,t|t^{-1}C_{1}t=C_{2}rangle } is an HNN-extension where C1, C2 are isomorphic finite subgroups of H then G is a finitely generated group and e(G) > 1.

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: The group G admits a splitting G=H∗CK as a free product with amalgamation where C is a finite group such that C ≠ H and C ≠ K. The group G is an HNN extension {displaystyle scriptstyle G=langle H,t|t^{-1}C_{1}t=C_{2}rangle } where and C1, C2 are isomorphic finite subgroups of H.

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 e(G) = ∞ if and only if G admits a proper free product decomposition G = A∗B with both A and B nontrivial.