Stallings Theorem About Ends of Groups - Applications and Generalizations

Applications and Generalizations

• Among the immediate applications of Stallings' theorem was a proof by Stallings of a long-standing conjecture that every finitely generated group of cohomological dimension one is free and that every torsion-free virtually free group is free.
• Stallings' theorem also implies that the property of having a nontrivial splitting over a finite subgroup is a quasi-isometry invariant of a finitely generated group since the number of ends of a finitely generated group is easily seen to be a quasi-isometry invariant. For this reason Stallings' theorem is considered to be one of the first results in geometric group theory.
• Stallings' theorem was a starting point for Dunwoody's accessibility theory. A finitely generated group G is said to be accessible if the process of iterated nontrivial splitting of G over finite subgroups always terminates in a finite number of steps. In Bass-Serre theory terms that the number of edges in a reduced splitting of G as the fundamental group of a graph of groups with finite edge groups is bounded by some constant depending on G. Dunwoody proved that every finitely presented group is accessible but that there do exist finitely generated groups that are not accessible. Linnell showed that if one bounds the size of finite subgroups over which the splittings are taken then every finitely generated group is accessible in this sense as well. These results in turn gave rise to other versions of accessibility such as Bestvina-Feighn accessibility of finitely presented groups (where the so-called "small" splittings are considered), acylindrical accessibility, strong accessibility, and others.
• Stallings' theorem is a key tool in proving that a finitely generated group G is virtually free if and only if G can be represented as the fundamental group of a finite graph of groups where all vertex and edge groups are finite (see, for example,).
• Using Dunwoody's accessibility result, Stallings' theorem about ends of groups and the fact that if G is a finitely presented group with asymptotic dimension 1 then G is virtually free one can show that for a finitely presented word-hyperbolic group G the hyperbolic boundary of G has topological dimension zero if and only if G is virtually free.
• Relative versions of Stallings' theorem and relative ends of finitely generated groups with respect to subgroups have also been considered. For a subgroup H≤G of a finitely generated group G one defines the number of relative ends e(G,H) as the number of ends of the relative Cayley graph (the Schreier coset graph) of G with respect to H. The case where e(G,H)>1 is called a semi-splitting of G over H. Early work on semi-splittings, inspired by Stallings' theorem, was done in the 1970s and 1980s by Scott, Swarup, and others. The work of Sageev and Gerasomov in the 1990s showed that for a subgroup H≤G the condition e(G,H)>1 correpsonds to the group G admitting an essential isometric action on a CAT(0)-cubing where a subgroup commensurable with H stabilizes an essential "hyperplane" (a simplicial tree is an example of a CAT(0)-cubing where the hyperplanes are the midpoints of edges). In certain situations such a semi-splitting can be promoted to an actual algebraic splitting, typically over a subgroup commensurable with H, such as for the case where H is finite (Stallings' theorem). Another situation where an actual splitting can be obtained (modulo a few exceptions) is for semi-splittings over virtually polycyclic subgroups. Here the case of semi-splittings of word-hyperbolic groups over two-ended (virtually infinite cyclic) subgroups was treated by Scott-Swarup and by Bowditch. The case of semi-splittings of finitely generated groups with respect to virtually polycyclic subgroups is dealt with by the algebraic torus theorem of Dunwoody-Swenson.

• A number of new proofs of Stallings' theorem have been obtained by others after Stallings' original proof. Dunwoody gave a proof based on the ideas of edge-cuts. Later Dunwoody also gave a proof of Stallings' theorem for finitely presented groups using the method of "tracks" on finite 2-complexes. Niblo obtained a proof of Stallings' theorem as a consequence of Sageev's CAT(0)-cubing relative version, where the CAT(0)-cubing is eventually promoted to being a tree. Niblo's paper also defines an abstract group-theoretic obstruction (which is a union of double cosets of H in G) for obtaining an actual splitting from a semi-splitting. It is also possible to prove Stallings' theorem for finitely presented groups using Riemannian geometry techniques of minimal surfaces, where one first realizes a finitely presented group as the fundamental group of a compact 4-manifold (see, for example, a sketch of this argument in the survey article of Wall). Gromov outlined a proof (see pp. 228–230 in ) where the minimal surfaces argument is replaced by an easier harmonic analysis argument and this approach was pushed further by Kapovich to cover the original case of finitely generated groups.