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).
Read more about this topic: Stallings Theorem About Ends Of Groups, Ends of Groups
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