2-ended groups are virtually cyclic: reference

Question

It is often said (e.g., in this post) that "It is standard that two-ended groups are virtually infinite cyclic". However, I cannot find a simple proof for this fact anywhere. Can anyone give me a reference?

Answer 1

Section 11.6 of Meirer's book Groups, Graphs and Trees gives a very friendly and self-contained proof of the following result (Theorem 11.33). The proof basically plays around with symmetric differences and cosets and intersections.

Theorem. Suppose $G$ has $2$ ends. Then there is a subgroup $H\leq G$ of index at most $2$ in $G$ and a surjection $H\twoheadrightarrow \mathbb{Z}$ with finite kernel.

Corollary 11.34 of Meier's book then explains how to get from the above lemma to the result you wish.

The theorem is a reworking of the following slightly stronger result of Wall; see Lemma 4.1 in C. T. C. Wall, Poincaré Complexes: I. Annals of Mathematics, Vol. 86, No. 2, 1967, pp. 213-245 (download). However, Wall's proof applies the fact that that $2$-ended groups are virtually-$\mathbb{Z}$ (thanks to the OP for pointing this out!). Here, $\pi$ is a group and $\mathbb{Z}_2*\mathbb{Z}_2\cong D_{\infty}$:

Theorem. Suppose that $\pi$ has 2 ends. Then there is a finite normal subgroup $F$ of $\pi$, such that the quotient group is isomorphic to $\mathbb{Z}$ or to $\mathbb{Z}_2*\mathbb{Z}_2$.