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A virtually infinite cyclic group $G$ is quasi-isometric to $\mathbb{Z}$ and thus has two ends; by Stallings theorem, $G$ acts (without inversion) on a tree with finite edge-stabilizers.

But the construction of Stallings theorem works more generally for groups with more than one end; is it possible to find a simpler construction for virtually infinite cyclic groups?

There is a purely algebraic classification of the virtually infinite cyclic groups which can be reformulated as simple HNN extensions and amalgations over finite groups, so (using Bass-Serre theory) I hope that it is possible to construction a simple action of such a group on a tree.

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Yes if you assume a group has a normal subgroup $Z$ of finite index which is infinite cyclic then it's much easier. If $Z$ is central, then a classical theorem shows that the derived subgroup is finite. So modding out by the derived subgroup, you get an abelian group, and eventually get that the group has a homomorphism onto $\mathbf{Z}$ (with finite kernel, unique up to sign) and thus acts on a linear tree.

Otherwise, the centralizer $Q$ of $Z$ has index 2 in $G$. By the previous argument, there is a unique homomorphism (up to sign) of $Q$ onto $\mathbf{Z}$. So its kernel $K$ is finite and normal in $G$. A simple calculation shows that $G/K$ is isomorphic to the infinite dihedral group, which also acts on a linear tree in the way you expect.

To summarize: every virtually infinite cyclic group is either finite-by-$\mathbf{Z}$ or finite-by-$D_\infty$.