Splitting over infinite subgroups By celebrated theorem of Stallings we know that groups with more than one end split over finite groups.
Now my question is 
Which infinite groups can split over an infinite cyclic subgroup?
On the other hand let $M$ be a manifold which is nice enough.
Every closed curve on $M$ induces a splitting of $\pi_1(M)$ over $\mathbb Z$.
Somehow there is a geometric characterization of such groups.
 A: There is a coarse geometric characterisation analogous to Stallings's theorem of the property of splitting over an infinite cyclic subgroups due to Papasoglu, at least for finitely presented groups. This is the main result of his paper Quasi-Isometry Invariance of Group Splittings: 
https://www.jstor.org/stable/3597318?seq=1#metadata_info_tab_contents
Essentially, unless the finitely presented group is commensurable to a surface group, it admits a non-trivial splitting over an infinite virtually cyclic subgroup if and only if its Cayley graph contains a coarsely separating line. To make this precise, let $(X, d_X)$ be the Cayley graph equipped with the word metric. We say that a path-connected subset $L$ with induced metric $d_L$ is a quasi-line if it is quasi-isometric to $\mathbb{R}$ and for any sequences $(x_n)$ and $(y_n)$ in $L$ with $d_L(x_n, y_n) \to \infty$ we have $d_X(x_n,y_n) \to \infty$. Then the group admits a non-trivial splitting over an infinite virtually cyclic subgroup if and only if the Cayley graph contains a quasi-line $L$ such that $X - L$ contains at least two connected components that are not contained in any finite neighbourhood of $L$.
Note that we do need to rule out groups commensurable to surface groups: all of their Cayley graphs contain separating quasi-lines, and most such groups admit splittings over infinite cyclic subgroups, but the triangle groups do not. 
