Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings
We define analogues of the graphs of free splittings and of (maximally )cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as $G=G_1\ast\dots\ast G_k\ast F$ where $F$ denotes a finitely generated free group, we identify the Gromov boundary of the graph of (G,[G_1]\dots,[G_k] (maximally )cyclic splittings with the space of equivalence classes of \mathcal{Z}(max) averse trees in the boundary of the corresponding outer space. A tree is \emph{\mathcal{Z}(max) averse} if it is not compatible with any tree $T' that is itself compatible with a (G,\mathcal{F} (maximally )cyclic splitting. Two \mathcal{Z}(max) averse trees are \emph{equivalent} if they are both compatible with a common tree in the boundary of the corresponding outer space.