We define analogues of the graphs of free splittings, of cyclic splittings,and of maximally-cyclic splittings of $F_N$ for free products of groups, andshow 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 freegroup, we identify the Gromov boundary of the graph of relative cyclicsplittings with the space of equivalence classes of $\mathcal{Z}$-averse treesin the boundary of the corresponding outer space. A tree is\emph{$\mathcal{Z}$-averse} if it is not compatible with any tree $T'$, that isitself compatible with a relative cyclic splitting. Two $\mathcal{Z}$-aversetrees are \emph{equivalent} if they are both compatible with a common tree inthe boundary of the corresponding outer space. We give a similar description ofthe Gromov boundary of the graph of maximally-cyclic splittings.
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机译：我们定义了组的自由积的自由分裂图，循环分裂图和最大循环分裂图$ F_N $的类似物，并显示了其双曲性。给定一个可数的组$ G $，其拆分为$ G = G_1 \ ast \ dots \ ast G_k \ ast F $，其中$ F $表示有限生成的自由组，我们用空间标识了相对循环分裂图的Gromov边界对应的外层空间边界中$ \ mathcal {Z} $的等价类的集合。如果树与任何树$ T'$不兼容，则它本身就是\ emph {$ \ mathcal {Z} $-averse}，它本身也与相对循环分裂兼容。如果两个$ \ mathcal {Z} $-aversetrees与对应的外层空间边界中的公共树兼容，则它们\ emph {equivalent}。我们给出了最大循环分裂图的Gromov边界的类似描述。
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