Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings (Q2813664)

Let \(G_1,\ldots ,G_k\) be nontrivial countable groups and \(F_N\) the free group of rank \(N\). Define \(G=G_1*\cdots *G_k*F_N\). Denote by \({\mathcal F}=\{ [ G_1 ],\ldots ,[ G_k ]\}\) the finite collection of \(G\)-conjugacy classes of the subgroups \(G_1,\ldots ,G_k\) which is called a \textit{free factor system} of \(G\).NEWLINENEWLINEThe author defines analogues of the graphs of free splittings, of cyclic splittings and of maximally cyclic splittings of \(F_N\) for such a free factor system. The Gromov boundary of the graph of relative cyclic splittings is identified with the space of equivalence classes of \({\mathcal Z}\)-averse trees in the boundary of the corresponding outer space. A tree is \({\mathcal Z}\)-\textit{averse} if it is not compatible with any tree that is itself compatible with a relative cyclic splitting.NEWLINENEWLINEA similar description is given of the Gromov boundary of the graph of maximally cyclic splittings.