Boundaries of relative factor graphs and subgroup classification for automorphisms of free products

Given a countable group $G$ splitting as free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the $Out(G,\mathcal{F})$ all outer automorphisms that preserve conjugacy classes each $G_i$. We show every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains relatively fully irreducible automorphism, or else it virtually preserves class proper factor relative to decomposition (the finite generation hypothesis on $H$ can be dropped $G=F_N$, more generally when is toral hyperbolic). In first case, nonperipheral in $G$, an atoroidal automorphism. The key geometric tool obtain these description Gromov boundaries versions graph $\mathrm{FF}$ and $\mathcal{Z}$-factor $\mathcal{Z}\mathrm{F}$, spaces equivalence arational trees (respectively trees). also identify loxodromic isometries with elements $Out(G,\mathcal{F})$, $\mathcal{Z}\mathrm{F}$ automorphisms.