Given a countable group splitting as a free product , we establish classification results for subgroups of the group of all outer automorphisms of that preserve the conjugacy class of each . We show that every finitely generated subgroup either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on can be dropped for , or more generally when is toral relatively hyperbolic). In the first case, either virtually preserves a nonperipheral conjugacy class in , or else contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph and the –factor graph , as spaces of equivalence classes of arational trees and relatively free arational trees, respectively. We also identify the loxodromic isometries of with the fully irreducible elements of , and loxodromic isometries of with the fully irreducible atoroidal outer automorphisms.
Vincent Guirardel. Camille Horbez. "Boundaries of relative factor graphs and subgroup classification for automorphisms of free products." Geom. Topol. 26 (1) 71 - 126, 2022. https://doi.org/10.2140/gt.2022.26.71