Geometry of free extensions of free groups via automorphisms with fixed points on the complex of free factors

We give conditions of an extension of a free group to be hyperbolic and relatively hyperbolic using the dynamics of the action of Out.F / on the complex of free factors combined with weak attraction theory.We work with subgroups of exponentially growing outer automorphisms and instead of using a standard ping-pong argument with loxodromics, we allow fixed points for the action and investigate the geometry of the extension group when the fixed points of the automorphisms on the complex of free factors are sufficiently far apart.20E06, 20F28, 20F67, 57M07