Sublinear projection tracking in acylindrically hyperbolic groups

We study projection phenomena in word metrics of finitely generated acylindrically hyperbolic groups. For a loxodromic WPD element acting on a hyperbolic space, we prove that shortest projection in the word metric to the corresponding cyclic subgroup sublinearly tracks the pullback of shortest projection to its axis in the hyperbolic space. As applications, we obtain effective upper bounds for growth functions and construct proper quotients whose growth rates converge to that of the original group. We further prove a growth--cogrowth inequality for confined subgroups in both acylindrically hyperbolic groups and Morse local-to-global groups with Morse elements.