Uniform exponential growth for groups with proper product actions on hyperbolic spaces

This paper studies the locally uniform exponential growth and product set growth for a finitely generated group $G$ acting properly on a finite product of hyperbolic spaces. Under the assumption of coarsely dense orbits or shadowing property on factors, we prove that any finitely generated non-virtually abelian subgroup has uniform exponential growth. These assumptions are fulfilled in many hierarchically hyperbolic groups, including mapping class groups, specially cubulated groups and BMW groups. Moreover, if $G$ acts weakly acylindrically on each factor, we show that, with two exceptional classes of subgroups, $G$ has uniform product set growth. As corollaries, this gives a complete classification of subgroups with product set growth for any group acting discretely on a simply connected manifold with pinched negative curvature, for groups acting acylindrically on trees, and for 3-manifold groups.