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This paper studies the locally uniform exponential growth and product setgrowth for a finitely generated group $G$ acting properly on a finite productof hyperbolic spaces. Under the assumption of coarsely dense orbits orshadowing property on factors, we prove that any finitely generatednon-virtually abelian subgroup has uniform exponential growth. Theseassumptions are full-filled in many hierarchically hyperbolic groups, includingmapping 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 productset growth for any group acting discretely on a simply connected manifold withpinched negative curvature, for groups acting acylindrically on trees, and for3-manifold groups.

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