Growth gaps and exponential genericity in acylindrically hyperbolic groups

Let $G$ be a finitely generated group acting by isometries on a geodesic metric space with two independent strongly contracting WPD elements. We prove that, for every finite generating set of $G$, strongly contracting WPD elements for this action are exponentially generic in word-metric balls. The proof establishes a growth gap for elements with small displacement relative to word length in an auxiliary projection complex. For finitely generated acylindrically hyperbolic groups, we establish growth tightness and cogrowth tightness for every finite generating set. We also show that the number of conjugacy classes meeting a ball of radius $n$ is comparable to the cardinality of that ball divided by $n$.