  Let $G$ be an acylindrically hyperbolic group on a $\delta$-hyperbolic space
$X$. Assume there exists $M$ such that for any finite generating set $S$ of
$G$, the set $S^M$ contains a hyperbolic element on $X$. Suppose that $G$ is
equationally Noetherian. Then we show the set of the growth rates of $G$ is
well-ordered (Theorem 1.1). The conclusion was known for hyperbolic groups, and
this is a generalization.
  Our result applies to all lattices in simple Lie groups of rank-1 (Theorem
1.3), and more generally, some family of relatively hyperbolic groups (Theorem
1.2). It also applies to the fundamental group, of exponential growth, of a
closed orientable $3$-manifold except for the case that the manifold has
Sol-geometry (Theorem 5.7).
