The rates of growth in an acylindrically hyperbolic group

Abstract

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. 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, and more generally, relatively hyperbolic groups under some assumption. 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.