The rates of growth in an acylindrically hyperbolic group

TL;DR

This paper proves that the set of growth rates in certain acylindrically hyperbolic groups is well-ordered, extending known results from hyperbolic groups to a broader class including some lattices and 3-manifold groups.

Contribution

It generalizes the well-ordering of growth rates from hyperbolic groups to acylindrically hyperbolic groups under specific conditions, including applications to lattices and 3-manifold groups.

Findings

Growth rates form a well-ordered set in the specified groups

Applicable to lattices in simple Lie groups of rank-1

Includes fundamental groups of most 3-manifolds except Sol-geometry

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 (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).