The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

TL;DR

This paper introduces the ambiguity function as a new invariant for finitely generated groups and proves the existence of the relative exponential growth rate for certain subgroups of acylindrically hyperbolic groups and right-angled Artin groups.

Contribution

It establishes the linear bound of the ambiguity function in acylindrically hyperbolic groups and proves the existence of the relative exponential growth rate for specific subgroups.

Findings

Ambiguity function is linearly bounded in acylindrically hyperbolic groups.

Relative exponential growth rate exists for subgroups containing a loxodromic element.

Relative exponential growth rate exists for all finitely generated subgroups of right-angled Artin groups.

Abstract

We introduce a new invariant of finitely generated groups, the ambiguity function, and prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate $\lim \limits_{n \rightarrow \infty} \sqrt[n]{\vert B^{X}_H(n) \vert}$ of a subgroup $H$ of an acylindrically hyperbolic group $G$ exists with respect to every finite generating set $X$ of $G$, if $H$ contains a loxodromic element of $G$. Further we prove that the relative exponential growth rate of every finitely generated subgroup $H$ of a right-angled Artin group $A_{\Gamma}$ exists with respect to every finite generating set of $A_{\Gamma}$.