The relative exponential growth rate of subgroups of acylindrically hyperbolic 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 $\underset{n\to \infty }{\lim }\sqrt[n]{|B_H^X(n)|}$ 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 $A_{\Gamma }$.

arXiv comment23 pages