  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 $A_{\Gamma}$.
