The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

Abstract

Abstract We introduce a new invariant of finitely generated groups, the ambiguity function, and we 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 n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated 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 Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .