The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo>lim</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:msub> <m:mo>⁡</m:mo> <m:mroot> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:msubsup> <m:mi>B</m:mi> <m:mi>H</m:mi> <m:mi>X</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>n</m:mi> </m:mroot> </m:mrow> </m:math> limn→∞√[n]{{1₁}{BXH(n)}} 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:math> AΓ exists with respect to every finite generating set of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:math> AΓ .