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 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 exists with respect to every finite generating set of .
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: WI 4079/4
Funding statement: The author was partially supported by the DFG grant WI 4079/4 within the SPP 2026 Geometry at infinity.
This article grew out of my master’s thesis, which I wrote at the Heinrich Heine University under the supervision of Prof. Oleg Bogopolski. I would like to thank him for introducing me to the subject of hyperbolic groups and supporting me during my bachelor and master studies.
Communicated by: Alexander Olshanskii
 R. Flores, D. Kahrobaei and T. Koberda, Algorithmic problems in right-angled Artin groups: Complexity and applications, J. Algebra 519 (2019), 111–129. 10.1016/j.jalgebra.2018.10.023Search in Google Scholar
 R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent Random Systems, Adv. Probab. Related Topics 6, Dekker, New York (1980), 285–325. Search in Google Scholar
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