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The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

Eduard Schesler
From the journal Journal of Group Theory

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 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 exists with respect to every finite generating set of A Γ .

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.

Acknowledgements

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.

  1. Communicated by: Alexander Olshanskii

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Received: 2020-11-16
Revised: 2021-08-13
Published Online: 2021-10-05
Published in Print: 2022-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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