Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 5, 2021

The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

Eduard Schesler
From the journal Journal of Group Theory


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.


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


[1] J. Behrstock and R. Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012), no. 2, 339–356. 10.1007/s00208-011-0641-8Search in Google Scholar

[2] M. Bestvina and M. Feighn, A hyperbolic Out ⁢ ( F n ) -complex, Groups Geom. Dyn. 4 (2010), no. 1, 31–58. 10.4171/GGD/74Search in Google Scholar

[3] B. H. Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008), no. 2, 281–300. 10.1007/s00222-007-0081-ySearch in Google Scholar

[4] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss. 319, Springer, Berlin, 1999. 10.1007/978-3-662-12494-9Search in Google Scholar

[5] R. Charney and L. Paris, Convexity of parabolic subgroups in Artin groups, Bull. Lond. Math. Soc. 46 (2014), no. 6, 1248–1255. 10.1112/blms/bdu077Search in Google Scholar

[6] J. M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), no. 3, 301–309. 10.1016/0022-1236(82)90090-8Search in Google Scholar

[7] M. Cordes, J. Russell, D. Spriano and A. Zalloum, Regularity of morse geodesics and growth of stable subgroups, preprint (2020), Search in Google Scholar

[8] R. Coulon, F. Dal’Bo and A. Sambusetti, Growth gap in hyperbolic groups and amenability, Geom. Funct. Anal. 28 (2018), no. 5, 1260–1320. 10.1007/s00039-018-0459-6Search in Google Scholar

[9] F. Dahmani, D. Futer and D. T. Wise, Growth of quasiconvex subgroups, Math. Proc. Cambridge Philos. Soc. 167 (2019), no. 3, 505–530. 10.1017/S0305004118000440Search in Google Scholar

[10] 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

[11] 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

[12] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. 10.1007/s002220050343Search in Google Scholar

[13] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. 10.4310/jdg/1214501132Search in Google Scholar

[14] A. Y. Olshanskii, Subnormal subgroups in free groups, their growth and cogrowth, Math. Proc. Cambridge Philos. Soc. 163 (2017), no. 3, 499–531. 10.1017/S0305004117000081Search in Google Scholar

[15] D. Osin, On acylindrical hyperbolicity of groups with positive first ℓ 2 -Betti number, Bull. Lond. Math. Soc. 47 (2015), no. 5, 725–730. 10.1112/blms/bdv047Search in Google Scholar

[16] D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. 10.1090/tran/6343Search in Google Scholar

[17] E. Rips, Subgroups of small cancellation groups, Bull. Lond. Math. Soc. 14 (1982), no. 1, 45–47. 10.1112/blms/14.1.45Search in Google Scholar

[18] R. Sharp, Relative growth series in some hyperbolic groups, Math. Ann. 312 (1998), no. 1, 125–132. 10.1007/s002080050214Search in Google Scholar

[19] A. Sisto, Contracting elements and random walks, J. Reine Angew. Math. 742 (2018), 79–114. 10.1515/crelle-2015-0093Search in Google Scholar

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

Scroll Up Arrow