Abstract
A function ℕ → ℕ is near exponential if it is bounded above and below by functions of the form \({2^{{n^c}}}\) for some c > 0. In this article we develop tools to recognize the near exponential residual finiteness growth in groups acting on rooted trees. In particular, we show the near exponential residual finiteness growth for certain branch groups, including the first Grigorchuk group, the family of Gupta–Sidki groups and their variations, and Fabrykowski–Gupta groups. We also show that the family of Gupta–Sidki p-groups, for p ≥ 5, have super-exponential residual finiteness growth.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abert, Representing graphs by the non-commuting relation, Publ. Math. Debrecen 69 (2006), 261–269.
L. Bartholdi and R. I. Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J. 28 (2002), 47–90.
K. Bou-Rabee, Quantifying residual finiteness, J. Algebra 323 (2010), 729–737.
K. Bou-Rabee and T. Kaletha, Quantifying residual finiteness of arithmetic groups, Compos. Math. 148 (2012), 907–920.
K. Bou-Rabee and D. B. McReynolds, Extremal behavior of divisibility functions, Geom. Dedicata 175 (2015), 407–415.
K. Bou-Rabee and B. Seward, Arbitrarily large residual finiteness growth, J. Reine Angew. Math. 710 (2016), 199–204.
K. Bou-Rabee and D. Studenmund, Full residual finiteness growths of nilpotent groups, Israel J. Math. 214 (2016), 209–233.
R. I. Grigorcuk, On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen. 14 (1980), 53–54.
R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 939–985.
R. I. Grigorchuk, Just infinite branch groups, in New horizons in pro-p groups, Progr. Math., Vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 121–179.
R. Grigorchuk, Solved and unsolved problems around one group, in Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., Vol. 248, Birkhäuser, Basel, 2005, pp. 117–218.
N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), 385–388.
N. Gupta and S. Sidki, Some infinite p-groups, Algebra i Logika 22 (1983), 584–589.
N. Gupta and S. Sidki, Extension of groups by tree automorphisms, in Contributions to group theory, Contemp. Math., Vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 232–246.
V. Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, Vol. 117, American Mathematical Society, Providence, RI, 2005.
E. Pervova, Profinite completions of some groups acting on trees, J. Algebra 310 (2007), 858–879.
A. Thom, About the length of laws for finite groups, Israel J. Math. 219 (2017), 469–478.
J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971), 373–391.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by NSF grant DMS-1405609.
Supported by Swiss NSF grant 200021 144323 and P2GEP2 162064.
Rights and permissions
About this article
Cite this article
Bou-Rabee, K., Myropolska, A. Groups with near exponential residual finiteness growth. Isr. J. Math. 221, 687–703 (2017). https://doi.org/10.1007/s11856-017-1570-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1570-3