Hyperfiniteness for group actions on trees

Kunnawalkam Elayavalli, Srivatsav; Oyakawa, Koichi; Shinko, Forte; Spaas, Pieter

doi:10.1090/proc/16851

Hyperfiniteness for group actions on trees
HTML articles powered by AMS MathViewer

by Srivatsav Kunnawalkam Elayavalli, Koichi Oyakawa, Forte Shinko and Pieter Spaas;

Proc. Amer. Math. Soc. 152 (2024), 3657-3664

DOI: https://doi.org/10.1090/proc/16851

Published electronically: July 19, 2024

HTML | PDF

Abstract:

We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfiniteness of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.

References

A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI 10.1017/s014338570000136x
Clinton T. Conley, Steve C. Jackson, Andrew S. Marks, Brandon M. Seward, and Robin D. Tucker-Drob, Borel asymptotic dimension and hyperfinite equivalence relations, Duke Math. J. 172 (2023), no. 16, 3175–3226. MR 4679959, DOI 10.1215/00127094-2022-0100
R. Dougherty, S. Jackson, and A. S. Kechris, The structure of hyperfinite Borel equivalence relations, Trans. Amer. Math. Soc. 341 (1994), no. 1, 193–225. MR 1149121, DOI 10.1090/S0002-9947-1994-1149121-0
Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR 578656, DOI 10.1090/S0002-9947-1977-0578656-4
Greg Hjorth, A converse to Dye’s theorem, Trans. Amer. Math. Soc. 357 (2005), no. 8, 3083–3103. MR 2135736, DOI 10.1090/S0002-9947-04-03672-4
Jingyin Huang, Marcin Sabok, and Forte Shinko, Hyperfiniteness of boundary actions of cubulated hyperbolic groups, Ergodic Theory Dynam. Systems 40 (2020), no. 9, 2453–2466. MR 4130811, DOI 10.1017/etds.2019.5
S. Jackson, A. S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Log. 2 (2002), no. 1, 1–80. MR 1900547, DOI 10.1142/S0219061302000138
C. Karpinski, Hyperfiniteness of boundary actions of relatively hyperbolic groups, arXiv:2212.00236, 2022.
A. S. Kechris, Unitary representations and modular actions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 326 (2005), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13, 97–144, 281–282 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 140 (2007), no. 3, 398–425. MR 2183218, DOI 10.1007/s10958-007-0449-y
A. S. Kechris, The theory of countable Borel equivalence relations, Cambridge Tracts in Mathematics, Vol. 234, Cambridge University Press, Cambridge, 2024.
Timothée Marquis and Marcin Sabok, Hyperfiniteness of boundary actions of hyperbolic groups, Math. Ann. 377 (2020), no. 3-4, 1129–1153. MR 4126891, DOI 10.1007/s00208-020-02001-9
P. Naryshkin and A. Vaccaro, Hyperfiniteness and Borel asymptotic dimension of boundary actions of hyperbolic groups, arXiv:2306.02056, 2023.
Denis V. Osin, Groups acting acylindrically on hyperbolic spaces, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 919–939. MR 3966794
Koichi Oyakawa, Hyperfiniteness of boundary actions of acylindrically hyperbolic groups, Forum Math. Sigma 12 (2024), Paper No. e32, 31. MR 4715159, DOI 10.1017/fms.2024.24
Piotr Przytycki and Marcin Sabok, Unicorn paths and hyperfiniteness for the mapping class group, Forum Math. Sigma 9 (2021), Paper No. e36, 10. MR 4252215, DOI 10.1017/fms.2021.34
Z. Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997), no. 3, 527–565. MR 1465334, DOI 10.1007/s002220050172
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
Henry Wilton and Pavel Zalesskii, Profinite properties of graph manifolds, Geom. Dedicata 147 (2010), 29–45. MR 2660565, DOI 10.1007/s10711-009-9437-3