Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T21:13:04.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperfiniteness of boundary actions of cubulated hyperbolic groups

Part of: Set theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  25 March 2019

JINGYIN HUANG ,
MARCIN SABOK  and
FORTE SHINKO
JINGYIN HUANG
Affiliation:
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street W, Montreal, QC, H3A 0B9Canada email [email protected], [email protected], [email protected]
MARCIN SABOK
Affiliation:
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street W, Montreal, QC, H3A 0B9Canada email [email protected], [email protected], [email protected] Instytut Matematyczny PAN, Śniadeckich 8, 00-656Warszawa, Poland
FORTE SHINKO
Affiliation:
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street W, Montreal, QC, H3A 0B9Canada email [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.

Keywords

group actionstopological dynamicshyperbolic groupshyperfinite equivalence relationsGromov boundary

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 20F65: Geometric group theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2453 - 2466
Check for updates
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, S.. Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups. Topology 33(4) (1994), 765–783.Google Scholar
Agol, I.. The virtual Haken conjecture. Doc. Math. 18 (2013), 1045–1087, with an appendix by Agol, Daniel Groves and Jason Manning.Google Scholar
Bridson, M. R. and Haefliger, A.. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 319. Springer, Berlin, 1999.Google Scholar
Bergeron, N. and Wise, D. T.. A boundary criterion for cubulation. Amer. J. Math. 134(3) (2012), 843–859.Google Scholar
Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1(4) (1982), 431–450 1981.Google Scholar
Caprace, P.-E. and Mühlherr, B.. Reflection triangles in Coxeter groups and biautomaticity. J. Group Theory 8(4) (2005), 467–489.Google Scholar
Conley, C. and Miller, B.. Measure reducibility of countable borel equivalence relations. Ann. Math. 185(2) (2017), 347–402.Google Scholar
Dougherty, R., Jackson, S. and Kechris, A. S.. The structure of hyperfinite Borel equivalence relations. Trans. Amer. Math. Soc. 341(1) (1994), 193–225.Google Scholar
Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.Google Scholar
Gao, S.. Invariant Descriptive Set Theory (Pure and Applied Mathematics (Boca Raton), 293). CRC Press, Boca Raton, FL, 2009.Google Scholar
Gao, S. and Jackson, S.. Countable abelian group actions and hyperfinite equivalence relations. Invent. Math. 201(1) (2015), 309–383.Google Scholar
Gromov, M.. Hyperbolic groups. Essays in Group Theory (Mathematical Sciences Research Institute Publications, 8). Springer, New York, 1987, pp. 75–263.Google Scholar
Harrington, L. A., Kechris, A. S. and Louveau, A.. A Glimm–Effros dichotomy for Borel equivalence relations. J. Amer. Math. Soc. 3(4) (1990), 903–928.Google Scholar
Haglund, F. and Wise, D. T.. Special cube complexes. Geom. Funct. Anal. 17(5) (2008), 1551–1620.Google Scholar
Haglund, F. and Wise, D. T.. A combination theorem for special cube complexes. Ann. of Math. (2) 176(3) (2012), 1427–1482.Google Scholar
Hagen, M. F. and Wise, D. T.. Cubulating hyperbolic free-by-cyclic groups: the general case. Geom. Funct. Anal. 25(1) (2015), 134–179.Google Scholar
Hagen, M. F. and Wise, D. T.. Cubulating hyperbolic free-by-cyclic groups: the irreducible case. Duke Math. J. 165(9) (2016), 1753–1813.Google Scholar
Jackson, S., Kechris, A. S. and Louveau, A.. Countable Borel equivalence relations. J. Math. Log. 2(1) (2002), 1–80.Google Scholar
Kapovich, I. and Benakli, N.. Boundaries of hyperbolic groups. Combinatorial and Geometric Group Theory (New York, 2000/Hoboken, NJ, 2001) (Contemporary Mathematics, 296). American Mathematical Society, Providence, RI, 2002, pp. 39–93.Google Scholar
Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156). Springer, New York, 1995.Google Scholar
Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence (Lecture Notes in Mathematics, 1852). Springer, Berlin, 2004.Google Scholar
Kahn, J. and Markovic, V.. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. Ann. of Math. (2) 175(3) (2012), 1127–1190.Google Scholar
Marquis, T.. On geodesic ray bundles in buildings. Geom. Dedicata (2018), doi:10.1007/s10711-018-0401-y.Google Scholar
Niblo, G. and Reeves, L.. Groups acting on CAT(0) cube complexes. Geom. Topol. 1 (1997), 1–7.Google Scholar
Ollivier, Y. and Wise, D. T.. Cubulating random groups at density less than 1/6. Trans. Amer. Math. Soc. 363(9) (2011), 4701–4733.Google Scholar
Sageev, M.. Ends of group pairs and non-positively curved cube complexes. Proc. Lond. Math. Soc. 3(3) (1995), 585–617.Google Scholar
Sageev, M.. CAT(0) cube complexes and groups. Geometric Group Theory (IAS/Park City Mathematics Series, 21). American Mathematical Society, Providence, RI, 2014, pp. 7–54.Google Scholar
Touikan, N.. On geodesic ray bundles in hyperbolic groups. Proc. Amer. Math. Soc. 146 (2018), 4165–4173.Google Scholar
Vershik, A. M.. The action of PSL(2, Z) in R1 is approximable. UspekhiM at. Nauk 33(1(199)) (1978), 209–210.Google Scholar
Wise, D. T.. Cubulating small cancellation groups. Geom. Funct. Anal. 14(1) (2004), 150–214.Google Scholar
Wise, D.. The Structure of Groups with a Quasiconvex Hierarchy, in preparation, 2017.Google Scholar