Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-24T04:22:55.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Unicorn paths and hyperfiniteness for the mapping class group

Part of: Ergodic theory Set theory

Published online by Cambridge University Press:  29 April 2021

Piotr Przytycki  and
Marcin Sabok
Piotr Przytycki
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke St. W, Montreal, Quebec, CanadaH3A 0B9; E-mail: piotr.przytycki@mcgill.ca; marcin.sabok@mcgill.ca.
Marcin Sabok
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke St. W, Montreal, Quebec, CanadaH3A 0B9; E-mail: piotr.przytycki@mcgill.ca; marcin.sabok@mcgill.ca.

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let S be an orientable surface of finite type. Using Pho-on’s infinite unicorn paths, we prove the hyperfiniteness of orbit equivalence relations induced by the actions of the mapping class group of S on the Gromov boundaries of the arc graph and the curve graph of S. In the curve graph case, this strengthens the results of Hamenstädt and Kida that this action is universally amenable and that the mapping class group of S is exact.

MSC classification

Secondary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 03E15: Descriptive set theory
Type
Topology
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e36
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Adams, S., ‘Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups’, Topology 33(4) (1994), 765–783.CrossRefGoogle Scholar
Adams, S., Elliott, G. A. and Giordano, T., ‘Amenable actions of groups’, Trans. Amer. Math. Soc. 344(2) (1994), 803–822.CrossRefGoogle Scholar
Anantharaman-Delaroche, C., ‘Systèmes dynamiques non commutatifs et moyennabilité’, Math. Ann. 279(2) (1987), 297–315.CrossRefGoogle Scholar
Anantharaman-Delaroche, C., ‘Amenability and exactness for dynamical systems and their ${C}^{\ast }$-algebras’, Trans. Amer. Math. Soc. 354(10) (2002), 4153–4178.CrossRefGoogle Scholar
Anantharaman-Delaroche, C. and Renault, J., Amenable Groupoids, Monographies de L’Enseignement Mathématique, vol. 36 (L’Enseignement Mathématique, Geneva, 2000). With a foreword by Skandalis, G. and Appendix B by Germain, E..Google Scholar
Bestvina, M., Guirardel, V. and Horbez, C., ‘Boundary amenability of $\mathrm{Out}\left({F}_N\right)$’, ‘Ann. Sci. Éc. Norm. Supér. (4), to appear. Preprint, 2021, arXiv:1705.07017.Google Scholar
Bestvina, M. and Reynolds, P., ‘The boundary of the complex of free factors’, Duke Math. J. 164(11) (2015), 2213–2251.Google Scholar
Birman, J. S. and Series, C., ‘Geodesics with bounded intersection number on surfaces are sparsely distributed’, Topology 24(2) (1985), 217–225.Google Scholar
Connes, A., Feldman, J. and Weiss, B., ‘An amenable equivalence relation is generated by a single transformation’, Ergodic Theory Dynam. Systems 1(4) (1981), 431–450.CrossRefGoogle 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
Gao, S., Invariant Descriptive Set Theory, Pure and Applied Mathematics (Boca Raton), vol. 293 (CRC Press, Boca Raton, FL, 2009).Google Scholar
Ghys, É. and de la Harpe, P. (eds.), Sur les groupes hyperboliques d’aprés Mikhael Gromov, Progress in Mathematics, vol. 83 (Birkhäuser Boston, Boston, MA, 1990). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988.CrossRefGoogle Scholar
Guirardel, V., Horbez, C. and Lécureux, J., ‘Cocycle superrigidity from higher rank lattices to $\mathrm{Out}\left({F}_N\right)$’, Preprint, 2020, arXiv:2005.07477.Google Scholar
Hamenstädt, U., ‘Train tracks and the Gromov boundary of the complex of curves’, in Spaces of Kleinian Groups, London Math. Soc. Lecture Note Ser., vol. 329 (Cambridge University Press, Cambridge, UK, 2006), 187–207.Google Scholar
Hamenstädt, U., ‘Geometry of the mapping class groups, I: Boundary amenability’, Invent. Math. 175(3) (2009), 545–609.CrossRefGoogle Scholar
Hamenstädt, U., ‘The boundary of the free factor graph and the free splitting graph’, Preprint, 2014, arXiv:1211.1630.Google Scholar
Hensel, S., Przytycki, P. and Webb, R. C. H., ‘1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs’, J. Eur. Math. Soc. (JEMS) 17(4) (2015), 755–762.CrossRefGoogle Scholar
Huang, J., Shinko, F. and Sabok, M., ‘Hyperfiniteness of boundary actions of cubulated hyperbolic groups’, Ergodic Theory Dynam. Systems 40(9) (2020), 2453–2466.CrossRefGoogle Scholar
Jackson, S., Kechris, A. S. and Louveau, A., ‘Countable Borel equivalence relations’, J. Math. Log. 2(1) (2002), 1–80.CrossRefGoogle Scholar
Kaimanovich, V. A., ‘Boundary amenability of hyperbolic spaces’, in Discrete Geometric Analysis, Contemp. Math., vol. 347 (American Mathematical Society, Providence, RI, 2004), 83–111.Google Scholar
Kechris, A. S., ‘Amenable versus hyperfinite Borel equivalence relations’, J. Symb. Log. 58(3) (1993), 894–907.CrossRefGoogle Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156 (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
Kechris, A. S. and Miller, B. D., Topics in Orbit Equivalence, Lecture Notes in Mathematics, vol. 1852 (Springer–Verlag, Berlin, 2004).CrossRefGoogle Scholar
Kida, Y., The Mapping Class Group from the Viewpoint of Measure Equivalence Theory, Mem. Amer. Math. Soc., vol. 196(916) (2008), viii+190.Google Scholar
Klarreich, E., ‘The boundary at infinity of the curve complex and the Relative Teichmüller Space’, Preprint, 1999, arXiv:1803.10339.Google Scholar
Lécureux, J., ‘Amenability of actions on the boundary of a building’, Int. Math. Res. Not. IMRN 17 (2010), 3265–3302.Google Scholar
Marquis, T., ‘On geodesic ray bundles in buildings’, Geom. Dedicata 202 (2019), 27–43.CrossRefGoogle Scholar
Marquis, T. and Sabok, M., ‘Hyperfiniteness of boundary actions of hyperbolic groups’, Math. Ann. 377 (2020), 1129—1153.CrossRefGoogle Scholar
Masur, H. A. and Minsky, Y. N., ‘Geometry of the complex of curves, I: Hyperbolicity’, Invent. Math. 138(1) (1999), 103–149.CrossRefGoogle Scholar
Masur, H. A. and Schleimer, S., ‘The geometry of the disk complex’, J. Amer. Math. Soc. 26(1) (2013), 1–62.CrossRefGoogle Scholar
Moore, J. T., ‘A brief introduction to amenable equivalence relations’, in Trends in Set Theory, Contemp. Math., vol. 752 (American Mathematical Society, Providence, RI, 2020), 153–163.CrossRefGoogle Scholar
Nevo, A. and Sageev, M., ‘The Poisson boundary of $\mathsf{CAT}(0)$cube complex groups’, Groups Geom. Dyn. 7(3) (2013), 653–695.CrossRefGoogle Scholar
Ozawa, N., ‘Boundary amenability of relatively hyperbolic groups’, Topology Appl. 153(14) (2006), 2624–2630.CrossRefGoogle Scholar
Ozawa, N., Amenable Actions and Applications, International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006), 1563–1580.Google Scholar
Pho-on, W., ‘Infinite unicorn paths and Gromov boundaries’, Groups Geom. Dyn. 11(1) (2017), 353–370.CrossRefGoogle Scholar
Schleimer, S. (2013), unpublished manuscript.Google Scholar
Touikan, N., ‘On geodesic ray bundles in hyperbolic groups’, Proc. Amer. Math. Soc. 146(10) (2018), 4165–4173.CrossRefGoogle Scholar
Veršik, A. M., ‘The action of $\mathsf{PSL}\left(2,\mathsf{Z}\right)$ in ${\mathsf{R}}^1$ is approximable’, Uspekhi Mat. Nauk 33(1(199)) (1978), 209–210.Google Scholar
Zimmer, R. J., ‘Amenable ergodic group actions and an application to Poisson boundaries of random walks’, J. Funct. Anal. 27(3) (1978), 350–372.Google Scholar
Zimmer, R. J., Ergodic Theory and Semisimple Groups, Monographs in Mathematics, vol. 81 (Birkhäuser Verlag, Basel, 1984).CrossRefGoogle Scholar