Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-16T02:00:02.327Z Has data issue: false hasContentIssue false
  • English
  • Français

Big mapping class groups and rigidity of the simple circle

Part of: Smooth dynamical systems: general theory Topological dynamics Low-dimensional dynamical systems Special aspects of infinite or finite groups

Published online by Cambridge University Press:  03 June 2020

DANNY CALEGARI  and
LVZHOU CHEN
DANNY CALEGARI
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois, USA email dannyc@math.uchicago.edu, lzchen@math.uchicago.edu
LVZHOU CHEN
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois, USA email dannyc@math.uchicago.edu, lzchen@math.uchicago.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\unicode[STIX]{x1D6E4}$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\unicode[STIX]{x1D6E4}$ on the circle is either trivial or semiconjugate to a unique minimal action on the so-called simple circle.

Keywords

group actionslow-dimensional dynamicsbig mapping class groupsrigidity

MSC classification

Primary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37E10: Maps of the circle
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37E45: Rotation numbers and vectors 20F05: Generators, relations, and presentations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 1961 - 1987
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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

Aramayona, J., Fossas, A. and Parlier, H.. Arc and curve graphs for infinite-type surfaces. Proc. Amer. Math. Soc. 145(11) (2017), 4995–5006.CrossRefGoogle Scholar
Bavard, J.. Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire. Geom. Topol. 20(1) (2016), 491–535.CrossRefGoogle Scholar
Bavard, J. and Walker, A.. The Gromov boundary of the ray graph. Trans. Amer. Math. Soc. 370(11) (2018), 7647–7678.CrossRefGoogle Scholar
Bavard, J. and Walker, A.. Two simultaneous actions of big mapping class groups. Preprint, 2018, arXiv: 1802.02715.Google Scholar
Calegari, D.. Circular groups, planar groups, and the Euler class. Proc. Casson Fest (Geometry and Topology Monographs, 7) . Geometry and Topology Publications, Coventry, 2004, pp. 431–491.Google Scholar
Calegari, D.. scl (MSJ Memoirs, 20) . Mathematical Society of Japan, Tokyo, 2009.CrossRefGoogle Scholar
Calegari, D.. Big mapping class groups and dynamics. Blog post https://lamington.wordpress.com/2009/06/22/big-mapping-class-groups-and-dynamics/.Google Scholar
Calegari, D.. Complex Dynamics and Big Mapping Class Groups. Monograph, in preparation.Google Scholar
Calegari, D. and Chen, L.. Normal subgroups of big mapping class groups, in preparation.Google Scholar
Durham, M., Fanoni, F. and Vlamis, N.. Graphs of curves on infinite-type surfaces with mapping class group actions. Ann. Inst. Fourier 68(6) (2018), 2581–2612.CrossRefGoogle Scholar
Farb, B. and Margalit, D.. A Primer on Mapping Class Groups (Princeton Mathematical Series, 49) . Princeton University Press, Princeton, NJ, 2012.Google Scholar
Ghys, É.. Groupes d’homéomorphismes du cercle et cohomologie bornée. Lefschetz Centennial Conf., Part III (Mexico City, 1984) (Contemporary Mathematics, 58) . American Mathematical Society, Providence, RI, 1987, pp. 81–106.Google Scholar
Mann, K. and Wolff, M.. Rigidity of mapping class group actions on $S^{1}$ . Geom. Topol. to appear.Google Scholar
Mather, J.. The vanishing of the homology of certain groups of homeomorphisms. Topology 10 (1971), 297–298.CrossRefGoogle Scholar
Patel, P. and Vlamis, N.. Algebraic and topological properties of big mapping class groups. Algebr. Geom. Topol. 18(7) (2018), 4109–4142.CrossRefGoogle Scholar
Segal, G.. Classifying spaces related to foliations. Topology 17(4) (1978), 367–382.Google Scholar
Sergiescu, V. and Tsuboi, T.. A remark on homeomorphisms of the Cantor set. Proc. Geometric Study of Foliations (Tokyo 1993). World Scientific, Singapore, 1994, pp. 431–436.Google Scholar
Yagasaki, T.. Homotopy types of homeomorphism groups of noncompact 2-manifolds. Topology Appl. 108(2) (2000), 123–136.CrossRefGoogle Scholar
Zimmermann, B. P.. On finite groups acting on spheres and finite subgroups of orthogonal groups. Sib. Electron. Math. Rep. 9 (2012), 1–12.Google Scholar