Algebraic & Geometric Topology

$C^1$ actions on the mapping class groups on the circle

Kamlesh Parwani

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Abstract

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C1 action of the mapping class group of S on the circle is trivial.

The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C1 faithful actions on the circle. We also prove that for n6, any C1 action of Aut(Fn) or Out(Fn) on the circle factors through an action of 2.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 2 (2008), 935-944.

Dates
Received: 22 February 2008
Revised: 19 March 2008
Accepted: 28 March 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796850

Digital Object Identifier
doi:10.2140/agt.2008.8.935

Mathematical Reviews number (MathSciNet)
MR2443102

Zentralblatt MATH identifier
1155.37028

Subjects
Primary: 37E10: Maps of the circle
Secondary: 57M60: Group actions in low dimensions

Keywords
mapping class groups Kazhdan groups actions on the circle

Citation

Parwani, Kamlesh. $C^1$ actions on the mapping class groups on the circle. Algebr. Geom. Topol. 8 (2008), no. 2, 935--944. doi:10.2140/agt.2008.8.935. https://projecteuclid.org/euclid.agt/1513796850


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References

  • U Bader, A Furman, A Shaker, Superrigidity, Weyl groups, and actions on the circle
  • M R Bridson, K Vogtmann, Homomorphisms from automorphism groups of free groups, Bull. London Math. Soc. 35 (2003) 785–792
  • M Burger, N Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. $($JEMS$)$ 1 (1999) 199–235
  • D Calegari, Dynamical forcing of circular groups, Trans. Amer. Math. Soc. 358 (2006) 3473–3491
  • P Dehornoy, A fast method for comparing braids, Adv. Math. 125 (1997) 200–235
  • B Deroin, V Kleptsyn, A Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math. 199 (2007) 199–262
  • B Farb, J Franks, Groups of homeomorphisms of one-manifolds I: actions of nonlinear groups
  • J Franks, personal communication (2001)
  • É Ghys, Actions de réseaux sur le cercle, Invent. Math. 137 (1999) 199–231
  • É Ghys, Groups acting on the circle, Enseign. Math. $(2)$ 47 (2001) 329–407
  • N V Ivanov, Mapping class groups, from: “Handbook of geometric topology”, North-Holland, Amsterdam (2002) 523–633
  • M Korkmaz, Low-dimensional homology groups of mapping class groups: a survey, Turkish J. Math. 26 (2002) 101–114
  • A Navas, Actions de groupes de Kazhdan sur le cercle, Ann. Sci. École Norm. Sup. $(4)$ 35 (2002) 749–758
  • A Navas, Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle, Comment. Math. Helv. 80 (2005) 355–375
  • H Short, B Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. $(2)$ 46 (2000) 279–312
  • F Taherkhani, The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite subgroups, Experiment. Math. 9 (2000) 261–274
  • W P Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974) 347–352
  • D Witte, Arithmetic groups of higher ${\bf Q}$–rank cannot act on $1$–manifolds, Proc. Amer. Math. Soc. 122 (1994) 333–340