## Algebraic & Geometric Topology

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

Kamlesh Parwani

#### 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 $n≥6$, any $C1$ action of $Aut(Fn)$ or $Out(Fn)$ on the circle factors through an action of $ℤ∕2ℤ$.

#### Article information

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

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

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

#### 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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