2007, 1(3): 443-464. doi: 10.3934/jmd.2007.1.443

Fixed points of Abelian actions

1. 

Department of Mathematics, Northwestern University, Evanston, Illinois, United States

2. 

Department of Mathematics and Computer Science, Herbert H. Lehman College (CUNY), New York, United States

3. 

Department of Mathematics, Eastern Illinois University, Illinois, United States

Received  October 2006 Revised  March 2007 Published  April 2007

We prove that if $\mathfrak{F}$ is an abelian group of $C^1$ diffeomorphisms isotopic to the identity of a closed surface $S$ of genus at least two, then there is a common fixed point for all elements of $\mathfrak{F}$. If $\mathfrak{F}$ is an abelian group of $C^1$ diffeomorphisms (not necessarily isotopic to the identity) of a closed surface $S$ of genus at least two, then $\mathfrak{F}$ has a subgroup of finite index all of whose elements share a common fixed point.
Citation: John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443
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