# American Institute of Mathematical Sciences

July  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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