Fixed points of Abelian actions

John Franks; Michael Handel; Kamlesh Parwani

doi:10.3934/jmd.2007.1.443

Article Contents

2007, Volume 1, Issue 3: 443-464. Doi: 10.3934/jmd.2007.1.443

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Fixed points of Abelian actions

1.
Department of Mathematics, Northwestern University, Evanston, Illinois
2.
Department of Mathematics and Computer Science, Herbert H. Lehman College (CUNY), New York
3.
Department of Mathematics, Eastern Illinois University, Illinois

Received: September 30, 2006

Revised: February 28, 2007

Published: March 2007

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Abstract

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.

Keywords:

Abelian group actions on surfaces,
fixed points.

Mathematics Subject Classification: Primary: 37E30, 57M60; Secondary: 57S25, 55M20.

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