Fixed points of Abelian actions
John Franks - Department of Mathematics, Northwestern University, Evanston, Illinois, United States (email) 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.
Received: October 2006; Revised: March 2007; Available Online: April 2007. |