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Fixed points of Abelian actions. (English) Zbl 1130.37023
Summary: We prove that if \(\mathcal F\) 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 \(F\). If \(F\) 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 \(F\) has a subgroup of finite index all of whose elements share a common fixed point.

MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
57M60 Group actions on manifolds and cell complexes in low dimensions
57S25 Groups acting on specific manifolds
55M20 Fixed points and coincidences in algebraic topology
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