Abstract
We prove that if X is a compact, oriented, connected 4-dimensional smooth manifold, possibly with boundary, satisfying \(\chi (X)\ne 0\), then there exists a natural number C such that any finite group G acting smoothly and effectively on X has an abelian subgroup A generated by two elements which satisfies \([G:A]\le C\) and \(\chi (X^A)=\chi (X)\). Furthermore, if \(\chi (X)<0\) then A is cyclic. This answers positively, for any such X, a question of Etienne Ghys. We also prove an analogous result for manifolds of arbitrary dimension and non-vanishing Euler characteristic, but restricted to pseudofree actions.
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