Finite group actions on homology spheres and manifolds with nonzero Euler characteristic

Abstract

Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that $Diff(X)$ is Jordan. This means that there exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an abelian subgroup whose index in $G$ is at most $C$. Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler characteristic are Jordan.

17 pages; v4: the previous version v3 has been substantially revised and split in two parts (roughly coinciding with arXiv:1403.0383v2 and arXiv:1310.6565); this is one of the two parts; a corollary on algebraic actions on smooth real affine manifolds has been added; v5: final version, accepted for publication by Journal of Topology

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Cited by in corpus (16)

• Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
• Automorphisms of pointless surfaces
• Jordan property for automorphism groups of compact spaces in Fujiki's class $C$
• Symmetries of flat manifolds, Jordan property and the general Zimmer program
• Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds
• Hamiltonian no-torsion
• Automorphism groups of $P^1$-bundles over a non-uniruled base
• Integral foliated simplicial volume and circle foliations
• Finite groups acting symplectically on $T^2\times S^2$
• Finite subgroups of the birational automorphism group are 'almost' nilpotent of class at most two
• Finite groups of bimeromorphic selfmaps of non-uniruled Kähler threefolds
• Special $p$-groups acting on compact manifolds
• Automorphism groups of Moishezon threefolds
• Some properties of the group of birational maps generated by the automorphisms of ${\mathbb{P}}_{\mathbb{C}}^n$ and the standard involution
• Actions of large finite groups on aspherical manifolds
• Finite subgroups of the birational automorphism group are `almost' nilpotent