Diffeomorphism Groups of Compact 4-manifolds are not always Jordan

Abstract

We show that if $M$ is a compact smooth manifold diffeomorphic to the total space of an orientable $S^2$ bundle over the torus $T^2$, then its diffeomorphism group does not have the Jordan property, i.e., Diff$(M)$ contains a finite subgroup $G_n$ for any natural number $n$ such that every abelian subgroup of $G_n$ has index at leat $n$. This gives a counterexample to an old conjecture of Ghys.

arXiv comment4 pages

References in corpus (4)

• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Finite group actions on manifolds without odd cohomology
• Finite subgroups of diffeomorphism groups
• Finite group actions on 4-manifolds with nonzero Euler characteristic

Cited by in corpus (16)

• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Jordan property for automorphism groups of compact spaces in Fujiki's class $C$
• Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds
• Symmetries of flat manifolds, Jordan property and the general Zimmer program
• Finite group actions on 4-manifolds with nonzero Euler characteristic
• Automorphism groups of Inoue and Kodaira surfaces
• Automorphism groups of $P^1$-bundles over a non-uniruled base
• Finite groups acting symplectically on $T^2\times S^2$
• Finite groups of bimeromorphic selfmaps of non-uniruled Kähler threefolds
• On the number of stabilizer subgroups in a finite group acting on a manifold
• Finite subgroups of the birational automorphism group are 'almost' nilpotent of class at most two
• Isometry groups of closed Lorentz 4-manifolds are Jordan
• Actions of large finite groups on aspherical manifolds
• Finite subgroups of the birational automorphism group are `almost' nilpotent
• Automorphism groups of Moishezon threefolds
• Special $p$-groups acting on compact manifolds