Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
  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. 
Bal\'azs Csik\'os, L\'aszl\'o Pyber, Endre Szab\'o
math.DG math.GR