Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds

A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csik\'os, Pyber and Szab\'o in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any $\epsilon>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $\Gamma$ all of whose abelian subgroups have at most $|\Gamma|^{\epsilon}$ elements. Finally, we also recover some results on nonexistence of effective actions of compact connected semisimple Lie group on manifolds.