NON JORDAN GROUPS OF DIFFEOMORPHISMS AND ACTIONS OF COMPACT LIE GROUPS ON MANIFOLDS

Abstract

A recent preprint of Csikos, Pyber and Szabo [CPS] proves that the diffeomorphism group of T2 × S2 is not Jordan. The purpose of this paper is to generalize the arguments in [CPS] in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any ∈ > 0 there exist compact manifolds admitting effective smooth actions of arbitrarily large finite groups Γ all of whose abelian subgroups have at most |Γ|∈ elements. Finally, we also recover some results on the nonexistence of effective actions of compact connected semisimple Lie groups on manifolds.