Rigidity of Trivial Actions of Abelian-by-cyclic Groups

The question of existence and stability of global fixed points for group actions has been studied in many different contexts. In the setting of actions of Lie groups it was shown by Lima [8] that n commuting vector fields on a genus g surface, Σg, of non-zero Euler characteristic have a common singularity. This implies that any action of the abelian Lie group R on Σg has a global fixed point. It was later shown by Plante [10] that any action of a nilpotent Lie group on a surface with non-zero Euler characteristic has a global fixed point. The study of stability of global fixed points for group actions is also related to the study of foliations. Given a foliation of a manifold with a compact leaf L and a transverse disk D, the holonomy map along L defines an action of π1(L) on the disk D. Perturbations of group actions are related to the study of foliations, for given a nearby leaf L′ diffeomorphic to L, the holonomy along L′ defines a new action that is a perturbation of the original. The Thurston stability theorem gives conditions for local stability of C foliations of a compact manifold. Methods of Thurston were then modified by Langevin and Rosenberg [7], Stowe [11] [12], and Schweitzer [13] to establish results regarding stability of global fixed points, and leaves of fibrations. Inspired by the ideas of Lima [8], Bonatti [1] used methods similar to Thurston’s to show that any Z action on surfaces with non-zero Euler characteristic generated by diffeomorphisms C close to the identity has a global fixed point. Using similar techniques, Druck, Fang and