Rigidity of trivial actions of abelian-by-cyclic groups

Abstract

Let ${\Gamma }_A$ denote the abelian-by-cyclic group associated to an integer-valued, non-singular matrix $A$. We show that if $A$ has no eigenvalues of modulus one, then there are no faithful $C^1$ perturbations of the trivial action $ ι: Γ_A \to \diff$, where $M$ is a compact manifold.

11 pages