Rigidity of trivial actions of abelian-by-cyclic groups

Let ${\mathrm{\Gamma }}_A$ 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 $\iota :{\mathrm{\Gamma }}_A\to \text{diff}$, where $M$ is a compact manifold.