Rigidity of trivial actions of abelian-by-cyclic groups

TL;DR

This paper proves that for certain abelian-by-cyclic groups, trivial actions on compact manifolds are rigid and cannot be smoothly perturbed if the associated matrix has no eigenvalues of modulus one.

Contribution

It establishes a rigidity result for trivial actions of abelian-by-cyclic groups under specific spectral conditions on the defining matrix.

Findings

No faithful $C^1$ perturbations of trivial actions when matrix has no eigenvalues of modulus one

Rigidity depends on spectral properties of the associated matrix

Trivial actions are structurally stable under these conditions

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 $ \iota: \Gamma_A \to \diff$, where $M$ is a compact manifold.