# 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.

## Key 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.