Small $C^1$ actions of semidirect products on compact manifolds

TL;DR

This paper proves that small $C^1$ actions of certain semidirect product groups on compact manifolds are necessarily abelian, under conditions related to eigenvalues of induced actions on cohomology, generalizing previous results.

Contribution

It extends McCarthy's result to a broader class of groups, showing that under specific eigenvalue conditions, small $C^1$ actions are abelian.

Findings

Small $C^1$ actions are abelian under eigenvalue conditions.

Generalization from abelian-by-cyclic groups to more complex extensions.

Eigenvalues of the induced action determine the triviality of small actions.

Abstract

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds.