Let
be a compact
fibered
–manifold,
presented as a mapping torus of a compact, orientable surface
with
monodromy
,
and let
be a compact Riemannian manifold. Our main result is that if the induced action
on
has no eigenvalues on the unit circle, then there exists a neighborhood
of the trivial action
in the space of
actions of
on
such that
any action in
is abelian. We will prove that the same result holds in the generality
of an infinite cyclic extension of an arbitrary finitely generated group
provided that the conjugation action of the cyclic group on
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.
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