  We construct a finitely generated subgroup of
$\text{Diff}^{\infty}(\mathbb{S}^3 \times \mathbb{S}^1)$ where every element is
conjugate to an isometry but such that the group action itself is far from
isometric (the group has "exponential growth of derivatives"). As a corollary,
one obtains a locally constant $\text{Diff}^{\infty}(\mathbb{S}^3 \times
\mathbb{S}^1)$ valued cocycle over a hyperbolic dynamical system which has
elliptic behavior over its periodic orbits but which preserves a measure with
non-zero top Fiber Lyapunov exponent. Additionally, we provide new examples of
Banach cocycles not satisfying the periodic approximation property as first
shown by Kalinin-Sadovskaya.
