Examples of diffeomorphism group cocycles with no periodic approximation

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.