Simplicity of the contactomorphism group of finite regularity

For a given coorientable contact manifold $(M^{2n+1},\xi)$, we consider the group $ \operatorname{Cont}_c^{(r,\delta)}(M,\alpha)$ consisting of $C^{r,\delta}$ contactomorphisms with compact support which is equipped with $C^{r,\delta}$-topology of H\"older regularity $(r,\delta)$ for $r \geq 1$ and $0 <\delta \leq 1$. We prove that for all H\"older class exponents with $r > n + 2$ or $r = n+1, \, \frac12 < \delta \leq 1$ (resp. $r < n+1$ or $r = n+1$ and $ 0< \delta <\frac12$), the group is a perfect (and so a simple) group. In particular, $\operatorname{Cont}_c^r(M,\xi)$ is simple for all integer $r \geq 1$. For the case of $\operatorname{Cont}_c^{(r,\delta)}(M,\alpha)$ of general H\"older regularity, we prove the simplicity for all pairs $(r,\delta)$ leaving only the case of $(r,\delta) = (n+1,\frac12)$ open.