Finitely generated groups of diffeomorphisms with prescribed critical regularity in every dimension

Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere $\mathbb{S}^n$ or the real projective space $\mathbb{RP}^n$. Given $α\ge 1$, we construct a finitely generated group $G_{n,α}$ that embeds into the group of $C^α$-diffeomorphisms of $M$, but does not embed into the group of $C^β$-diffeom