Free products and the algebraic structure of diffeomorphism groups

Let $M$ be a compact one–manifold, and let ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{1+\mathrm{b}\mathrm{v}}(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation. We prove that if $G$ is a group which is not virtually metabelian, then $(G\times \mathbb{Z})\ast \mathbb{Z}$ is not realized as a subgroup of ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{1+\mathrm{b}\mathrm{v}}(M)$. This gives the first examples of finitely generated groups $G,H\le {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{+}^{\mathrm{\infty }}(M)$ such that $G\ast H$ does not embed into ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{1+\mathrm{b}\mathrm{v}}(M)$. By contrast, for all countable groups $G,H\le {\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}^{+}(M)$ there exists an embedding $G\ast H\to $\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}$^+(M)$. We deduce that many common groups of homeomorphisms do not embed into ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{1+\mathrm{b}\mathrm{v}}(M)$, for example the free product of $\mathbb{Z}$ with Thompson’s group $F$. We also complete the classification of right-angled Artin groups which can act smoothly on $M$ and in particular, recover the main result of a joint work of the authors with Baik. Namely, a right-angled Artin group $A(\mathrm{\Gamma })$ either admits a faithful $C^{\mathrm{\infty }}$ action on $M$, or $A(\mathrm{\Gamma })$ admits no faithful $C^{1+\mathrm{b}\mathrm{v}}$ action on $M$. In the former case, $A(\Gamma)\cong\prod_i G_i$where$G_i$ is a free product of free abelian groups. Finally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on $S^1$.