Free products and the algebraic structure of diffeomorphism groups

Abstract: Let $M$ be a compact one--manifold, and let $\mathrm{Diff}^{1+\mathrm{bv}}(M)$ 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 $(G\times\mathbb{Z})*\mathbb{Z}$ is not realized as a subgroup of $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. This gives the first examples of finitely generated groups $G,H\le \mathrm{Diff}_+^\infty(M)$ such that $G\ast H$ does not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. By contrast, for all countable groups $G,H\le\mathrm{Homeo}^+(M)$ there exists an embedding $G\ast H\to \mathrm{Homeo}^+(M)$. We deduce that many common groups of homeomorphisms do not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$, 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 $A(\Gamma)$ either admits a faithful $C^{\infty}$ action on $M$, or $A(\Gamma)$ admits no faithful $C^{1+\mathrm{bv}}$ 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 $S^1$.