  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 $S^1$.
