  We address the problem of computing the critical regularity of groups of
homeomorphisms of the interval. Our main result is that if $H$ and $K$ are two
non-solvable groups then a faithful $C^{1,\tau}$ action of $H\times K$ on a
compact interval $I$ is {\em not overlapping} for all $\tau>0$, which by
definition means that there must be non-trivial $h\in H$ and $k\in K$ with
disjoint support. As a corollary we prove that the right-angled Artin group
$(F_2\times F_2)*\mathbb{Z}$ has critical regularity one, which is to say that
it admits a faithful $C^1$ action on $I$, but no faithful $C^{1,\tau}$ action.
This is the first explicit example of a group of exponential growth which is
without nonabelian subexponential growth subgroups, whose critical regularity
is finite, achieved, and known exactly. Another corollary we get is that
Thompson's group $F$ does not admit a faithful $C^1$ overlapping action on $I$,
so that $F*\mathbb{Z}$ is a new example of a locally indicable group admitting
no faithful $C^1$--action on $I$.
