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We address the problem of computing the critical regularity of groups ofhomeomorphisms of the interval. Our main result is that if $H$ and $K$ are twonon-solvable groups then a faithful $C^{1,\tau}$ action of $H\times K$ on acompact interval $I$ is {\em not overlapping} for all $\tau>0$, which bydefinition means that there must be non-trivial $h\in H$ and $k\in K$ withdisjoint 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 thatit 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 iswithout nonabelian subexponential growth subgroups, whose critical regularityis finite, achieved, and known exactly. Another corollary we get is thatThompson'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 admittingno faithful $C^1$--action on $I$.

Direct products, overlapping actions, and critical regularity