Direct products, overlapping actions, and critical regularity

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 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 $.