||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,C-tau action of HxK on a compact interval I is not overlapping for all tau > 0, which by definition means that there must be non-trivial h is an element of H and k is an element of K with disjoint support. As a corollary we prove that the right-angled Artin group (F-2 x F-2) (*) Z has critical regularity one, which is to say that it admits a faithful C-1 action on I, but no faithful C-1,C-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 * Z is a new example of a locally indicable group admitting no faithful C-1 action on I.