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 C1,τ action of H×K on a compact interval I is not overlapping for all τ>0, which by definition means that there must be non-trivial h ϵ H and k ϵ K with disjoint support. As a corollary we prove that the right-angled Artin group (F2× F2) *Z has critical regularity one, which is to say that it admits a faithful C1 action on I, but no faithful C1,¿ 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 C1 overlapping action on I, so that F *Z is a new example of a locally indicable group admitting no faithful C1 action on I. © 2021, American Institute of Mathematical Sciences. All rights reserved.