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TITLE DIRECT PRODUCTS, OVERLAPPING ACTIONS, AND CRITICAL REGULARITY
KIAS AUTHORS Kim, Sang-Hyun
JOURNAL JOURNAL OF MODERN DYNAMICS, 2021
ARCHIVE  
ABSTRACT 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.
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