Direct products, overlapping actions, and critical regularity

TL;DR

This paper investigates the regularity thresholds for group actions on intervals, showing certain complex groups cannot act smoothly beyond a specific regularity, and providing explicit examples of groups with finite critical regularity.

Contribution

It establishes new bounds on the regularity of group actions, introduces explicit examples of groups with finite critical regularity, and explores the non-overlapping nature of actions for non-solvable groups.

Findings

Non-solvable groups' actions are non-overlapping for all 

The group   *  is shown to have critical regularity one

Thompson's group F cannot have a faithful $C^1$ overlapping action on the interval

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 $C^{1,\tau}$ action of $H\times K$ on a compact interval $I$ is {\em 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 $I$.