Nonsmoothable, locally indicable group actions on the interval

TL;DR

This paper introduces a new criterion based on local orbit order structures to identify groups of homeomorphisms of the interval that cannot be smoothed into C^1 diffeomorphisms, expanding understanding of group actions.

Contribution

It provides a stronger criterion for nonsmoothability of group actions on the interval, leading to new examples of locally indicable groups not conjugate to C^1 diffeomorphisms.

Findings

Local order structure of orbits indicates nonsmoothability.

New examples of non-C^1 group actions on the interval.

Extension of Thurston stability theorem insights.

Abstract

By the Thurston stability theorem, a group of C^1 orientation-preserving diffeomorphisms of the closed unit interval is locally indicable. We show that the local order structure of orbits gives a stronger criterion for nonsmoothability that can be used to produce new examples of locally indicable groups of homeomorphisms of the interval that are not conjugate to groups of C^1 diffeomorphisms.