Non-smoothability for a class of groups of piecewise linear homeomorphisms of the interval

TL;DR

This paper proves that certain groups of piecewise linear homeomorphisms of the interval cannot have highly regular faithful actions on the line, using circle reduction and Herman-Yoccoz theory.

Contribution

It introduces a novel approach combining circle reduction and Herman-Yoccoz theory to establish non-smoothability results for specific groups.

Findings

No sufficiently regular faithful actions exist for the considered groups.

Reduction to circle case simplifies the analysis.

Application of Herman-Yoccoz theory provides the key argument.

Abstract

For a certain class of groups of piecewise linear homeomorphisms of the interval, we prove that they admit no sufficiently regular faithful action on the line. Building on previous work of Brum, Matte Bon, Rivas, and the author [arXiv:2104.14678], the new ingredient is an observation from a recent work of Hyde and Tatch Moore [arXiv:2103.14911], which allows to reduce the problem to the case of the circle, and then apply Herman-Yoccoz theory.