Distortion in groups of generalized piecewise-linear transformations

TL;DR

This paper investigates the distortion properties of elements in a hierarchy of groups of piecewise-linear transformations, revealing that elements can change from undistorted to distorted as the group complexity increases.

Contribution

It explicitly constructs elements that are undistorted in one group but become distorted in a larger group within the hierarchy.

Findings

Existence of elements undistorted in  but distorted in .

Explicit construction of such elements.

Distortion properties vary along the chain of groups.

Abstract

For each natural number $n$, we consider the subgroup $\mathcal{R}_n$ of Homeo$_+[0,1]$ made by the elements that are linear except for a subset whose Cantor-Bendixson rank is less than or equal to $n$. These groups of generalized piecewise-linear transformations yield an ascending chain of groups as we increase $n$. We study how the notion of distorted element changes along this chain. Our main result establishes that for each natural number $n$, there exits an element that is undistorted of $\mathcal{R}_n$ yet distorted in $\mathcal{R}_{n+1}$. Actually, such an element is explicitly constructed.