Distortion in groups of generalized piecewise-linear transformations

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.

18 pages, 6 figures