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.