Distortion in Groups of Generalized Piecewise-Affine Transformations

For each natural number n, we consider the subgroup Rn of Homeo+[0,1] made by the elements that are affine except for a subset whose Cantor-Bendixson rank is less than or equal to n. These groups of generalized piecewise-affine 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 exists an element that is undistorted of Rn yet distorted in Rn+1. Actually, such an element is explicitly constructed. © The Author(s), under exclusive licence to Springer Nature B.V. 2025.