Classifying Stein's groups

In this paper, we provide a comprehensive classification of Stein's groups, which generalize the well-known Higman-Thompson groups. Stein's groups are defined as groups of piecewise linear bijections of an interval with finitely many breakpoints and slopes belonging to specified additive and multiplicative subgroups of the real numbers. Our main result establishes a classification theorem for these groups under the assumptions that the slope group is finitely generated and the additive group has rank at least 2. We achieve this by interpreting Stein's groups as topological full groups of ample groupoids. A central concept in our analysis is the notion of $H^1$-rigidity in the cohomology of groupoids. In the case where the rank of the additive group is 1, we adopt a different approach using attracting elements to impose strong constraints on the classification.