I will report on the work done in the PhD thesis of my student João Carnevale, where he studied group actions on the line or the circle with the property that the number of fixed points of any non-trivial element does not exceed some uniform fixed constant N. This extends classical results by Hölder and Solodov, and gives a more general framework for the work of Kovačević on Möbius-like actions without the convergence property. In particular, we will discuss a combination theorem, which allows to control the number of fixed points when one glues together two actions in a controlled way.