Examples of groups acting on the circle with at most 2 fixed points.

The main goal of this talk is to address the question: "Which are the groups of orientable homeomorphism on the circle, such that every element fixes at most 2 points?". The classical examples are the rotation group SO(2) and the projective linear group PSL(2,R). However, in 1999, using tools like orbital opening and the Ping-Pong lemma, Kovačević showed that they are not the only examples. In this talk will be presented some new examples using the same tools, in addition to some partial results that point to a possible classification of all groups that answer the question.