Rotation numbers in Thompson-Stein groups and applications
Isabelle Liousse

Stein-Thompson groups are finitely presented groups of piecewise linear cicle
homeomorphisms in which slopes and denominators of break points can be divisible by an arbitrary (but prescribed) finite set of multiplically independant integers.

First, we will present some properties of rotation numbers for Stein-Thompson groups.

Then we will show how these dynamical properties imply results of
PL-rigidity, non isomorphicity, non exoticity of automorphisms, non existence of distorded element and non smoothability.

Finally, we will explain how D. Zhuang constructed the first examples of finitely presented groups containing elements with irrational stable commutator length, using existence of elements with irrational rotation numbers in Stein-Thompson groups.