Extensible positive loops and vanishing of symplectic cohomology

I will discuss the relationship between positive loops of contactomorphisms of a fillable contact manifold and the symplectic cohomology (SH) of the filling. The main result is that the existence of a positive loop which is "extensible" implies SH vanishes. We also discuss the relationship between non-extensible loops and exotic mapping classes in the symplectomorphism group. The results have a relative (Lagrangian) formulation: an extensible positive loop of fillable Legendrians implies the wrapped Floer cohomology of the filling vanishes. As an application, one obtains a criterion for a positive loop to be non-extensible; interestingly enough, non-extensible loops provide a nice construction of exotic symplectic mapping classes (in the absolute case) and exotic Lagrangian fillings (in the relative case).