Contactomorphism groups and Legendrian flexibility

We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold $(V, ξ)$ has a supporting open book whose pages are flexible Weinstein manifolds, then the connected component G of the identity in its automorphism group is a uniformly simple group: for every non-trivial element g, every other element is a product of at most 128( V + 1) conjugates of g± 1. In particular any conjugation invariant norm on this group is bounded. We also prove the later statement still holds for the universal cover of G.