  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
$(V, \xi)$ 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 $128(\dim V + 1)$ conjugates of $g^{\pm
1}$. In particular any conjugation invariant norm on this group is bounded. We
also prove the later statement still holds for the universal cover of $G$.
