Contactomorphism groups and Legendrian flexibility

Abstract

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 $G$.

38 pages, 5 figures v2: corrects one silly mistake, updates references and acknowledgments. Submitted version