On Symplectomorphisms of the Symplectization of a Compact Contact Manifold

Abstract

Let $(N,\alpha)$ be a compact contact manifold and $(N \times {\mathbb R}$, $d(e^t\alpha))$ its symplectization. We show that the group $G$ which is the identity component in the group of symplectic diffeomorphisms $\phi$ of $(N\times {\mathbb R}, d(e^t\alpha))$ that cover diffeomorphisms $\underline {\phi}$ of $ N\times S^1$ is simple, by showing that $G$ is isomorphic to the kernel of the Calabi homomorphism of the associated locally conformal symplectic structure.