On Maximal Tori in the Contactomorphism Groups of Regular Contact Manifolds

By a theorem of Banyaga the group of diffeomorphisms of a manifold P preserving a regular contact form α is a central S extension of the commutator of the group of symplectomorphisms of the base B = P/S. We show that if T is a Hamiltonian maximal torus in the group of symplectomorphism of B, then its preimage under the extension map is a maximal torus not only in the group Diff (P, α) of diffeomorphisms of P preserving α but also in the much bigger group of contactomorphisms Diff (P, ξ), the group of diffeomorphism of P preserving the contact distribution ξ = kerα. We use this (and the work of Hausmann, and Tolman on polygon spaces) to give examples of contact manifolds (P, ξ = kerα) with maximal tori of different dimensions in their group of contactomorphisms.