QUASI-morphisms and L ^p-metrics on groups of volume-preserving diffeomorphisms

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form μ. We show that every homogeneous quasi-morphism on the identity component Diff ₀(M, μ) of the group of volume-preserving diffeomorphisms of M, which is induced by a quasi-morphism on the fundamental group π ₁(M), is Lipschitz with respect to the L ^p-metric on Diff ₀(M, μ). As a consequence, assuming certain conditions on π ₁(M), we construct bi-Lipschitz embeddings of finite dimensional vector spaces into Diff ₀(M, μ).