Barcodes and quasi-isometric embeddings into Hamiltonian diffeomorphism groups

Abstract

We construct an embedding $Phi$ of $[0,1]^{infty}$ into $Ham(M, omega)$, the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold $(M, omega)$. We then prove that $Phi$ is in fact a quasi-isometry. After imposing further assumptions on $(M, omega)$, we adapt our methods to construct a similar embedding of $R oplus [0,1]^{infty}$ into either $Ham(M, omega)$ or $widetilde{Ham}(M, omega)$, the universal cover of $Ham(M, omega)$. Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. We conclude by proving the boundedness of the boundary depth function $beta$ restricted to the set of autonomous Hamiltonian diffeomorphisms of $(S^2, omega)$. The majority of our proofs rely heavily on a continuity result for barcodes (as presented in the work of M. Usher and J. Zhang) associated to filtered Floer homology viewed as a persistence module.