A quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer's metric

TL;DR

This paper constructs a quasi-isometric embedding of an infinite-dimensional cube into the Hamiltonian diffeomorphism group with Hofer's metric, revealing geometric properties of these groups through Floer homology techniques.

Contribution

It introduces a new quasi-isometric embedding of infinite-dimensional spaces into Hamiltonian diffeomorphism groups, utilizing filtered Floer homology and barcode continuity results.

Findings

Established a quasi-isometry between $[0,1]^{ty}$ and $Ham(M, \u03a9)$

Extended embeddings to universal covers of Hamiltonian groups under certain conditions

Proved new results on filtered Floer chain complexes for radially symmetric Hamiltonians

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 $\mathbb{R} \oplus [0,1]^{\infty}$ 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. 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.