Complexity of Hofer's geometry in higher dimensional manifolds

Abstract

This paper establishes robust obstructions to representing Hamiltonian diffeomorphisms as $k$-th powers ($k \geq 2$) or embedding them in flows for certain higher-dimensional symplectic manifolds $(M,\omega)$, including surface bundles. We prove that in the Hamiltonian group $(\mathrm{Ham}(M,\omega), d_H)$ equipped with the Hofer metric, there exist arbitrarily large balls that are disjoint from the set of $k$-th powers. Furthermore, we demonstrate that the free group on two generators embeds into every asymptotic cone of $(\mathrm{Ham}(M,\omega), d_H)$, revealing the large-scale geometric complexity of the Hamiltonian group.