Complexity of Hofer's geometry in higher dimensional manifolds

Paper

Abstract

This paper establishes robust obstructions to representing Hamiltonian diffeomorphisms as

-th powers (

) or embedding them in flows for certain higher-dimensional symplectic manifolds

, including surface bundles. We prove that in the Hamiltonian group

equipped with the Hofer metric, there exist arbitrarily large balls that are disjoint from the set of

-th powers. Furthermore, we demonstrate that the free group on two generators embeds into every asymptotic cone of

, revealing the large-scale geometric complexity of the Hamiltonian group.