Complexity of Hofer's geometry in higher-dimensional manifolds

This paper establishes robust obstructions to representing Hamiltonian diffeomorphisms as $ k $-th powers ($ k \geq 2 $) or embedding them in Hamiltonian 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 of two generators embeds into every asymptotic cone of $ (\mathrm{Ham}(M, \omega), d_H) $, revealing the large-scale geometric complexity of the Hamiltonian group.