Complexity of Hofer’s geometry in some higher dimensional manifolds

The group of Hamiltonian diffeomorphisms , equipped with the Hofer metric , is a central object in symplectic topology. A landmark result by Polterovich and Shelukhin established the profound geometric complexity of this group for surfaces and their products, showing that high powers are sparse in the metric space. More recently, Álvarez-Gavela et al. demonstrated that the free group embeds quasi-isometrically into Ham of surfaces, revealing its large-scale non-commutativity.

In this talk, I will review these results and present a generalization to some higher-dimensional symplectic manifolds, including surface bundles. We prove robust obstructions that prevent a Hamiltonian diffeomorphism from being a 
-th power (for ) or from being embedded in a flow. We also show that every asymptotic cone of 
for our higher-dimensional manifolds contains an embedded free group.