Zhijing Wendy Wang, a mathematician at the University of Chicago, presents this symplectic geometry seminar talk for the joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar. The subject is the group of Hamiltonian diffeomorphisms equipped with the Hofer metric, a central object in symplectic topology. Wang reviews the landmark results of Polterovich and Shelukhin on the sparseness of high powers in this metric space for surfaces, and the more recent work of Alvarez-Gavela and collaborators showing that free groups embed quasi-isometrically into the Hamiltonian group of surfaces. She then presents her own generalization of these results to higher-dimensional symplectic manifolds, including surface bundles, proving obstructions that prevent certain diffeomorphisms from being k-th powers or from embedding in a flow. The talk closes with a result showing that every asymptotic cone of these higher-dimensional groups contains an embedded free group. It is a technical research seminar aimed at specialists in symplectic geometry and geometric group theory.