Complexity of Hofer's geometry in higher dimensional manifolds

TL;DR

This paper investigates the geometric complexity of Hamiltonian diffeomorphism groups in higher dimensions, showing large-scale obstructions to certain algebraic structures and embedding properties within the Hofer metric framework.

Contribution

It establishes new obstructions to representing Hamiltonian diffeomorphisms as powers and demonstrates the embedding of free groups into asymptotic cones, revealing complex large-scale geometry.

Findings

Existence of large Hofer metric balls disjoint from $k$-th powers.

Embedding of free groups into asymptotic cones.

Obstructions to representing diffeomorphisms as powers or flows.

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.