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arXiv · 2512.10503

Complexity of Hofer's geometry in higher dimensional manifolds

Zhijing Wendy Wang
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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 $(M,ω)$, including surface bundles. We prove that in the Hamiltonian group $(\mathrm{Ham}(M,ω), d_H)$ 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 $(\mathrm{Ham}(M,ω), d_H)$, revealing the large-scale geometric complexity of the Hamiltonian group.

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Zhijing Wendy Wang. 2025-12-13. Complexity of Hofer's geometry in higher dimensional manifolds. https://doi.org/10.3934/jmd.2026013

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