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 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.

💡 Research Summary

The paper investigates the large‑scale geometry of the Hamiltonian diffeomorphism group Ham(M, ω) equipped with Hofer’s metric, focusing on symplectic manifolds that satisfy a set of topological and geometric conditions denoted by (). Condition () requires the manifold to be closed (or a Stein domain) and symplectically aspherical, to contain two Lagrangian tori T₁ and T₂ that intersect transversely at a single point, to have the inclusion π₁(T₁∪T₂)→π₁(M) injective (so that the fundamental group contains a free product ℤⁿ * ℤⁿ), and to be α‑atoroidal for any loop α generated by curves on the two tori. Concrete examples satisfying (*) are given: plumbing of cotangent bundles of two n‑dimensional tori, surface bundles S¹⋉ψS² with appropriate monodromy, Luttinger‑surgery modifications of such bundles, and symplectic sums of these constructions.

The first main result (Theorem 1.1) states that for any integer k≥2 there exists a sequence {fₙ} ⊂ Ham(M, ω) such that the Hofer distance from fₙ to the set of k‑th powers Hamₖ(M, ω) tends to infinity as n→∞. Consequently the same holds for the set of autonomous Hamiltonian diffeomorphisms Aut(M, ω). The proof proceeds by constructing a family of “linked twist maps” on a neighbourhood of T₁∪T₂ that are modelled on the plumbing domain. Near each torus a Hamiltonian twist τ₁ or τ₂ is defined; these coincide with the time‑1 map of a geodesic flow near the zero section and are the identity elsewhere. For a word w in the free group F₂, the map τ(N,w) is a finite composition of the twists according to w.

A detailed Floer‑theoretic analysis is carried out. Specific free homotopy classes γ_N are identified in which τ(N,w) has only finitely many periodic points. The action functional of these points is computed, and a uniform lower bound δ_N>0 on the action differences between a generator and any of its Floer boundaries is established. This uses the Conley–Zehnder index calculation (Section 5) and the notion of boundary depth β_α(φ). Because Hofer’s norm bounds the minimal action difference (Propositions 3.1 and 3.2), the existence of a positive δ_N forces the Hofer distance from τ(N,w) to any k‑th power to grow at least linearly with N, proving Theorem 1.1.

Proposition 1.2 further shows that the constructed maps have rich dynamics: the number of k‑periodic points grows at least exponentially in k, and there is an open set U⊂M such that for a sequence of periodic points {xₙ} the orbits {τ(N,w)ⁱ(xₙ)} become dense in U as n increases. This demonstrates that the examples are genuinely high‑dimensional and not merely product constructions whose dynamics are confined to a low‑dimensional factor.

The second main theorem (Theorem 1.3) concerns the asymptotic cone of (Ham(M, ω), d_H). For any non‑principal ultrafilter U, there exists a faithful homomorphism F₂→Cone_U(Ham(M, ω), d_H). The construction uses the same family τ(N,w) and the fact that the Hofer norm of τ(N,w) grows linearly with N. By rescaling distances by 1/N and passing to the ultralimit, each word w defines a distinct element in the cone, and the relations among words are preserved, yielding an embedding of the free group. This shows that the asymptotic cone of the Hamiltonian group is highly non‑abelian and contains free subgroups, indicating a rich large‑scale geometric structure.

The paper is organized as follows: Section 2 gives explicit constructions of manifolds satisfying (*), including surface bundles, Luttinger surgeries, and symplectic sums. Section 3 reviews the necessary Floer theory, defining the action functional, filtered Floer homology, boundary depth, and the relation between action gaps and Hofer norm. Section 4 constructs the linked twist maps τ(N,w) and analyses their periodic points and action estimates. Section 5 computes Conley–Zehnder indices and establishes the lower bound on action differences. The final sections combine these ingredients to prove Theorems 1.1 and 1.3.

In summary, the work extends previous results on the scarcity of autonomous or k‑th power Hamiltonian diffeomorphisms from surfaces to a broad class of higher‑dimensional symplectic manifolds, and it reveals that the Hamiltonian group’s asymptotic geometry is far more intricate than previously known, containing embedded free groups. This opens new directions for studying rigidity versus flexibility phenomena in symplectic topology, the interaction between Floer‑theoretic invariants and large‑scale metric geometry, and the potential for further exotic embeddings in the Hamiltonian diffeomorphism group.