Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

TL;DR

This paper demonstrates that free groups with two generators can be embedded into all asymptotic cones of the Hamiltonian diffeomorphism group of certain symplectic surfaces, revealing deep geometric group properties.

Contribution

It establishes the embedding of free groups into asymptotic cones of Hamiltonian diffeomorphisms for genus g ≥ 4 surfaces, extending to products with aspherical manifolds.

Findings

Free groups embed into asymptotic cones of Hamiltonian diffeomorphisms

Embedding persists under stabilization with aspherical manifolds

Results hold for symplectic surfaces of genus g ≥ 4

Abstract

Given a symplectic surface $(\Sigma, \omega)$ of genus $g \ge 4$, we show that the free group with two generators embeds into every asymptotic cone of $(\mathrm{Ham}(\Sigma, \omega), d_\mathrm{H})$, where $d_\mathrm{H}$ is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds.