arXiv:2610.31940v1 Announce Type: new Abstract: We settle several questions about Hofer's geometry of the two-sphere. We prove that the space of equators has infinite Hofer diameter, construct simple commutators of arbitrarily large Hofer norm, and exhibit a Hamiltonian diffeomorphism whose iterates escape to infinity in Hofer distance from the set of autonomous diffeomorphisms. Finally, we construct a bi-Lipschitz group monomorphism from the free group of rank two, equipped with its conjugation-invariant generator norm, into the Hamiltonian group equipped with Hofer's metric. The proof rests on an interplay between Lagrangian Floer theory in the fourth symmetric product of the sphere and combinatorial group theory. This paper was written with substantial assistance from AI.
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