Quasimorphisms on surfaces and continuity in the Hofer norm (Q6608223)

Quasimorphisms are a fundamental tool in symplectic geometry. They are particularly useful in studying large-scale geometric properties of Hamiltonian groups endowed with the Hofer metric: a bi-invariant metric that measures the cost of deforming Hamiltonians into one another, and is unique among \(C^\infty\)-continuous Finsler metrics. There are plenty of constructions, and among these the Calabi homomorphism, and the Calabi quasimorphism on the sphere, are the most useful ones, because they are even Lipschitz with respect to the Hofer metric.\N\NThis paper examines two other important constructions of quasimorphisms on Hamiltonian groups, and shows that they are not Lipschitz. The first is a construction by \textit{L. Polterovich} [NATO Sci. Ser. II, Math. Phys. Chem. 217, 417--438 (2006; Zbl 1089.53066)], which given a quasimorphism \(r\) on the fundamental group of \(M\) produces a quasimorphism \(\rho\) on Ham\((M)\). The second is a construction of \textit{J.-M. Gambaudo} and \textit{É. Ghys} [Topology 36, No. 6, 1355--1379 (1997; Zbl 0913.58003)], which given a quasimorphism \(r\) on the braid group with \(k\) strands in \(M\) produces a quasimorphism \(\rho\) on Ham\((M)\). The former construction can be seen as the case \(k = 1\) of the latter construction.\N\NThe proof is similar in both cases. It is first carried out for surfaces, by constructing finer and finer sequences of embedded pairs of pants and estimating the quasimorphisms on compositions of flows of Hamiltonians supported on each of these pairs of pants. This construction can be lifted to symplectic ball bundles, and then extended to all higher-dimensional symplectic manifolds via the symplectic neighbourhood theorem.\N\NThese arguments prove that the quasimorphisms are not Lipschitz. This then implies that they are not continuous, thanks to Lemma \(2\) in the paper. This connects the two properties for any group equipped with a bi-invariant path metric, and could be of independent interest (the statement is for Lie groups, but as far as I can tell this is never used in the proof).