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Room P3.10, Mathematics Building

On the Bounded Isometry Conjecture

F. Lalonde and L. Polterovich study the isometries of the group of Hamiltonian diffeomorphisms with respect to the Hofer metric. They defined a symplectic diffeomorphism $\phi$ to be bounded, if the Hofer norm of $[\phi, h]$ remains bounded as $h$ varies on $\operatorname{Ham}(M, \omega)$. The set of bounded symplectic diffeomorphisms, $BI_0 (M, \omega),$ of $(M, \omega)$ is a group that contains all Hamiltonian diffeomorphisms. They conjectured that these two groups are equal, $\operatorname{Ham}(M, \omega) = BI_0 (M, \omega)$ for every closed symplectic manifold. They prove this conjecture in the case when the symplectic manifold is a product of closed surfaces of positive genus. In this talk we give an outline of a new class of manifolds for which bounded isometry conjecture holds.