Abstract: 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 ψ to be bounded, if the Hofer norm of [ψ, h] remains bounded as h varies on Ham(M, ω). The set of bounded symplectic diffeomorphisms, BI0 (M), of (M, ω) is a group that contains all Hamiltonian diffeomorphisms. They conjectured that these two groups are equal, Ham(M, ω) = BI0 (M, ω) 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.
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