Abstract
Shnirelman has proved that the group of volume-preversing diffeomorphisms of the cube in R 3 has finite diameter and has announced the result that this is false for the square. Despite the fact that Shnirelman formulated his theorem only for the 3-dimensional cube, his proof can be modified for the case of the group of volume preserving diffeomorphisms of any compact simply-connected Riemannian manifold of dimension > 2. It turns out that the situation with the group of symplectic diffeomorphisms is completely different. We prove that the diameter of the symplectomorphism group of any compact exact symplectic manifold (necessarily with boundary) is infinite
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