Computability and the growth rate of symplectic homology

For each n greater than 7 we explicitly construct a sequence of Stein manifolds diffeomorphic to complex affine space of dimension n so that there is no algorithm to tell us in general whether a given such Stein manifold is symplectomorphic to the first one or not. We prove a similar undecidability result for contact structures on the 2n - 1 dimensional sphere. We can generalize these results by replacing com- plex affine space with any smooth affine variety of dimension n and the 2n - 1 dimensional sphere with any smooth affine variety intersected with a sufficiently large sphere. We prove these theorems by using an invariant called the growth rate of symplectic homology to reduce these problems to an undecidability result for groups.

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