Rigidity and Flexibility in P-Adic Symplectic Geometry

Our main goal in this paper is to extend mathematical ideas of Gromov concerning the phenomenon of symplectic squeezing from real to p-adic geometry. Let n⩾ 2 be an integer and let p be a prime number. We prove that the analog of Gromov’s non-squeezing theorem does not hold for p-adic embeddings: for any p-adic absolute value R, the entire p-adic space (ℚ_p)^2n is symplectomorphic to the p-adic cylinder Z_p^2n(R) of radius R, showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the p-adic affine analog of Gromov’s result still holds. We will also show that in the nonlinear situation, if the p-adic embeddings are equivariant with respect to a torus action, then non-squeezing holds, which generalizes a recent result by Figalli, Palmer and the second author. This allows us to introduce equivariant p-adic analytic symplectic capacities, of which the p-adic equivariant Gromov width is an example.