Rigidity and flexibility in p-adic symplectic geometry

Abstract 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 $$n\geqslant 2$$ 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 $$(\mathbb {Q}_p)^{2n}$$ ( Q p ) 2 n is symplectomorphic to the p -adic cylinder $$\textrm{Z}_p^{2n}(R)$$ Z p 2 n ( 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.