Rigidity and flexibility in $p$-adic symplectic geometry

TL;DR

This paper explores the properties of $p$-adic symplectic geometry, revealing both flexibility in embeddings unlike the real case and certain rigidity results, along with the introduction of equivariant symplectic capacities.

Contribution

It demonstrates the failure of Gromov's non-squeezing theorem in the $p$-adic setting, establishes some rigidity results, and introduces equivariant $p$-adic symplectic capacities.

Findings

$p$-adic space is symplectomorphic to $p$-adic cylinders of any radius

$p$-adic affine Gromov's non-squeezing still holds

Equivariant embeddings satisfy non-squeezing, leading to new capacities

Abstract

Let $n\ge 2$ 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 $(\mathbb{Q}_p)^{2n}$ is symplectomorphic to the $p$-adic cylinder $\mathrm{Z}_p^{2n}(R)$ 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 non-linear 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.