A symplectic Hilbert-Smith conjecture

2024/03/11 by Egor Shelukhin, Shelukhin, Egor
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Topology (math.AT) #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2403.07195

openalex publication_date 2024/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove new cases of the Hilbert-Smith conjecture for actions by natural homeomorphisms in symplectic topology. Specifically, we prove that the group of p-adic integers \mathbb Z_p does not admit non-trivial continuous actions by Hamiltonian homeomorphisms, the C⁰ limits of Hamiltonian diffeomorphisms, on symplectically aspherical symplectic manifolds. For a class of symplectic manifolds, including all standard symplectic tori, we deduce that a locally compact group acting faithfully by homeomorphisms in the C⁰ closure of time-one maps of symplectic isotopies must be a Lie group. Our methods of proof differ from prior approaches to the question and involve barcodes and power operations in Floer cohomology. They also apply to other natural metrics in symplectic topology, notably Hofer's metric. An appendix by Leonid Polterovich uses this to deduce obstructions on Hamiltonian actions by semi-simple p-adic analytic groups.

Related

• A Hofer-like metric on the group of symplectic diffeomorphisms
• A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms on fixed points
• Lectures on Groups of Symplectomorphisms
• Group actions on p-adic symplectic manifolds
• The Hilbert-Smith Conjecture