A symplectic Hilbert-Smith conjecture

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 $C^0$ 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 $C^0$ 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.